Answer:
a) FALSE we obtain a NOT significant results after conduct the hypothesis tes.
b) FALSE, the standard error is not associated to a certain % of people included in the study
c) FALSE. if we want to reduce the standard error we need to increase the sample size or data
d) FALSE, always if we have a higher confidence level the confidence interval associated to this level would be wider than for a lower confidence interval
Step-by-step explanation:
Data given and notation n
n represent the random sample taken
[tex]\hat p=0.52[/tex] estimated proportion of U.S. adult Twitter user get at least some news on Twitter
[tex]p_o=0.5[/tex] is the value that we want to test
[tex]\alpha=0.01[/tex] represent the significance level
Confidence=99% or 0.99
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that more than half U.S. adult twitter users get some news throught Twitter:
Null hypothesis:[tex]p\leq 0.5[/tex]
Alternative hypothesis:[tex]p > 0.5[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Calculate the statistic
The standard error is given:
[tex] SE = \sqrt{\frac{p_o (1-p_o)}{n}}=0.024[/tex]
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.52 -0.5}{0.024}=0.833[/tex]
Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level assumed is [tex]\alpha=0.01[/tex]. The next step would be calculate the p value for this test.
Since is a right tailed test the p value would be:
[tex]p_v =P(z>0.833)=0.2024[/tex]
So the p value obtained was a very high value and using the significance level given [tex]\alpha=0.01[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.
Now let's idendity the statements
a) FALSE we obtain a NOT significant results after conduct the hypothesis tes.
b) FALSE, the standard error is not associated to a certain % of people included in the study
c) FALSE. if we want to reduce the standard error we need to increase the sample size or data
d) FALSE, always if we have a higher confidence level the confidence interval associated to this level would be wider than for a lower confidence interval
The statement a) is true because the data is statistically significant. Statements b), c) and d) are false because: standard error doesn't reflect the portion of the population included in the study; standard error reduces with more data; and a higher confidence level results in a wider interval, not narrower.
Explanation:a) This statement is true. The data can be considered statistically significant as long as the percentage is not within the standard error range below 50%. In this case, 52% minus 2.4% (standard error) = 49.6% which is below 50%. However, at a significance level of alpha = 0.01, it would be statistically significant since it is more than one standard error away from 50%.
b) This statement is false. The standard error is not an indication of what portion of the population is included in the study. Instead, it's a measure of statistical accuracy of the estimate.
c) This statement is false. The standard error of the estimate would typically decrease as we collect more data. As more data is collected, estimates are usually more accurate.
d) This statement is false. A 90% confidence interval is narrower than a 99% confidence interval. The larger the confidence level, the wider the interval to factor in more uncertainty.
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Aimee sells hand-embroidered dog apparel over the Internet. Her annual revenue is $128,000 per year, the explicit costs of her business are $42,000, and the opportunity costs of her business are $30,000. What are the implicit costs of her business?
Answer:
200,000
Step-by-step explanation:
add 128,000+42,000+30,000
The implicit cost of her business is $56000
What is the implicit cost?Implicit costs are a specific type of opportunity cost, the cost of resources already owned by the firm that could have been put to some other use.
Given that, the annual revenue of Aimee is $128,000, the explicit costs of her business are $42,000, and the opportunity costs of her business are $30,000.
We need to find the implicit costs of her business,
Therefore,
Economic profit = total revenues – explicit costs – implicit costs.
30000 = 128000-42000-implicit costs.
Implicit costs = 56000
Hence, the implicit cost of her business is $56000
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A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times and determines that 14 of the plates have blistered.
Does this data provide compelling evidence for concluding that more than 10% of all plates blister under such circumstances?
Use Alpha =0.10.
A. What is the parameter of interest?
B. State the null and alternative hypotheses.
C. Calculate the test statistic.
D. Find the rejection region.
E. Make a decision and interpret.
F. Find a p-value corresponding to the test and compare with your decision in E.
Answer:
a) Parameter of interest [tex] p[/tex] representing the true proportion of the plates have blistered.
b) Null hypothesis:[tex]p\leq 0.1[/tex]
Alternative hypothesis:[tex]p > 0.1[/tex]
c) [tex]z=\frac{0.14 -0.1}{\sqrt{\frac{0.1(1-0.1)}{100}}}=1.33[/tex]
d) For this case we need to find a value in the normal standard distribution that accumulates 0.1 of the area in the right tail and for this case is:
[tex] z_{critc}= 1.28[/tex]
e) For this case since our calculated value is higher than the critical value 1.33>1.28 we have enough evidence to reject the null hypothesis and we can conclude that the true proportion is significantly higher than 0.1
f) [tex]p_v =P(z>1.33)=0.0917[/tex]
If we compare the p value and the significance level given [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the true proportion is higher than 0.1 or 10%
Step-by-step explanation:
Data given and notation
n=100 represent the random sample taken
Part a
Parameter of interest [tex] p[/tex] representing the true proportion of the plates have blistered.
X=14 represent the number of the plates have blistered.
[tex]\hat p=\frac{14}{100}=0.14[/tex] estimated proportion of the plates have blistered.
[tex]p_o=0.1[/tex] is the value that we want to test
[tex]\alpha=0.1[/tex] represent the significance level
Confidence=90% or 0.90
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Part b: Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that more than 10% of all plates blister under such circumstances.:
Null hypothesis:[tex]p\leq 0.1[/tex]
Alternative hypothesis:[tex]p > 0.1[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Part c: Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.14 -0.1}{\sqrt{\frac{0.1(1-0.1)}{100}}}=1.33[/tex]
Part d: Rejection region
For this case we need to find a value in the normal standard distribution that accumulates 0.1 of the area in the right tail and for this case is:
[tex] z_{critc}= 1.28[/tex]
Part e
For this case since our calculated value is higher than the critical value 1.33>1.28 we have enough evidence to reject the null hypothesis and we can conclude that the true proportion is significantly higher than 0.1
Part f
Since is a right taild test the p value would be:
[tex]p_v =P(z>1.33)=0.0917[/tex]
If we compare the p value and the significance level given [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the true proportion is higher than 0.1 or 10%
In automobile mileage and gasoline-consumption testing, 13 automobiles were road tested for 300 miles in both city and highway driving conditions. The following data were recorded for miles-per-gallon performance. City 16.2 16.7 15.9 14.4 13.2 15.3 16.8 16.0 16.1 15.3 15.2 15.3 16.2 Highway 19.1 20.3 18.0 18.3 18.9 17.1 16.9 18.3 18.7 20.8 19.1 18.2 18.4 Use the mean, median, and mode to make a statement about the difference in performance for city and highway driving. Compute the mean for the miles-per-gallon performance for city and highway driving. (Round your answers to two decimal places.) city 15.9 miles per gallon highway 15.3 miles per gallon Is the mean mileage better on the highway than in the city? Yes No Compute the median for the miles-per-gallon performance for city and highway driving. city miles per gallon highway miles per gallon Is the median mileage better on the highway than in the city? Yes No Compute the mode(s) for the miles-per-gallon performance for city and highway driving. (Enter your answers as a comma-separated list.) city miles per gallon highway miles per gallon Is the modal mileage better on the highway than in the city? Yes No Make a statement about the difference in performance for city and highway driving conditions. Miles-per-gallon performance is in highway driving condition than in city driving conditions.
Answer:
A. No
City driving mean requires less miles than Highway driving mean.
B. No
City driving median requires less miles than Highway driving median.
C. No
City driving mode requires less miles than Highway driving mode
D.
The driving conditions in the City is better than the driving conditions in the Highway because all measures of centers are lesser than those of Highway.
Step-by-step explanation:
Identify the polynomial.
a 2 + b - cd 3
Answer: 2a + b - 3cd
Step-by-step explanation:
a 2 + b - cd 3
2a + b - 3cd
On July 1, 2015, Frank Corp. Purchased $100,000 of 8% bonds at face value. Interest is paid annually on June 30. If the accounting year for Frank ends at December 31, 2015, what will be reported with respect to the bonds on that date
Answer:
interest income in amount of $4000 will be accrued
Step-by-step explanation:
given data
principal = $100,000
rate = 8 percent = 0.08
time period = 6 month ( July - December ) = [tex]\frac{1}{2}[/tex] year
solution
we get here interest at December 31, 2015 that is express as
interest = principal × rate × time ..................1
put here value and we will get
interest = $100,000 × 0.08 × [tex]\frac{1}{2}[/tex]
interest = $100,000 × 0.08 × 0.5
interest = $4000
so interest income in amount of $4000 will be accrued
You have $3,500 on a credit card that charges a 13% interest rate. If you want to pay off the credit card in 4 years, how much will you need to pay each month (assuming you don't charge anything new to the card)?
To pay off your credit card debt of $3,500 at 13% interest rate in 4 years, you will have to calculate the monthly payments by using a certain formula. After defining the parameters (monthly payment, monthly interest rate, present value or current loan amount, and time), all you have to do is put the values in it and calculate the monthly payment.
Explanation:This question refers to a financial problem in the subject of Mathematics, particularly pertaining to the concept of
simple interest
. To find out how much you need to pay each month to clear out your credit card debt, you would need to understand how the 13% interest is applied to your debt of $3,500. The formula for finding monthly payments is
P = [r*PV(1 + r)^t]/[(1 + r)^t – 1]
, where:
P is your monthly payment,r is your monthly interest rate, calculated by taking your annual rate (0.13 in this case) and dividing it by 12,PV is your present value or current loan amount, which is $3500 in this case, andt is time, the number of periods in which payments will be made, which is 4*12 = 48 months in this case.After defining the parameters, all you have to do is put the values in the formula and calculate the monthly payment.
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To calculate the monthly payments on a credit card debt of $3,500 with a 13% interest rate over 4 years, you can use the formula for calculating the monthly payment on a fixed-rate loan. The monthly payment comes out to be approximately $94.78.
Explanation:To calculate the monthly payments on a credit card debt, you can use the formula for calculating the monthly payment on a fixed-rate loan. The formula is:
Monthly Payment = (Principal * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Months))
In this case, the principal is $3,500, the monthly interest rate is 13% divided by 12 (0.13/12), and the number of months is 4 years multiplied by 12 months (4 * 12). Plugging in these values, we get:
Monthly Payment = (3500 * 0.13/12) / (1 - (1 + 0.13/12)^(-4 * 12))
Solving this equation will give us the monthly payment amount. Using a calculator, the monthly payment comes out to be approximately $94.78.
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Find the parametric equations for the line through the point P(2,4,4) that is perpendicular to the plane −1x+1y−4z=1. a) Use the variable t and write these equations so that t=0 corresponds to the point P.
Answer:
[tex]\frac{x-2}{-1} =\frac{y-4}{1} =\frac{z-4}{-4} =t[/tex]
Step-by-step explanation:
Given that a line passes through P(2,4,4)
Also the line is perpendicular to the plane
[tex]-1x+1y-4z=1.[/tex]
From the equation of the plane we can say that normal to the plane has direction ratios as (-1,1,-4)
Since the required line is also perpendicular to the plane, the direction ratios of the required line is
(-1,1,4)
It passes through (2,4,4)
If Q(x,y,z) are general points on the line then
Direction ratios of PQ are = (x-2, y-4, z-4)
These are proportional to (-1,1,4)
So parametric form of the line is
[tex]\frac{x-2}{-1} =\frac{y-4}{1} =\frac{z-4}{-4} =t[/tex]
Whem t=0 we get the point P.
Consider the set consisting of the first 10 positive whole numbers (that is, 1-10). Complete parts (a) through (c) below.
(a) Determine explicitly the numbers in the set that are at least 4.
(b) Determine explicitly the numbers in the set that are at most 5
(c) Determine explicitly the numbers in the set that are between 4 and 6, inclusive.
Answer:
a) {4,5,6,7,8,9,10}
b) {1,2,3,4,5}
c) {4,5,6}
The time required for a citizen to complete the 2010 U.S. Census "long" form is normally distributed with a mean of 40 minutes and a standard deviation of 10 minutes. The lowest 10 percent of the citizens would need at least how many minutes to complete the form
Answer:
The lowest 10 percent of the citizens would need at least 52.8 minutes to complete the form.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 40, \sigma = 10[/tex]
The slowest 10 percent of the citizens would need at least how many minutes to complete the form
This is the value of X when Z has a pvalue of 0.9. So it is X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 40}{10}[/tex]
[tex]X - 40 = 1.28*10[/tex]
[tex]X = 52.8[/tex]
The lowest 10 percent of the citizens would need at least 52.8 minutes to complete the form.
Final answer:
To find the time needed for the lowest 10 percent of citizens to complete the 2010 U.S. Census 'long' form, we use the z-score formula to calculate the value. The lowest 10 percent of citizens would need at least 26.8 minutes to complete the form.
Explanation:
To find how many minutes would be needed for the lowest 10 percent of citizens to complete the 2010 U.S. Census 'long' form, we need to use the z-score formula.
First, we calculate the z-score using the formula: z = (x - mean) / standard deviation, where x is the value we want to find the z-score for, mean is the mean of the distribution, and the standard deviation is the standard deviation of the distribution.
Next, we use a z-table or a calculator to find the z-score that corresponds to a cumulative probability of 0.1. The z-table gives us a z-score of -1.28. We can rearrange the z-score formula to solve for x: x = (z * standard deviation) + mean. Plugging in the values, we get: x = (-1.28 * 10) + 40 = 26.8. Therefore, the lowest 10 percent of citizens would need at least 26.8 minutes to complete the form.
A building engineer analyzes a concrete column with a circular cross section. The circumference of the column is 18 \pi18π18, pi meters.
What is the area AAA of the cross section of the column?
Give your answer in terms of pi.
The area of the cross section of the column is [tex]18 \pi \ m^2[/tex]
Explanation:
Given that a building engineer analyzes a concrete column with a circular cross section.
Also, given that the circumference of the column is [tex]18 \pi[/tex] meters.
We need to determine the area of the cross section of the column.
The area of the cross section of the column can be determined using the formula,
[tex]Area= \pi r^2[/tex]
First, we shall determine the value of the radius r.
Since, given that circumference is [tex]18 \pi[/tex] meters.
We have,
[tex]2 \pi r=18 \pi[/tex]
[tex]r=9[/tex]
Thus, the radius is [tex]r=9[/tex]
Now, substituting the value [tex]r=9[/tex] in the formula [tex]Area= \pi r^2[/tex], we get,
[tex]Area = \pi (9)^2[/tex]
[tex]Area = 81 \pi[/tex]
Thus, the area of the cross section of the column is [tex]18 \pi \ m^2[/tex]
Research suggests that children who eat hot breakfast at home perform better at school. Many argue that not only hot breakfast but also parental care of children before they go to school has an impact on children's performance. In this case, parental care is: Group of answer choices An independent variable A dependent variable A mediating variable A moderating variable
Answer:children at home prefer hot breakfast than hot school breakfast because at home you could put it in the refrigerator but at school you have to throw it in the trash before you go to trash.
Step-by-step explanation:
Parental care in this context is considered the independent variable because it influences children's school performance.
Explanation:In this scenario, parental care is considered an independent variable. The reason for this is because it is the variable that influences or predicts the outcome, which in this case, is children's performance in school. The independent variable is the one that is manipulated or controlled in a study to observe its effects on the dependent variable (children's school performance here). Examples of parental care might include ensuring the child eats a good breakfast, aiding with schoolwork, or providing emotional support.
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During the period of time that a local university takes phone-in registrations, calls come in at the rate of one every two minutes.
What is the expected number of calls in one hour?
Answer:
Let X the random variable who represent the number of occurences in a period of time for the calls.
For this case we have the following parameter [tex] \lambda = 1 \frac{call}{2 minutes}[/tex]
And we are interested in the expected number of calls in one hour.
We know that 1 hr = 60 mins so then the expected number of calls that arrive in one hour are:
[tex] \lambda = \frac{1 call}{2 minutes} * \frac{60 minutes}{1 hour} = 30 calls per hour[/tex]
Step-by-step explanation:
Definitions and concepts
The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:
[tex]P(X=x) =\lambda^x \frac{e^{-\lambda}}{x!}[/tex]
And the parameter [tex]\lambda[/tex] represent the average ocurrence rate per unit of time.
Solution to the problem
Let X the random variable who represent the number of occurences in a period of time for the calls.
For this case we have the following parameter [tex] \lambda = 1 \frac{call}{2 minutes}[/tex]
And we are interested in the expected number of calls in one hour.
We know that 1 hr = 60 mins so then the expected number of calls that arrive in one hour are:
[tex] \lambda = \frac{1 call}{2 minutes} * \frac{60 minutes}{1 hour} = 30 calls per hour[/tex]
In Applied Life Data Analysis (Wiley, 1982), Wayne Nelson presents the breakdown time of an insulating fluid between electrodes at 34 kV. The times, in minutes, are as follows: 0.13, 0.68, 0.91, 1.36, 2.74, 3.08, 4.07, 4.71, 4.96, 6.56, 7.29, 7.91, 8.37, 12.11, 31.61, 32.65, 33.78, 36.72, and 72.96. Calculate the sample mean and sample standard deviation. Round the answers to 3 decimal places.
Answer:
Mean = 14.347 and Standard Deviation = 18.89
Step-by-step explanation:
Mean = Sum of all the numbers/total numbers = 272.6/19 = 14.347
For standard deviation, at first we will find variance which is
variance = The sum of the squared differences between each data point and the mean, divided by the number of data points (n) - 1:
x= data points
n= total number of data points
= ∑(x_i - Mean (x))^2/(n-1)
variance = 357
Standard deviation = [tex]\sqrt{variance}[/tex]
SD = [tex]\sqrt{357}[/tex]
SD = 18.89
Answer:
The mean= 14.347
The standard deviation =18.390
Step-by-step explanation:
The mean or average of a set of scores = the summation of the set of scores divided by the summation of the frequency of the respective scores.
Adding up the scores we have:
0.13+0.68+0.91+1.36+2.74+3.08+4.07+4.71+4.96+6.56+7.29+7.91+8.37+12.11+31.61+32.65+33.78+36.72+72.96 = 272.60
The frequency or number of scores is 19, so dividing this figure by the number of items in the set, we have:
272/19 =14.347
Therefore the mean of the scores (time, in minutes) is 14.347(to 3 decimal places)
In order to calculate the standard deviation of the set, we need to find the summation of the squares of the different deviations from the mean.
Example, 0.13 is the first score in the data. We subtract the mean (14.35) from 0.13 and we'll have 14.22 which is the score's deviation from the mean. the next step is to find the square of the deviation,14.22 which will be 202.208.
We'll repeat this same process for the remainder of the scores and then sum up the squares of the deviations.Doing this, the summation of the squares of all the deviations from the mean score will be = 6425.67
Once this is calculated we then solve to obtain the standard deviation of the scores by applying the formula:
√summation of (x - mean deviations)/total number of scores
=√(6425.67/19)
= √ (338.193)
= 18.390( to 3 decimal places)
Therefore the mean and the standard deviation of the set of scores are 14.347 and 18.390 respectively
A well-mixed cookie dough will produce cookies with a mean of 66 chocolate chips apiece. What is the probability of getting a cookie with at least 55 chips? Round your answer to four decimal places.
Answer:
0.7262
Step-by-step explanation:
Let random variable = X
Probability of getting a cookie with at least 55 chips:
P(55≤ X ≤78), where X1 = 55 and X2 = 78
Let mean, U = 66 and standard deviation, S.D. = 10
Let Z = X - U/S.D; Z1 = X1 - U/S.D and Z2 = X2 - U/S.D
Z1 = 55 - 66/10 = - 1.1
Z2 = 78 - 66/10 = 1.2
P(X1≤ X ≤ X2) = P(X1 -U/S.D ≤ X - U/S.D ≤ X2 - U/S.D)
P(Z1 ≤ Z ≤ Z2) = P(0 ≤ Z ≤ Z1) + P(0 ≤ Z ≤ Z2)
Hence, (55 ≤ X ≤ 78) = P( - 1.1 ≤ Z ≤ 1.2)
∴ P(-1.1 ≤ Z ≤ 1.2) = P(0 ≤ Z ≤ 1.1) + (0 ≤ Z ≤ 1.2)
= 0.3413 + 0.3849 = 0.7262
Elise’s garden is below. Tomato plants require 2 square feet of space.
(Shape of garden is a Rectangle 6.3 feet by 12.6 feet )
How many tomato plants can Elise fit in her garden?
Answer:
Step-by-step explanation:
You should try going too jeeska
Answer:
Step-by-step explanation:
Elise’s garden is rectangular. The formula for determining the area of a rectangle is expressed as
Area = length × width
From the information given,
Length of garden = 6 feet
Width of garden = 12.6 feet
Area of garden = 6 × 12.6 = 75.6 square feet
Tomato plants require 2 square feet of space. This means that the number of tomato plants that Elise can fit in her garden is
75.6/2 = 37.8
Since the number of tomato plants must be whole number, then the number of tomato plants that Elise can fit in her garden is 37
A can in the shape of a right circular cylinder is required to have a volume of 700 cubic centimeters. The top and bottom are made up of a material that costs 8� per square centimeter, while the sides are made of material that costs 5� per square centimeter. Find a function that describes the total cost of the material as a function of the radius r of the cylinder
Answer:
the cost function is Cost=7000 m*$ /R + 50.265 $/m² * R²
Step-by-step explanation:
then the cost function is
Cost= cost of side area+ cost of top + cost of bottom = 2*π*R*L * 5$/m² +
π*R² * 8$/m² + π*R² * 8$/m²
since the volume V is
V=π*R²*L → V/(π*R²)=L
then
Cost=2*π*R*V/(π*R²) * 5$/m² + π*R² * 8$/m² + π*R² * 8$/m²
replacing values
Cost=2*700 m³ /R * 5$/m² + π*R² * 16$/m² = 7000 m*$ /R + 50.265 $/m² * R²
thus the cost function is
Cost=7000 m*$ /R + 50.265 $/m² * R²
Answer: 50.24r² + 7000/r
Step-by-step explanation:
The formula for determining the volume of a cylinder is expressed as
Volume = πr²h
Since the volume of the can is 700cm³, then
πr²h = 700
h = 700/πr²
The formula for determining the total surface area of a cylinder is expressed as
Total surface area = 2πr² + 2πrh
The surface area of the top and bottom of the can is 2πr².
Since top and bottom are made up of a material that costs $8 per square centimeter, then the cost is
2πr² × 8 = 16πr²
Since π = 3.14, the surface area of the top and bottom of the cylindrical can is
16 × 3.14 × r² = 50.24r²
The surface area of the side of the can is
2πrh = 2πr × 700/πr²
= 1400/r
Since the the sides are made of material that costs $5 per square centimeter, then the cost is
1400/r × 5 = 7000/r
The total cost of the material as a function of the radius, r of the cylinder is
50.24r² + 7000/r
How much time do Americans spend eating or drinking? Suppose for a random sample of 1001 Americans age 15 or older, the mean amount of time spent eating or drinking per day is 1.22 hours with a standard deviation of 0.65 hour. (a) A histogram of time spent ea ting and drinking each day is skewed right. Use this result to explain why a large sample size is needed to construct a confidence interval for the mean time spent eating and drinking each day. (b) There are over 200 million Americans age 15 or older. Explain why this, along with the fact that the data were obtained using a random sample, satisfies the requirements for constructing a confidence interval. (c) Determine and interpret a 95% confidence interval for the mean amount of time Americans age 15 or older spend eating and drinking each day. (d) Could the interval be used to estimate the mean amount of time a 9-year-old American spends eating and drinking each day? Explain.
Answer:
Step-by-step explanation:
a)
To construct a confidence interval ,the histogram should be bell shaped.It means that the it should be normally distributed with no skewness.To eliminate the skewness a large sample size is required so that the sample is normally distribute about the mean Also ,in order to construct a t-interval the sample data must come from the population that is normally distributed or the sample size is larger than 30.Since the question stated that the population distribution is skewed to the right \bar{x} is guaranteed to be normally distributed if [tex]n\geq 30[/tex]
b)The sample satisfies the normal distribution because the sample sixe is greater than 30 which is 1001.
c) [[tex]\bar{x}=1.22 \,s=.65\, n=1001[/tex]
Area in the right tail =2.5% or .025
degree of freedom =1001-1 =1000
[tex]t_{\alpha /2}=1.96[/tex]
95% confidence interval is :[tex]\bar{x}\pm t_{\alpha /2}s/\sqrt{n} [/tex]
lower bound is :[tex]1.22-1.96*.65/\sqrt{1001} =1.18 [/tex]
upper bound is :[tex]-1.22+1.96*.65/\sqrt{1001}=1.26 [/tex]
d)No because the sample data was obtined from Americans age 15 or older which cannot be applied to a different age group
Seventy percent of children who go to the doctor have fevers. Of those with fevers, 30% also have a rash. Of those without fevers, 20% have a rash. What is the probability that a child at the doctor's office with a rash does not have a fever
Answer:
The probability that a child with a rash does not have a fever is 22%
Step-by-step explanation:
1. Probability of having fever:
[tex]P(fever)=0.70[/tex]
2. Probability of not having fever:
[tex]P(not fever)=1-P(fever)\\P(not fever)=1-0.70\\P(not fever)=0.30[/tex]
3. Probability of fave fevers and a rash:
[tex]P(fever and rash)=(0.70)(0.30)=0.21[/tex]
4. Probability of having a rash but not a fever:
[tex]P(rash and not fever)=(0.30)(0.20)=0.60[/tex]
5. Probability of having a rash:
[tex]P(rash)=P(rash and fever)+P(rash and no fever)\\P(rash)=0.21+0.06=0.27[/tex]
6. Probability a child with a rash does not have a fever
[tex]P=\frac{P(rash and not fever)}{P(rash)} =\frac{0.06}{0.27} =0.22[/tex]
22% of the child at the doctor's office with a rash does not have a fever.
Final answer:
To find the probability that a child with a rash does not have a fever, we analyze the given percentages, calculate how many children have a rash with and without fever, and then find the ratio of those without a fever to the total number with a rash, resulting in a probability of approximately 22.22%.
Explanation:
The question asks: What is the probability that a child at the doctor's office with a rash does not have a fever? To solve this, let's start by analyzing the given percentages.
70% of children who go to the doctor have fevers.
Of those with fevers, 30% also have a rash.
Of those without fevers, 20% have a rash.
To find the probability that a child with a rash does not have a fever, we need to calculate the proportion of children with a rash who are fever-free compared to all children with a rash.
Step-by-step Calculation:
Assuming 100 children visit the doctor: 70 will have fevers, and 30 will not.
Of the 70 with fevers, 21 (30% of 70) have a rash.
Of the 30 without fevers, 6 (20% of 30) have a rash.
In total, 27 children have a rash (21 with fever + 6 without).
The probability a child with a rash does not have a fever is the number of children with a rash but no fever divided by the total number of children with a rash: 6/27.
This calculation shows that the probability of a child having a rash but no fever is 6/27 or approximately 22.22%.
Read the paragraph and answer the question below: Results from two CNN/USA Today/Gallup polls, one conducted in March 2003 and one in November 2003, were recently presented online. Both polls involved samples of 1001 adults, aged 18 years and older. In the March sample, 45% of those sampled claimed to be fans of professional baseball whereas 51% of those polled in November claimed to be fans. Construct a 99% confidence interval for the proportion of adults who professed to be baseball fans in November 2003 (after the World Series). Interpret this interval.
Answer:
99% confidence interval for the proportion of adults who professed to be baseball fans in November 2003 is (46.9%, 55.1%)
This interval means that the lower limit of the proportion of adults who professed to be baseball fans is 46.9% and the upper limit of the proportion is 55.1%
Step-by-step explanation:
Confidence interval = P' + or - t×sqrt[P'(1-P') ÷ n]
P' is sample proportion = 51% = 0.51
n = 1001
confidence level = 99%
t-value corresponding to 99% confidence interval and infinity degree of freedom is 2.576
t × sqrt[P'(1-P') ÷ n] = 2.576 × sqrt[0.51(1-0.51) ÷ 1001] = 2.576 × 0.0158 = 0.041
Lower limit = P' - 0.041 = 0.51 - 0.041 = 0.469 = 46.9%
Upper limit = P' + 0.041 = 0.51 + 0.041 = 0.551 = 55.1%
99% Confidence interval is (46.9%, 55.1%)
The interval means that the proportion of adults who professed to be baseball fans in November 2003 is between 46.9% and 55.1%
Translate the statement into a confidence interval for p. Approximate the level of confidence.
In a survey of 1000 U.S. adults, 19% are concerned that their taxes will be audited by the Internal Revenue Service. The survey's margin of error is plus or minus 3%. (Source: Rasmussen Reports)
Approximate the level of confidence.
The interval is approximately a _______% confidence interval.
Answer:
The interval is approximately a 98% confidence interval
Step-by-step explanation:
From the question : Error, E= 3% = 0.03, Total population, n=1000, number of people that are concerned that their taxes will be audited, p = 19% = 0.19
E^{2}=z_{\alpha/2}^{2}\cdot \frac{p(1-p)}{n}
0.03^{2}=z_{\alpha/2}^{2}\cdot \frac{0.19(1-0.19)}{1000}
z_{\alpha/2}^{2}=5.848
z_{\alpha/2}=2.418
Area right to 2.418 is 0.0078. So
\alpha/2=0.0078
Therefore \alpha=0.0156\approx 0.02
Thus, the interval is approximately a 98% confidence interval.
Answer:
98 percent
Step-by-step explanation:
plato
Sandra just finished planting avacados,
carrots, radishes, tomatoes, and spinach in her new garden. The
garden is a circle whose diameter is 50 yards. If she planted equal
regions of each vegetable, what is the area of Sandra's garden that
has carrots? .
Answer:392.5
Step-by-step explanation:
50/2=25
25*25=625
625*3.14=1962.5
1962.5/5=392.5
Im right look at pic for proof
A professor planned to give an examination in a large class on the Monday before Thanksgiving vacation. Some students asked whether he could change the date because so many of their classmates had at least one other exam on that date. They speculated that at least 40% of the class had this problem. The professor agreed to poll the class, and if there was convincing evidence that the proportion with at least one other exam on that date was greater than .40, he would change the date. Of the 250 students in the class, 109 reported that they had another exam on that date. What proportion of the class reported that they had another exam on the date?
Answer:
0.436
Step-by-step explanation:
Given that a professor planned to give an examination in a large class on the Monday before Thanksgiving vacation
Some students asked whether he could change the date because so many of their classmates had at least one other exam on that date.
They speculated that at least 40% of the class had this problem.
The professor agreed to poll the class, and if there was convincing evidence that the proportion with at least one other exam on that date was greater than .40, he would change the date.
No of students in total = 250
Reported they had another exam = 109
proportion of the class reported that they had another exam on the date
= [tex]\frac{109}{250} =0.436[/tex]
Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a certain article, the mean of the x distribution is about $48 and the estimated standard deviation is about $7.
(a) Consider a random sample of n = 60 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?
The sampling distribution of x is approximately normal with mean μx = 48 and standard error σx = $0.12. The sampling distribution of x is approximately normal with mean μx = 48 and standard error σx = $7. The sampling distribution of x is not normal. The sampling distribution of x is approximately normal with mean μx = 48 and standard error σx = $0.90.
Is it necessary to make any assumption about the x distribution? Explain your answer.
It is necessary to assume that x has an approximately normal distribution. It is not necessary to make any assumption about the x distribution because μ is large. It is necessary to assume that x has a large distribution. It is not necessary to make any assumption about the x distribution because n is large.
(b) What is the probability that x is between $46 and $50? (Round your answer to four decimal places.)
3
(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $46 and $50? (Round your answer to four decimal places.)
Answer:
Step-by-step explanation:
Hello!
X: amount spent in a supermarket impulse buying in a 10 min unplanned shopping interval by one customer.
It is known that the mean of this variable is μ= $48 and its standard deviation is δ=$7
a.
The Central limit theorem states that if there is a population with probability function f(X;μ,δ²) from which a random sample of size n is selected. Then the distribution of the sample mean tends to the normal distribution with mean μ and variance δ²/n when the sample size tends to infinity.
As a rule, a sample of size greater than or equal to 30 is considered sufficient to apply the theorem and use the approximation.
X[bar]≈N(μ;σ²/n)
The mean of the sampling distribution is μ= $48
The standard deviation of the sampling distribution is σ/√n= $7/√60= $0.90
It is not necessary to make any assumption about the distribution of X since n=60 is considered large enough, you can directly approximate the sampling distribution to normal regardless of the distribution of X.
b.
To calculate this probability you have to use the approximation of the sampling distribution:
Z= (X[bar]-μ)/(σ/√n)≈N(0;1)
μ= $48
σ/√n= $0.90
P(46≤X[bar]≤50)= P(X[bar]≤50) - P(X[bar]≤46)
P(Z≤(50-48)/0.90) - P(Z≤(46-48)/0.90)
P(Z≤2.22) - P(Z≤-2.22)= 0.987 - 0.013= 0.974
c.
If we assume that X has an approximately normal distribution, then you will use it's a mean and standard deviation to reach the asked probability.
Z= (X-μ)/δ≈N(0;1)
μ= $48
δ= $7
P(46≤X≤50)= P(X≤50) - P(X≤46)
P(Z≤(50-48)/7) - P(Z≤(46-48)/7)
P(Z≤0.29) - P(Z≤-0.29)= 0.61409 - 0.38591= 0.22818≅ 0.2282
I hope it helps!
The Central Limit Theorem ensures that the average amount spent by a sample of 60 customers due to impulse buying is normally distributed with a mean of $48 and a standard error of approximately $0.90.
Explanation:Regarding part (a) of the question, by applying the Central Limit Theorem (CLT), we can state that for the sample of n = 60 customers, the sampling distribution of the sample mean x will be approximately normal due to the large sample size, even if the distribution of x is not normal. The mean of the distribution μx remains the same at $48, and the standard deviation (often referred to as the standard error, σx) can be calculated by dividing the population standard deviation by the square root of the sample size (n), which is σx = $7/√60 ≈ $0.90. Thus, the correct statement is: 'The sampling distribution of x is approximately normal with mean μx = $48 and standard error σx = $0.90.'
A coffee company wants to make sure that their coffee is being served at the right temperature. If it is too hot, the customers could burn themselves. If it is too cold, the customers will be unsatisfied. The company has determined that they want the average coffee temperature to be 65 degrees C. They take a sample of 20 orders of coffee and find the sample mean to be equal to 70.2 C. What does mu represent for this problem?
In the given sample, the parameter [tex]\mu[/tex] represents the average temperature of the coffee in the population whose value is not known.
In statistics, a parameter means a numerical attribute of a population. A population parameter describes a particular aspect of the entire population, which is the complete set of individuals, items, or units of interest. The mean, standard deviation, and variance are some of the population parameters.
A coffee company wants to make sure that their coffee is being served at the right temperature. If it is too hot, the customers could burn themselves. If it is too cold, the customers will be unsatisfied. The company has determined that they want the average coffee temperature to be 65 degrees C. They take a sample of 20 orders of coffee and find the sample mean to be equal to 70.2 C. [tex]\mu[/tex] represents the average temperature of the coffee in the population, and the value is not known for this problem.
Also, [tex]\bar{X}[/tex] represents the sample mean here which is known and equals 70.2 C.
b
Therefore, [tex]\mu[/tex] represents the population mean and is not known in this problem.
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The symbol mu in the coffee company's problem represents the population mean, the desired average coffee temperature, which should be 65 degrees C.
In the context of the coffee company's quality control problem, the symbol mu represents the population mean, which is the average temperature of all cups of coffee served by the company. The sample mean, denoted as x-bar, is 70.2°C, which is the average temperature calculated from the sample of 20 orders of coffee. The goal of the company is to have mu equal to 65°C, as they want this to be the average serving temperature to ensure customer satisfaction and safety. If they find that x-bar is significantly different from mu, it may suggest that corrective actions are needed to reach the desired temperature.
A special deck of cards has ten cards. Four are green, three are blue, and three are red. When a card is picked, its color of it is recorded. An experiment consists of first picking a card and then tossing a coin. a. Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A). (Round to two decimal places)
Answer:
P(A) = 3/20
Step-by-step explanation:
P(A)=P(blue)P(head)=(3/10)(1/2)=3/20
as there are 10 cards in total, out of which 3 are blue so the probability to get the blue card is, P(blue) = 3/10. and the probability of getting a head when a coin is tossed is P(head) = 1/2.
So in total
P(A) = P(blue)*p(head) = (3/10)*(1/2) = 3/20 = 0.15
Final answer:
P(A) = 0.15.
Explanation:
To find P(A), the probability of event A happening, we need to consider two separate events: picking a blue card and tossing a head on a coin.
First, the probability of picking a blue card from the special deck is 3 out of 10, because there are 3 blue cards among the 10 total cards.
Next, the probability of tossing a head on a coin is 1 out of 2, since a coin has two sides and both outcomes (heads and tails) are equally likely.
To find the combined probability P(A) of the two independent events, we multiply the probabilities:
[tex]P(A) = P(Blue Card) \(\times\) P(Head Coin Toss)[/tex]
Thus:
[tex]P(A) = (3/10) \(\times\) (1/2) = 3/20 = 0.15[/tex]
Rounded to two decimal places, P(A) = 0.15.
New York City is one of the most expensive cities in the US for lodging. The mean hotel room rate is $244.00 per night; assume that room rates are normally distributed, with the standard deviation of $55.00 What is the probability that a hotel room costs between $250.00 and $285.00?
Answer:
22.96% probability that a hotel room costs between $250.00 and $285.00
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 244, \sigma = 55[/tex]
What is the probability that a hotel room costs between $250.00 and $285.00?
This is the pvalue of Z when X = 285 subtracted by the pvalue of Z when X = 250. So
X = 285
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{285 - 244}{55}[/tex]
[tex]Z = 0.75[/tex]
[tex]Z = 0.75[/tex] has a pvalue of 0.7734
X = 250
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{250 - 244}{55}[/tex]
[tex]Z = 0.11[/tex]
[tex]Z = 0.11[/tex] has a pvalue of 0.5438
0.7734 - 0.5438 = 0.2296
22.96% probability that a hotel room costs between $250.00 and $285.00
Trapezoid ABCD is rotated on 180° about the origin. Draw the image A'B'C'D' of the given trapezoid and determine which of the following statements are true?
Select all correct statements.
A. A'D' ∥B'C'
B. A'B' ∥ D'C'
C. AD∥ A'D'
D. A'B' ∥ AB
E. AD ∥ B'C'
Answer: B, C, D
Step-by-step explanation:
since it is rotated, the parallel sides stay parallel.
The trapezoid ABCD rotated to form A'B'C'D, A'B' ∥ D'C',
TransformationTransformation is the movement of a point from its initial location to a new location. Types of transformation are rotation, reflection, dilation and translation.
Rotation is a rigid transformation, hence it preserves the shape and size. If a point A(x, y) is rotated on 180° about the origin, the new point is A'(-x, -y).
Hence for the trapezoid ABCD rotated to form A'B'C'D, A'B' ∥ D'C',
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The Operations Manager likes your answer and thanks you for the analyses. He then asks you to do the ABC analysis. The manager adds that the company typically does ABC analysis such that the A segment (all A items togehter) accounts for at least 70% of value, B segment for at least 15% and C segment for at most 10% of value.
SKU Number of Items Value ($/item) Weight (lb/item)
A 220 67 52
B 225 3 59
C 245 21 60
D 145 7 48
E 230 26 42
F 240 83 49
Imagine, you want to first do the ABC analysis based on SKU value.
What percentage of the total number of items is accounted for by SKUs in segment B?
What percentage of the total value of inventory is accounted for by SKUs in segment B?
What percentage of the total weight of inventory is accounted for by SKUs in segment B?
Answer:
Requirements are computed below.
Step-by-step explanation:
For ABC analysis:
Step1: Calculate the total value of each item.
SKU Number of Items Value ($/item) Value of the product ($)
A 220 67 14740
B 225 3 675
C 245 21 5145
D 145 7 1015
E 230 26 5980
F 240 83 19920
Step2: Reorganize the table on the basis of total value in descending order, & calculate % share of total value of investment and cumulative % share of total value of investment.
SKU Number of Items Value ($/item) Value of the product ($) % of Total value Cumulative % of total value Segment
SKU Number of Items Value ($/item) Value of the product ($) % of Total value Cumulative % of total value Segment
F 240 83 19920 41.96 41.96 A
A 220 67 14740 31.05 73.01 A
E 230 26 5980 12.60 85.60 B
C 245 21 5145 10.84 96.44 B
D 145 7 1015 2.14 98.58 C
B 225 3 675 1.42 100.00 C
----------
% of total number of items in segment B = (230+245)/1305 = 36.4%
% of total value of inventory in segment B = (12.6 + 10.4)/100 = 23.43%
% of the total weight of inventory in segment B = (9660+14700)/67795 = 35.93%
SKU Number of Items Value ($/item) Weight (lb/item) Total weight
A 220 67 52 11440
B 225 3 59 13275
C 245 21 60 14700
D 145 7 48 6960
E 230 26 42 9660
F 240 83 49 11760
It is known that 70% of all brand A external hard drives work in a satisfactory manner throughout the warranty period (are "successes"). Suppose that n = 15 drives are randomly selected. Let X = the number of successes in the sample. The statistic X/n is the sample proportion (fraction) of successes. Obtain the sampling distribution of this statistic. [Hint: One possible value X/n is 0.2, corresponding to X = 3. What is the probability of this value (what kind of random variable is X)?] (Round your answers to three decimal places.)
Answer:
The probability distribution of [tex]\hat p[/tex] is, [tex]N(0.70, 0.032)[/tex].
Step-by-step explanation:
The random variable X is defined as the number of brand A external hard drives work in a satisfactory manner throughout the warranty period.
The sample selected is of size, n = 15.
The probability of selecting a hard drive that works in a satisfactory manner throughout is, p = 0.70.
Every hard drive works independently of the others.
The random variable X follows a Binomial distribution with parameters n = 15 and p = 0.70.
The probability mass function of X, the binomial random variable is:
[tex]P(X=x)={15\choose x}0.70^{x}(1-0.70)^{15-x};\ x=0,1,2,3...[/tex]
The probability distribution of X is shown below.
Now a statistic is defined as:
[tex]\hat p=\frac{X}{n}[/tex]
This statistic is known as the sample proportion.
A Normal approximation to Binomial can be used to approximate the distribution of sample proportion if the following conditions are satisfied:
np ≥ 5n(1 - p) ≥ 5Then the sampling distribution of [tex]\hat p[/tex] follows a Normal; distribution with mean [tex]p=0.70[/tex] and standard deviation [tex]\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.70(1-0.70)}{200}}=0.032[/tex].
Check the conditions:
[tex]np=15\times0.70=10.5\\n(1-p)=15\times (1-0.70)=4.5\approx5[/tex]
So, the probability distribution of [tex]\hat p[/tex] is, [tex]N(0.70, 0.032)[/tex].
The sampling distribution of X/n is approximately normally distributed with a mean of 0.7 and a standard deviation of 0.117.
To determine the sampling distribution of the statistic X/n, we need to consider the probability distribution of the number of successes (X) in the sample of n = 15 drives.
Since the probability of a success (a working hard drive) is p = 0.7 and the probability of a failure (a non-working hard drive) is q = 1 - p = 0.3, we can model X using the binomial distribution with parameters n = 15 and p = 0.7.
The probability of getting exactly x successes (working hard drives) in a sample of n = 15 drives is given by the binomial probability mass function (PMF):
P(X = x) = (nCx) * px * (1-p)^(n-x)
where:
nCx is the binomial coefficient, which represents the number of ways to choose x successes out of n trials.
px is the probability of success (working hard drive) on each trial.
(1-p)^(n-x) is the probability of failure (non-working hard drive) on each trial.
For x = 3 successes (working hard drives), we calculate the probability:
P(X = 3) = (15C3) * (0.7)^3 * (0.3)^12 ≈ 0.193
This means that the probability of observing 3 successes (working hard drives) in a sample of 15 drives is approximately 0.193.
The sample proportion (fraction) of successes, X/n, is a continuous random variable that can take values between 0 and 1. To determine the sampling distribution of X/n, we need to consider the distribution of X and how it transforms to X/n.
The distribution of X/n is approximately normal for large sample sizes (n ≥ 30), but for smaller sample sizes, it can be skewed. In this case, with n = 15, the distribution of X/n is slightly skewed, but it is still close to being normal.
The mean of the sampling distribution of X/n is equal to the population proportion, which is p = 0.7.
The standard deviation of the sampling distribution of X/n is given by:
σ(X/n) = √(p * q / n) ≈ √(0.7 * 0.3 / 15) ≈ 0.117
Therefore, the sampling distribution of X/n is approximately normally distributed with a mean of 0.7 and a standard deviation of 0.117.
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Find the largest interval which includes x = 0 for which the given initial-value problem has a unique solution. (Enter your answer using interval notation.) (x − 3)y'' + 4y = x, y(0) = 0, y'(0) = 1
Answer:
[tex](-\infty,3)[/tex]
Step-by-step explanation:
We are given that
[tex](x-3)y''+4y=x[/tex]
[tex]y''+\frac{4}{x-3}y=\frac{x}{x-3}[/tex]
y(0)=0
y'(0)=1
By comparing with
[tex]y''+p(x)y'+q(x)y=g(x)[/tex]
We get
[tex]p(x)=\frac{4}{x-3}[/tex]
[tex]g(x)=\frac{x}{x-3}[/tex]
q(x)=0
p(x),q(x) and g(x) are continuous for all real values of x except 3.
Interval on which p(x),q(x) and g(x) are continuous
[tex](-\infty,3)[/tex]and (3,[tex]\infty)[/tex]
By unique existence theorem
Largest interval which contains 0=[tex](-\infty,3)[/tex]
Hence, the larges interval on which includes x=0 for which given initial value problem has unique solution=[tex](-\infty,3)[/tex]
The largest interval on which includes x=0 for which given initial-value problem has unique solution is [tex](-\infty, 3)[/tex]
The given parameters are:
(x − 3)y'' + 4y = x,
y(0) = 0
y'(0) = 1
Divide the equation (x − 3)y'' + 4y = x through by (x - 3)
[tex]y'' + \frac{4y}{x - 3} = \frac{x}{x - 3}[/tex]
Compare the above equation to the following equation
y" + p(x) y' + q(x)y = g(x)
Then, we have:
[tex]p(x) = \frac{4y}{x - 3}[/tex]
[tex]q(x) = 0[/tex]
[tex]g(x) = \frac x{x - 3}[/tex]
The domains of functions p(x) and g(x) are all set of real values except 3
This is represented as:
[tex](-\infty, 3)\ u\ (3,\infty)[/tex]
Using the unique existence theorem, we have:
The largest interval that contains x = 0 is [tex](-\infty, 3)[/tex]
Hence, the largest interval on which includes x=0 for which given initial-value problem has unique solution is [tex](-\infty, 3)[/tex]
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