To determine if the nutrition label is accurate, a hypothesis test can be conducted using the provided sample data. Calculating the z-score and finding the p-value will determine if there is evidence that the label is inaccurate.
Explanation:In order to determine if there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips, we can conduct a hypothesis test using the given sample data. Let's state the null and alternative hypotheses:
Null Hypothesis (H0): The nutrition label provides an accurate measure of calories in the bags of potato chips.
Alternative Hypothesis (Ha): The nutrition label does not provide an accurate measure of calories in the bags of potato chips.
To test these hypotheses, we can calculate the z-score using the formula:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
In this case, the population mean is 130 calories (as stated on the nutrition label), the sample mean is 134 calories, the population standard deviation is 17 calories (as given), and the sample size is 35 bags (as given). Plugging in these values, we can calculate the z-score.
Once we have the z-score, we can find the p-value associated with it from a standard normal distribution table or using statistical software. If the p-value is less than the chosen significance level (usually 0.05), we reject the null hypothesis and conclude that there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips.
Without knowing the calculated p-value, we cannot make a decision based on the given random sample.
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Umbrella Corporation purchases a raw material in 55-gallon drums from a supplier. Records for the supplier indicate that the impurity level in the material per drum has a normal distribution with a mean of 3% and a standard deviation of 0.4%. An impurity level of 4% or more in any shipment requires that Umbrella return the entire drum to the supplier.
What is the probability that Umbrella has to return any given shipment?
Answer:
The probability that Umbrella returns any shipment is 0.0062.
Step-by-step explanation:
Let X = the impurity level in the material per drum
Then it is provided that,
[tex]X\sim N(\mu = 0.03,\ \sigma=0.004)[/tex]
Also if the impurity level is more than or equal to 4% or 0.04 in any shipment, then Umbrella returns the entire drum to the supplier.
Compute the probability that Umbrella returns any shipment as:
[tex]P(X\geq 0.04)=P(\frac{X-\mu}{\sigma}\geq \frac{0.04-0.03}{0.004}) \\=P(Z\geq 2.5)\\=1-P(Z<2.5)\\=1-0.99379\\=0.00621\\\approx0.0062[/tex]
Use the z-table fro left z-scores to determine the probability.
Thus, the probability that Umbrella returns any shipment is 0.0062.
The weight of a certain type of brick has an expectation of 1.12 kilograms with a variance of 0.0009 kilograms2. How many bricks would need to be selected so that the average weight has a standard deviation of no more than 0.005 kilograms
Answer:
[tex]n \geq 36[/tex]
Step-by-step explanation:
For this case we know the mean and the deviation:
[tex] \mu = 1.12 kg , \sigma= 0.0009[/tex]
The mean is given by this:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
If we find the expected value for the sample mean we have:
[tex] E(\bar X) = E(\frac{\sum_{i=1}^n X_i}{n}) =\frac{1}{n} \sum_{i=1}^n E(X_i)[/tex]
Since each observation in the sample [tex] X_1, X_2,...., X_{25}[/tex] have an expectation [tex] E(X_i) = \mu , i =1,2,...,25[/tex] so then we have that:
[tex] E(\bar X) = \frac{1}{n} n\mu = \mu[/tex]
Now for the variance of the sample mean we have this:
[tex]Var (\bar X) = Var(\frac{\sum_{i=1}^n X_i}{n})= \frac{1}{n^2} \sum_{i=1}^n Var(X_i)[/tex]
And again each observation have a variance [tex] \sigma^2_i = 0.0009 , i =1,2,...,25[/tex] then we have:
[tex]Var (\bar X) =\frac{1}{n^2} n(\sigma^2) =\frac{\sigma^2}{n}[/tex]
And then the standard deviation would be:
[tex] Sd(\bar X) =\sqrt{\frac{\sigma^2}{n}}= \frac{\sigma}{\sqrt{n}}[/tex]
And we want that the standard deviation for the sample no more than 0.005 so we have this condition:
[tex]\frac{\sigma}{\sqrt{n}} \leq 0.005[/tex]
And since we know the value of [tex] \sigma= \sqrt{0.0009}=0.03[/tex] we can solve for the value of n like this:
[tex] \frac{0.03}{0.005} \leq \sqrt{n}[/tex]
[tex] 6 \leq \sqrt{n}[/tex]
And if we square both sides we got:
[tex] n\geq 36[/tex]
Final answer:
To achieve an average weight standard deviation of no more than 0.005 kilograms for a certain type of brick with a variance of 0.0009 kilograms², 36 bricks need to be selected.
Explanation:
The question concerns the calculation of the number of bricks needed so that the average weight of these bricks has a standard deviation of no more than 0.005 kilograms. Given that the weight of a certain type of brick has an expectation (mean) of 1.12 kilograms and a variance of 0.0009 kilograms2, the first step is to understand that the variance of the average weight of n bricks is equal to the variance of a single brick divided by n.
To achieve a standard deviation of the average weight of no more than 0.005 kilograms, we use the formula for the standard deviation of the mean, which is √(variance/n). The variance given is 0.0009 kilograms2. Setting the equation √(0.0009/n) <= 0.005 and solving for n gives:
0.0009/n = 0.0052
0.0009/n = 0.000025
n = 0.0009 / 0.000025 = 36
Therefore, to ensure the standard deviation of the average weight is no more than 0.005 kilograms, 36 bricks would need to be selected.
Money Flow The rate of a continuous money flow starts at $1000 and increases exponentially at 5% per year for 4 years. Find the present value and accumulated amount if interest earned is 3.5% compounded continuously.
Answer:
Present value = $4,122.4
Accumulated amount = $4,742
Step-by-step explanation:
Data provided in the question:
Amount at the Start of money flow = $1,000
Increase in amount is exponentially at the rate of 5% per year
Time = 4 years
Interest rate = 3.5% compounded continuously
Now,
Accumulated Value of the money flow = [tex]1000e^{0.05t}[/tex]
The present value of the money flow = [tex]\int\limits^4_0 {1000e^{0.05t}(e^{-0.035t})} \, dt[/tex]
= [tex]1000\int\limits^4_0 {e^{0.015t}} \, dt[/tex]
= [tex]1000\left [\frac{e^{0.015t}}{0.015} \right ]_0^4[/tex]
= [tex]1000\times\left [\frac{e^{0.015(4)}}{0.015} -\frac{e^{0.015(0)}}{0.015} \right][/tex]
= 1000 × [70.7891 - 66.6667]
= $4,122.4
Accumulated interest = [tex]e^{rt}\int\limits^4_0 {1000e^{0.05t}(e^{-0.035t}} \, dt[/tex]
= [tex]e^{0.035\times4}\times4,122.4[/tex]
= $4,742
The present value and interest accumulated would be as follows:
Present Value = $ 4,122.4
Interest Accumulated = $ 4742
Given that,
Principal at the beginning of money flow = $1,000
Exponential interest rate = 5% per year
Time Period = 4 years
So,
The accumulated money flow's worth = [tex]1000e^{0.05t}[/tex]
The current value of the money can be determined by [tex]\int\limits^4_0 1000e^{0.05t}(e^{-0.035t}) {} \, dt[/tex]
On solving, we get
The present value = $ 4,122.4
Interest Accumulated = $4,742
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A bottling plant fills one-gallon jugs with milk. The label on a one gallon jug of milk states that the volume of milk is 128 fluid ounces (fl.oz.) Federal law mandates that the jug must contain no less than the stated volume. The actual amount of milk in the jugs is normally distributed with mean µ = 129 fl. Oz. and standard deviation ơ = 0.8 fl. Oz. Use this information to answer below questions.
1. Find the z-score corresponding to a jug containing 128 fl. Oz. of milk?
2. What is the probability that a randomly selected jug will contain less than 128 fl. Oz. of milk?
Final answer:
The z-score for a jug with 128 fl. Oz. of milk and the probability of a jug containing less than 128 fl. Oz. are calculated using the mean and standard deviation provided.
Explanation:
1. Find the z-score corresponding to a jug containing 128 fl. Oz. of milk:
Calculate the z-score using the formula: z = (x - µ) / ơSubstitute the values: z = (128 - 129) / 0.8Calculate to find the z-score.2. Probability that a jug contains less than 128 fl. Oz. of milk:
Convert the value to a z-score using the formula.Refer to the Z-table or calculator to find the probability.Suppose you and your 4 friends (5 people) are dressing up as the 6 main characters of the first Avengers movie: Iron Man, Hulk, Thor, Black Widow, Captain America and Hawkeye. (each question is independent of the others.) How many ways can you do this if all 5 people dress up as a different character?
Answer:
720
Step-by-step explanation:
If every person has to choose a different character, the first person to choose a character has 6 options, the second has 5, the third has 4, the fourth has 3, and the last person has only two options. Therefore, the total number of ways you can do this if all 5 people dress up as a different character is:
[tex]n=6*5*4*3*2\\n=720[/tex]
There are 720 ways.
Final answer:
There are 720 different ways for 5 people to dress up as the 6 main Avengers characters; this combinatorics problem uses permutations to find the answer.
Explanation:
Combinations for Avengers Characters
The question is about calculations of combinations, which falls under the subject of Mathematics. More specifically, this is a combinatorics problem that can typically be found at a High School level. We want to find out how many different ways 5 people can dress up as any of the 6 main Avengers characters, assuming each person dresses up as a different character. To solve this, we can use the concept of permutations because the order in which we assign the characters to the 5 friends does matter.
In this case, we have 6 characters to choose from, and we want to assign these characters to 5 friends. We are therefore looking for the number of permutations of 6 characters taken 5 at a time, which is calculated using the formula:
P(n, k) = n! / (n - k)!
Here, 'n' is the total number of characters, and 'k' is the number of people to dress up. Therefore, we have:
P(6, 5) = 6! / (6 - 5)! = 6! / 1! = 720 / 1 = 720
There are 720 different ways for the 5 friends to dress up as the Avengers characters.
Determine whether the relation R defined below is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. For each property, either explain why R has that property or give an example showing why it does not.
a) Let A = {1, 2, 3, 4} and let R = { (2, 3) }
b) Let A = {1, 2, 3, 4} and let R = { (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 4) }.
Answer:
See below
Step-by-step explanation:
Remember some definitions about binary relations. If R⊆S×S then
R is reflexive if (a,a)∈R for all a∈SR is irreflexive if (a,a)∉R for all a∈SR is symmetric if (a,b)∈R implies (b,a)∈R for all a,b∈SR is asymmetric if (a,b)∈R implies (b,a)∉R for all a,b∈SR is antisymmetric if (a,b)∈R and (b,a)∈R imply that a=b, for all a,b∈SR is transitive if (a,b)∈R and (b,c)∈R imply (a,c)∈R for all a,b,c∈Sa) R is not reflexive since (1,1)∉R.
R is irreflexive, since (a,a)∉R for all a=1,2,3,4
R is asymmetric: (2,3)∈R and (3,2)∉R (thus R is not symmetric).
R is antisymmetric, there are no cases to check. R is transitive, there are no cases to check.
b) R is reflexive, checking case by case, (a,a)∈R for all a=1,2,3,4. Hence R is not irreflexive.
R is not asymmetric: (1,2)∈R but (2,1)∈R. R is not symmetric, since (4,1)∈R but (1,4)∉R
R is not antisymmetric: (1,2)∈R and (2,1)∈R but 1≠2.
R is not transitive: (1,2)∈R and (2,4)∈R but (1,4)∉R.
60% of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the probability that at least one of the next three vehicles fail. (Give your answer as a decimal number with 3 digits of precision.)
Answer:
60%
Step-by-step explanation:
The percentage does not change.
Probability at least one of next 3 vehicles fail inspection: approximately 0.784 (to 3 decimal places).
To find the probability that at least one of the next three vehicles fail, we can calculate the probability of the complementary event, i.e., the probability that all three vehicles pass, and then subtract it from 1.
Given that 60% of vehicles pass the inspection, the probability that one vehicle fails the inspection is 1 - 0.60 = 0.40.
Since the vehicles pass or fail independently, the probability that all three vehicles pass is:
[tex]\[0.60 \times 0.60 \times 0.60 = 0.216\][/tex]
Now, the probability that at least one vehicle fails is:
[tex]\[1 - 0.216 = 0.784\][/tex]
So, the probability that at least one of the next three vehicles fail is approximately [tex]\(0.784\)[/tex] (to 3 decimal places).
Donna de paul is raising money for the homeless. She discovers that each church group requires 2 hours of let her writing and 1 hour of follow-up while for each labor union she needs 2 hours of letter written and 3 hours of follow-up. Donna can raise $150 from each church group and $200 from each Union Local and she has a maximum of 20 hours of letters written and a maximum of 16 hours of follow-up available per month. Determine the most profitable mixture of group she should contact in the most money she can lose in a month.
z=()x1+()x2
Answer:
x = 7
y = 3
z (max) = 4950/3 = 1650
Step-by-step explanation:
Let call
x numbers of church goup and
y numbers of Union Local
Then
First contraint
2*x + 2*y ≤ 20
Second one
1*x + 3*y ≤ 16
Objective Function
z = 150*x + 200*y
Then the system is
z = 150*x + 200*y To maximize
Subject to:
2*x + 2*y ≤ 20
1*x + 3*y ≤ 16
x ≥ 0 y ≥ 0
We will solve by using the Simplex method
z - 150 *x - 200*y = 0
2*x + 2*y + s₁ = 20
1*x + 3*y + 0s₁ + s₂ = 16
First Table
z x y s₁ s₂ Cte
1 -150 -200 0 0 = 0
0 2 2 1 0 = 20
0 1 3 0 1 = 16
First iteration:
Column pivot ( y column ) row pivot (third row) pivot 3
Second table
z x y s₁ s₂ Cte
1 -250/3 0 0 200/3 = 3200/3
0 - 4/3 0 -1 2/3 = -20/3
0 1/3 1 0 1/3 = -20/3
Second iteration:
Column pivot ( x column ) row pivot (second row) pivot -4/3
Third table
z x y s₁ s₂ Cte
1 0 0 750/12 700/6 = 4950/3
0 1 0 3/4 -1/2 = 7
0 0 1 -1/4 1/2 = 9/3
Donna should contact four church groups and four labor unions to maximize her fundraising, which would yield $1400. The maximum money she could 'lose' (or not earn) is $0 if she does not contact any group.
Explanation:We need to set up linear inequalities to represent the constraints described by Donna's situation. We will let x represent the number of church groups and y represent the number of labor unions she contacts.
The time required for letter writing can be described by the inequality: 2x + 2y <= 20, and the time for follow-ups is x + 3y <= 16.
We want to maximize the sum z = 150x + 200y which represents the total money she can raise.
To find the maximum, one method is to graph the feasible region defined by the constraints and find the vertices. Then evaluate the objective function at the vertices. In this case, the vertices are (0,0), (0,5), (8,0) and (4,4).
Upon calculation, contacting four church groups and four labor unions yields the highest amount of $1400.
As for the loss, given the nature of the task, the most money she could 'lose' in a month is just not raising any money at all, which would be $0 if she doesn't contact any group.
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How expensive is Maui? A newspaper gave the following costs in dollars per day for a random sample of condominiums located throughout the island of Maui. 88 50 66 60 360 55 500 71 41 350 60 50 250 45 45 125 235 65 60 110 (a) Compute the mean, median, and mode for the data. (Round your answers to two decimal places.)
The mean (average) of the data set is approximately 110.40, the median (the middle value when the data set is arranged in order from least to greatest) is 65, and there is no mode (the most frequently occurring number) in the data set.
Explanation:The subject of your question is Mathematics, specifically in the field of Statistics. To compute the mean, median, and mode, you would do the following:
Add up all numbers in the data set and divide by the number of items in that set. This is the Mean. Arrange the data set from smallest to largest and find the middle value. If there is an even number of items in the data set, the median is the average of the middle two numbers. This is the Median. The number that appears most frequently in your data set is the Mode.For the given data set your mean is approximately 110.40 , the median is 65 and there is no mode as no numbers in the data set are repeated.
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Let R be the region bounded by y=x^2, x=1, and y=0. use the shell method to find the volume of the solid generated when R is revolved about the line x= 2
Answer:
[tex]V = \frac{5\pi}{6}[/tex] or 2.62
Step-by-step explanation:
Since our region (on the left) is bounded by x = 1 and x = 0 (where [tex]y = x^2 = 0[/tex], if we take center at x = 2 then our radius will range from 1 to 2 (x=1 to x = 0). We can use the following integration to calculate the volume using shell method
[tex]V = \int\limits^2_1 {2\pi r h} \, dr[/tex]
where r = 2 - x so x = 2 - r and [tex]h = y = x^2 = (2-r)^2[/tex] for [tex]1 \leq r \leq 2[/tex]
[tex]V = \int\limits^2_1 {2\pi r(2-r)^2} \, dr[/tex]
[tex]V = \int\limits^2_1 {2\pi r(4 - 4r + r^2)} \, dr[/tex]
[tex]V = 2\pi \int\limits^2_1 {r^3 - 4r^2 +4r} \, dr[/tex]
[tex]V = 2\pi\left[\frac{r^4}{4} - \frac{4r^3}{3} + 2r^2\right]^2_1[/tex]
[tex]V = 2\pi\left[\left(\frac{2^4}{4} - \frac{4*2^3}{3} + 2*2^2\right) - \left(\frac{1^4}{4} - \frac{4*1^3}{3} + 2*1^2\right)\right][/tex]
[tex]V = 2\pi(4 - 32/3 +8 - 1/4 + 4/3 - 2)[/tex]
[tex]V = \pi(20 - 56/3 - 1/2)[/tex]
[tex]V = \pi\frac{120 - 112 - 3}{6}[/tex]
[tex]V = \frac{5\pi}{6}[/tex] or 2.62
A house hunter on Long Island estimates that 20% of the available houses in her price range are in acceptable condition. Furthermore, she has time to look at only one house each week. What is the probability that she will find an acceptable house in the first two weeks that she looks (round off to second decimal place)?
Answer:
36% probability that she will find an acceptable house in the first two weeks that she looks.
Step-by-step explanation:
For each house, there is a 20% probability of it being acceptable. And a 100-20 = 80% of not being acceptable
What is the probability that she will find an acceptable house in the first two weeks that she looks (round off to second decimal place)?
She only looks one house a week.
So this is the same as the probability of taking two or less weeks to find a house.
The are two outcomes
Finding an acceptable house in the first week, with 20% probability
Not finding an acceptable house in the first week, with 80% probability, and then finding an acceptable house in the second week, with 20% probability.
Probability:
[tex]P = 0.2 + 0.8*0.2 = 0.36[/tex]
36% probability that she will find an acceptable house in the first two weeks that she looks.
The correct answer is 0.36 or 36%.
To solve this problem, we need to calculate the probability that the house hunter does not find an acceptable house in the first two weeks and then subtract this probability from 1 to find the probability that she does find an acceptable house within that time frame.
Let's denote the probability of finding an acceptable house in a given week as[tex]\( p \)[/tex] and the probability of not finding an acceptable house in a given week as [tex]\( q \)[/tex]. Since 20% of the houses are in acceptable condition, [tex]\( p = 0.20 \) and \( q = 1 - p = 1 - 0.20 = 0.80 \).[/tex]
The probability that the house hunter does not find an acceptable house in the first week is[tex]\( q \).[/tex] The probability that she does not find an acceptable house in the first two weeks is the product of the probabilities that she does not find one in each of the two weeks separately, which is [tex]\( q \times q \) or \( q^2 \).[/tex]
Now, we calculate [tex]\( q^2 \):[/tex]
[tex]\[ q^2 = (0.80)^2 = 0.64 \][/tex]
This is the probability that she will not find an acceptable house in the first two weeks. To find the probability that she will find an acceptable house in the first two weeks, we subtract this value from 1:
[tex]\[ \text{Probability of finding an acceptable house in two weeks} = 1 - q^2 \][/tex]
[tex]\[ \text{Probability of finding an acceptable house in two weeks} = 1 - 0.64 \][/tex]
[tex]\[ \text{Probability of finding an acceptable house in two weeks} = 0.36 \][/tex]
It is known that the A matrix of a single-input-single-output state space system has 4 eigenvalues at -1, 1, 2, 3, and the D matrix is 0. Furthermore, when the input is u(t) = 1 and the initial states are all zero, the steady-state output of the system is 5. Find the controllable canonical form of the system
Answer:
The controllable canonical form of the system =
Y = (5 0 0 0) (x1)
(x2)
(x3)
(x4)
Step-by-step explanation:
The detailed explanation is as shown in the attached file.
Five urns are numbered 3,4,5,6 and 7, respectively. Inside each urn, there are n² dollars where n is the number on the urn.
The following experiment is performed:
An urn is selected at random. If its number is a prime number the experimenter receives the amount in the urn and the experiment is over. If its number is not a prime number, a second urn is selected from the remaining four and the experimenter receives the total amount in the two urns selected.
What is the probability that the experimenter ends up with exactly twenty- five dollars?
Answer:
0.25 or 25%
Step-by-step explanation:
3, 5 and 7 are prime numbers.
There are two possible outcomes for which the experimenter ends up with exactly twenty-five dollars:
A) Choosing urn 5 (5 x 5 = 25).
[tex]P(A) = \frac{1}{5}[/tex]
B) Choosing urn 4 and then urn 3 ([4 x 4] + [3 x 3] = 25).
[tex]P(B)= \frac{1}{5} *\frac{1}{4}=\frac{1}{20}[/tex]
The probability that the experimenter ends up with exactly $25 is:
[tex]P(x=\$25)=P(A)+P(B)= \frac{1}{5}+\frac{1}{20}\\P(x=\$25)=0.25=25\%[/tex]
A Gift for You makes floral arrangements and fruit baskets. The small business has a maximum of 40 hours per week available in the assembly department and a maximum of 10 hours per week in the packaging department. Each floral arrangement takes 20 minutes to assemble and 6 minutes to package. Each fruit basket takes 15 minutes to assemble and 2 minutes to package. The profit for each floral arrangement is $50 and the profit for each fruit basket is $35. The company wants to maximize their profit.a. Set up the linear programming problem by writing the objective function as well as the system of constraints. b. How many floral arrangements and fruit baskets must be sold to maximize profit? c. What is the maximum profit? Let x the number of floral arrangements and y the number of fruit baskets.Give the objective function a. Max P= 50x+ 35y b. Max P= 20x + 6y c. Min P= 15x+2yd. Max P= 35x +20ye. Min P= 35x+20yGive a constrainta. 20x+ 6y <=40b. 20x+15y<=40c. 6x+2y<=10d. 20x+ 15y<=24,00e. 6x+2y>=600How many of each should they sell to maximize their profit?a. 84 Floral Arrangements and 48 Fruit Baskets b. 0 Floral Arrangements and 160 Fruit Baskets c. 0 Floral Arrangements and 300 Fruit Baskets d. 100 Floral Arrangements and O Fruit Baskets e. 48 Floral Arrangements and 84 Fruit Baskets
Answer:
z(max) = 5000
x = 100
y = 0
Step-by-step explanation:
Let call
x floral arrangements and
y fruit baskets
Then Objective function is according to profits in each gift
z = 50*x + 35*y
Constraints:
1.- Hours available in Assembly department 40 in minutes is 2400 minutes
20*x + 15*y ≤ 2400
2.- Hours available in packaging department 10 in minutes is 600
6*x + 2*y ≤ 600
3.- x and y must be x ≥ 0 y ≥ 0
Then the system is:
z - 50*x - 50*y = 0 To maximize subject to:
20*x + 15*y ≤ 2400
6*x + 2*y ≤ 600
x ≥ 0 y ≥ 0
Simplex Method:
z x y s₁ s₂ Cte
1 -50 -35 0 0 = 0
0 20 15 1 0 = 2400
0 6 2 0 1 = 600
First iteration: 6 is a pivot we dvede R3 by 6
z x y s₁ s₂ Cte
1 0 15 0 50/6 = 5000
0 0 -50/6 -1 20/6 = -400
0 20 15 1 0 = 2400
0 1 2/6 0 1/6 = 100
We have done no negative number in the objective function we stop iteration and
z(max) = 5000
x = 100
y = 0
to add to R1 50*R3 [ 0 50 100/6 0 50/6 5000
to add to R2 20*R3 [ 0 20 40/6 0 20/6 2000
The Dow Jones Industrial Average has had a mean gain of 432 pear year with a standard deviation of 722. A random sample of 40 years is selected. What is the probability that the mean gain for the sample was between 250 and 500?
Answer:
66.98% probability that the mean gain for the sample was between 250 and 500.
Step-by-step explanation:
To solve this problem, it is important to know the Normal probability distribution and the Central limit theorem.
Normal probability distribution
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.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
In this problem, we have that:
[tex]\mu = 432, \sigma = 722, n = 40, s = \frac{722}{\sqrt{40}} = 114.16[/tex]
What is the probability that the mean gain for the sample was between 250 and 500?
This is the pvalue of Z when X = 500 subtracted by the pvalue of Z when X = 250.
So
X = 500
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{500 - 432}{114.16}[/tex]
[tex]Z = 0.6[/tex]
[tex]Z = 0.6[/tex] has a pvalue of 0.7257.
X = 250
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{250 - 432}{114.16}[/tex]
[tex]Z = -1.59[/tex]
[tex]Z = -1.59[/tex] has a pvalue of 0.0559.
So there is a 0.7257 - 0.0559 = 0.6698 = 66.98% probability that the mean gain for the sample was between 250 and 500.
The probability that the mean gain for a sample of 40 years is between 250 and 500 is approximately 0.6698, or 66.98%.
To determine the probability that the mean gain for a random sample of 40 years of the Dow Jones Industrial Average is between 250 and 500, we will use the concept of the sampling distribution of the sample mean. The steps involved are:
State the given information:
- Population mean [tex](\(\mu\))[/tex] = 432
- Population standard deviation [tex](\(\sigma\))[/tex] = 722
- Sample size n = 40
Find the standard error of the mean (SEM):
The standard error of the mean is calculated as:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{722}{\sqrt{40}} \approx 114.2 \][/tex]
Convert the sample means to z-scores:
To find the probability that the sample mean [tex](\(\bar{x}\))[/tex] is between 250 and 500, we convert these values to z-scores using the formula:
[tex]\[ z = \frac{\bar{x} - \mu}{\text{SEM}} \] For \(\bar{x} = 250\): \[ z = \frac{250 - 432}{114.2} \approx \frac{-182}{114.2} \approx -1.59 \] For \(\bar{x} = 500\):[/tex]
[tex]\[ z = \frac{500 - 432}{114.2} \approx \frac{68}{114.2} \approx 0.60 \][/tex]
Find the probability corresponding to the z-scores:
Using the standard normal distribution table or a calculator, we find the probabilities corresponding to these z-scores.
[tex]\(z = -1.59\), \(P(Z \leq -1.59) \approx 0.0559\) \\\(z = 0.60\), \(P(Z \leq 0.60) \approx 0.7257\)[/tex]
Calculate the probability that the mean gain is between 250 and 500:
\[tex][ P(250 \leq \bar{x} \leq 500) = P(Z \leq 0.60) - P(Z \leq -1.59) \] \[[/tex]
= 0.7257 - 0.0559 = 0.6698
Conclusion:
The probability that the mean gain for a sample of 40 years is between 250 and 500 is approximately 0.6698, or 66.98%.
A ball is dropped from a state of rest at time t=0.The distance traveled after t seconds is s(t)=16t^2 ft.
(a) How far does the ball travel during the time interval [3,3.5] ?Δs=____ft(b) Compute the average velocity over [3,3.5] .Δs/Δt= ____ ft/sec(c) Compute the average velocity over time intervals [3, 3.01] , [3, 3.001] , [3, 3.0001] , [2.9999, 3] , [2.999, 3] , [2.99, 3] .
Use this to estimate the object's instantaneous velocity at t=3 .V(3)= ____ ft/sec
The ball travels a distance of 52 ft during the time interval [3,3.5]. The average velocity over this interval is 104 ft/sec. When we calculate over increasingly smaller time intervals, it appears the instantaneous velocity at t=3 is 96 ft/sec.
Explanation:(a) We need to find the displacement Δs over the time interval [3,3.5]. For that, we subtract the position at time 3 from the position at time 3.5:
Δs = s(3.5) - s(3) = 16(3.5^2) - 16(3^2) = 196 - 144 = 52 ft.
(b) We calculate the average velocity by dividing Δs by Δt. That is Δs/Δt = 52/0.5 = 104 ft/sec.
(c) To estimate the instantaneous velocity at t=3, we have to compute the average velocities over smaller and smaller intervals centered at t=3. Let's calculate the average velocities for [3,3.01],[2.999,3]:
For [3,3.01]: Δs/Δt = [16*(3.01)^2 - 16*3^2]/0.01 = 96.12 ft/sec. For [2.999,3]: Δs/Δt = [16*3^2 - 16*(2.999)^2]/0.001 = 96 ft/sec.
Instantaneous velocity at t=3, v(3) can be estimated as the limit of these average velocities, which seems to be approaching about 96 ft/sec.
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(a) The ball travel Δs= 52 ft during the time interval [3,3.5].
(b) The average velocity over [3,3.5] is Δs/Δt= 104 ft/sec
(c) The object's instantaneous velocity at t=3 .V(3)= 96 ft/sec
Let's solve the problem step-by-step.
Given:
The distance traveled by the ball after t seconds is given by the function: [tex]s(t) = 16t^2[/tex]
(a) To find the distance traveled, we need to calculate the change in the distance function, which is
[tex]\Delta s = s(3.5) - s(3)[/tex]
First, compute [tex]s(3)[/tex]:
[tex]s(3) = 16(3)^2 = 16 \times 9 = 144[/tex] ft
Next, compute [tex]s(3.5)[/tex]:
[tex]s(3.5) = 16(3.5)^2 = 16 \times 12.25 = 196[/tex] ft
So,
[tex]\Delta s = 196 - 144 = 52[/tex] ft
(b) Average velocity is calculated by dividing the change in distance by the change in time, which is
[tex]\frac{\Delta s}{\Delta t} = \frac{52}{0.5} = 104[/tex] ft/sec
(c) For the time intervals, we compute each average velocity:
[3, 3.01]:
[tex]\Delta t = 0.01[/tex]
[tex]s(3.01) = 16(3.01)^2 = 16 \times 9.0601 = 144.9616[/tex] ft
[tex]\Delta s = 144.9616 - 144 = 0.9616[/tex] ft
[tex]\text{Average velocity} = \frac{0.9616}{0.01} = 96.16[/tex] ft/sec
[3, 3.001]:
[tex]\Delta t = 0.001[/tex]
[tex]s(3.001) = 16(3.001)^2 = 16 \times 9.006001 = 144.096016[/tex] ft
[tex]\Delta s = 144.096016 - 144 = 0.096016[/tex] ft
[tex]\text{Average velocity} = \frac{0.096016}{0.001} = 96.016[/tex] ft/sec
[3, 3.0001]:
[tex]\Delta t = 0.0001[/tex]
[tex]s(3.0001) = 16(3.0001)^2 = 16 \times 9.00060001 = 144.00960016[/tex] ft
[tex]\Delta s = 144.00960016 - 144 = 0.00960016[/tex] ft
[tex]\text{Average velocity} = \frac{0.00960016}{0.0001} = 96.0016[/tex] ft/sec
[2.9999, 3]:
[tex]\Delta t = 0.0001[/tex]
[tex]s(2.9999) = 16(2.9999)^2 = 16 \times 8.99940001 = 143.99040016[/tex] ft
[tex]\Delta s = 144 - 143.99040016 = 0.00959984[/tex] ft
[tex]\text{Average velocity} = \frac{0.00959984}{0.0001} = 95.9984[/tex] ft/sec
[2.999, 3]:
[tex]\Delta t = 0.001[/tex]
[tex]s(2.999) = 16(2.999)^2 = 16 \times 8.994001 = 143.904016[/tex] ft
[tex]\Delta s = 144 - 143.904016 = 0.095984[/tex] ft
[tex]\text{Average velocity} = \frac{0.095984}{0.001} = 95.984[/tex] ft/sec
[2.99, 3]:
[tex]\Delta t = 0.01[/tex]
[tex]s(2.99) = 16(2.99)^2 = 16 \times 8.9401 = 143.0416[/tex] ft
[tex]\Delta s = 144 - 143.0416 = 0.9584[/tex] ft
[tex]\text{Average velocity} = \frac{0.9584}{0.01} = 95.84[/tex] ft/sec
From these values, we can see that as the interval gets smaller, the average velocities are approaching a specific value. This helps us estimate the instantaneous velocity at [tex]t = 3[/tex].
A baseball team plays in a stadium that holds 56000 spectators. With the ticket price at $11 the average attendence has been 21000. When the price dropped to $9, the average attendence rose to 28000. Assume that attendence is linearly related to ticket price.
(a) Find the demand function, assuming that it is linear.
(b) How should ticket prices be set to maximize revenue?
Answer:
(a) [tex]p(x) = 17-\frac{x}{3,500}[/tex]
(b) $8.50
Step-by-step explanation:
(a) The slope of the demand function, p(x), is determined by:
[tex]m=\frac{11-9}{21,000-28,000}=-\frac{1}{3,500 }[/tex]
Applying the point (21,000; 11) to the general linear equation formula gives us the demand function:
[tex]p(x) - 11 = -\frac{1}{3,500}*(x-21,000)\\p(x) = 17-\frac{x}{3,500}[/tex]
(b) The revenue function, r(x), is given by:
[tex]r(x) =x*p(x) = 17x-\frac{x^2}{3,500}[/tex]
The value of x for which the derivate of the revenue function is zero gives us the attendance for which revenue is maximized:
[tex]\frac{dr(x)}{dx} =0= 17-\frac{2x}{3,500}\\x=29,750[/tex]
At an attendance of 29,750, the price is:
[tex]p = 17-\frac{29,750}{3,500}\\p=\$8.50[/tex]
Tickets should be set at a price of $8.50.
The demand function for the baseball team can be found using the given data points. The revenue function can be derived using the demand function, and its maximum can be determined by finding the critical points. The ticket price should be set at $8.50 to maximize revenue for the baseball team.
Explanation:(a) To find the demand function, we can use the point-slope form of a linear equation. Let's denote the ticket price as p and the attendance as a. We have two data points: (11, 21000) and (9, 28000). We can use the formula: a - a1 = m(p - p1) where (p1, a1) represents one of the given data points and m represents the slope. Let's use the first data point:
21000 - 28000 = m(11 - 9)
-7000 = 2m
m = -3500
So, the demand function is: a(p) = -3500p + b. To find the constant term b, we can substitute one of the data points into the equation. Let's use the first data point (11, 21000):
21000 = -3500(11) + b
21000 = -38500 + b
b = 59500
Therefore, the demand function is: a(p) = -3500p + 59500.
(b) Revenue is the product of the ticket price and the attendance. Let's denote the revenue as R and the ticket price as p. The revenue function can be written as: R(p) = p * a(p). Substituting the demand function into the revenue function, we have: R(p) = p * (-3500p + 59500). To maximize revenue, we can find the critical points of the function by taking the derivative and setting it equal to zero.
R'(p) = -7000p + 59500 = 0
-7000p = -59500
p = 8.5
The critical point is p = 8.5. To confirm that this gives a maximum, we can take the second derivative and check its sign.
R''(p) = -7000
Since the second derivative is negative, the critical point p = 8.5 corresponds to the maximum revenue. Therefore, the ticket price should be set at $8.50 to maximize revenue.
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Which of the following is equivalent to finding the zeros of a function
A. X-intercepts
B. Y-intercepts
C. Slope
D. Origin
Answer:
A. X-intercepts
Step-by-step explanation:
The zeros of a function are the values of x for which y is 0.
For example:
y = 2x - 4
Has zeros
2x - 4 = 0
2x = 4
x = 2
Which means that when x = 2, y = 0. Looking at the graphic of this function, it crosses the x line when x = 2. So the zeros are also called x intercepts.
So the correct answer is
A. X-intercepts
Answer:
X-intercepts
Step-by-step explanation:
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Determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0. (Enter your answers as a comma-separated list.) t2y'' − 2ty' + 2y = 0
Answer:
The only solution is r=2 or r=1
Step-by-step explanation:
Check the attachment
To determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0, substitute y = tr into the given differential equation and solve for r. The values of r that satisfy the equation are r = 1.
Explanation:To determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0, we need to substitute y = tr into the given differential equation and solve for r.
First, we find the first and second derivatives of y = tr:
y' = r
y'' = 0
Substituting these derivatives into the differential equation, we get:
t2(0) - 2t(r) + 2(tr) = 0
Simplifying the equation, we have:
(2 - 2r)t = 0
This equation is satisfied when t = 0 or r = 1. Therefore, the values of r for which the given differential equation has solutions of the form y = tr for t > 0 are r = 1.
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On a test, 74% of the questions are answered correctly. If 111 questions are correct, how many questions are on the test?
Answer:
150
Step-by-step explanation:
You have to divide the correct answers (111) by the total amount of questions (x) in order to find the 74% (0.74)
As per the unitary method, there are 150 questions on the test.
Let's start by breaking down the information we have been given. We know that 74% of the questions on the test were answered correctly, and the number of questions answered correctly is 111.
Step 1: Convert Percentage to Decimal
To make calculations easier, we need to convert the percentage into a decimal. To do this, we divide 74 by 100, which gives us 0.74.
Step 2: Set Up the Equation
Now we can set up an equation to solve for the total number of questions on the test. Let's denote the total number of questions as "x". Since 74% of the questions were answered correctly, we can say that 0.74x questions were answered correctly.
Step 3: Solve for Total Questions
We are given that 111 questions were answered correctly. So, we can set up the equation:
0.74x = 111
Step 4: Solve for "x"
To solve for "x", we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.74:
x = 111 / 0.74
x = 150
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A large disaster cleaning company estimates that 30 percent of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that it has contracted, calculate the mean number of jobs completed within the bid time.
Answer: 2.4
Step-by-step explanation:
Given : A large disaster cleaning company estimates that 30 percent of the jobs it bids on are finished within the bid time.
i..e The proportion of the jobs it bids on are finished within the bid time: p = 30%= 0.30
Sample size of jobs that it has contracted : n= 8
Then , the mean number of jobs completed within the bid time : [tex]\mu=np[/tex]
[tex]= 8\times0.30=2.4[/tex]
Hence, the mean number of jobs completed within the bid time is 2.4 .
Name the quadrant in which the angle below is located.
sine = 4 and cose is positive.
Answer:
first quadrant
Step-by-step explanation:
To know this we have to take into account that for the breast we have to look at the y-axis and in the cosine we will look at the x-axis
We have the positive and the negative part of the x-axis, now we notice that it says that the cosine is positive, then the quadrant will be with the positive x, these quadrants can be the first or the fourth.
Now if we look at sine = 4 is positive, this means that the axis y will also be positive, this means that it will be between the first and second quadrant.
Now if we look to meet the conditions of the sine and cosine, the angle has to be found in the first quadrant
gasoline wholesale distributor has bulk storage tanks holding a fixed supply. The tanks are filled every Monday. Of interest to the wholesaler is the proportion of this supply that is sold during the week. Over many weeks, this proportion has been observed to be modeled fairly well by a beta distribution with alpha = 4 and beta = 2. Find the probability that at least 90% of the stock will be sold in a given week? a. 0.07 b. 0.05 c. 0.09 d. 0.06 e. 0.08
Answer:
e. 0.08
Step-by-step explanation:
In the question above, a certain quantity of goods was supplied while a specific quantity of goods was sold per week. In a given week, if the number of proportion sold is X, therefore:
f(x) = {Γ(4+2)/Γ(4)Γ(2) x^3 (1-x), 0≤x≤1 ; 0, elsewhere
and
P(X greater than 0.9) = [tex]\int\limits^1_ {0.9} \, 20(x^{3} - x^{4}) dx[/tex] = 20*{(y^4/4)[1,0.9] - (y^5/5)[1,0.9]} = 20*{(0.25 - 0.164) - (0.20 - 0.118)} = 20*{0.086 - 0.0819} = 20*0.0041 = 0.082
Therefore the probability of the proportion sold is approximately 0.082
The probability that at least 90% of the stock will be sold in a given week is approximately 0.05. The correct answer option is b. 0.05
Here's how to calculate it:
1. Given the beta distribution with parameters alpha = 4 and beta = 2, we want to find [tex]\( P(X > 0.9) \)[/tex], where X represents the proportion of stock sold in a week.
2. Since [tex]\( P(X > 0.9) = 1 - P(X \leq 0.9) \)[/tex], we need to find the cumulative distribution function (CDF) of the beta distribution and then subtract it from 1.
3. Using the incomplete beta function formula [tex]\( I_{x}(\alpha, \beta) \)[/tex], we have:
[tex]\[ P(X > 0.9) = 1 - I_{0.9}(4, 2) \][/tex]
4. Calculating [tex]\( I_{0.9}(4, 2) \)[/tex] using software or a calculator gives approximately 0.95.
5. Subtracting this from 1:
[tex]\[ P(X > 0.9) = 1 - 0.95 = 0.05 \][/tex]
So, the correct answer is b. 0.05.
The measures of the angles of ABC are given by the expressions in the table A( 6X - 1 ) ° B 20° C ( x + 14 ) ° what are the measures of angles A and C ? Enter your answers in the boxes mA = mC =
Answer:
Step-by-step explanation:
The sum of the angles in a triangle is 180 degrees. This means that in triangle ABC,
Angle A + angle B + angle C = 180
Therefore,
6x - 1 + 20 + x + 14 = 180
6x + x + 20 + 14 - 1 = 180
7x + 33 = 180
Subtracting 33 from the left hand side and the right hand side of the equation, it becomes
7x + 33 - 33 = 180 - 33
7x = 147
Dividing the left hand side and the right hand side of the equation by 7, it becomes
7x/7 = 147/7
x = 21
Therefore
Angle A = 6x - 1 = 6 × 21 - 1
Angle A = 125 degrees
Angle C = x + 14 = 21 + 14
Angle C = 35 degrees.
Preston and Joel are both solving the equation 2x=14. Preston is not sure what to do because he does not know a power of 2 that equals 14. Joel uses his calculator to graph y=2x and y=14 and find the point of intersection. Will Joel's method work?
Answer:
yes
Step-by-step explanation:
You can always separate an equation into two parts and see where those graphs intersect.
Joel's method works well.
_____
Additional comments
Preston should know that the invention of logarithms makes it easy to solve equations like this. x = log₂(14) = log(14)/log(2) ≈ 3.8073549.
As for Joel's method, I prefer to subtract the right side to get the equation ...
2^x -14 = 0
Then graphing y = 2^x -14, I look for the x-intercept. Most graphing calculators make it easy to find x- and y-intercepts. Not all make it easy to find points of intersection between different curves.
Answer:
Yes, the graph intersects around (3.807,14), so 3.807 is a good estimate of the solution to 2^x=14.
Step-by-step explanation:
A construction company needs to remove 2 1/6 tons of dirt from a construction site. They can remove 710 tons of dirt each hour. What is the total number of hours it will take to remove the dirt?
Answer:
13/4260 tons
Step-by-step explanation:
We have the rate at which they remove tons of dirt per hour. We also know that total that needs to be removed. We can determine the time by dividing the amount of tons that need to be removed by the rate:
[tex]=(13/6)/\cdot{710}=13/4260[/tex]
will take 13/4260 hours to remove the dirt
A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions: a. In how many ways can one component of each type be selected
Answer:
120 ways
Step-by-step explanation: If there are three different categories of components, that is, component A, comprising four types; component B comprising, five types and component C comprising, six types. Three types are to be selected, that is, one type from each category. The number of ways one component of each type be selected is:
Component A * Component B * Component C = 4 X 5 X 6 = 120
You are applying for a job at two companies. Company A offers starting salaries with mu equals μ=$27,000 and σ=$3,000. Company B offers starting salaries with μ=$27,000 and sσ=$7,000. From which company are you more likely to get an offer of 33,000 ormore?
Answer:
For the company A : [tex] P(X>33000)= P(Z> \frac{33000-27000}{3000}) =P(Z>2) = 1-P(Z<2)= 0.0228[/tex]
For the company B: [tex] P(X>33000)= P(Z> \frac{33000-27000}{7000}) =P(Z>0.857) = 1-P(Z<0.857)= 0.196[/tex]
So as we can see we have a higher probability for the company B.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Company A
Let X the random variable that represent the salaries of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(27000,3000)[/tex]
Where [tex]\mu=27000[/tex] and [tex]\sigma=3000[/tex]
For this case if we find the probability for [tex] P(X>33000)[/tex] using the z score given by:
[tex] z= \frac{x -\mu}{\sigma}[/tex]
And if we use this formula we got:
[tex] P(X>33000)= P(Z> \frac{33000-27000}{3000}) =P(Z>2) = 1-P(Z<2)= 0.0228[/tex]
Company B
Let X the random variable that represent the salaries of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(27000,7000)[/tex]
Where [tex]\mu=27000[/tex] and [tex]\sigma=7000[/tex]
For this case if we find the probability for [tex] P(X>33000)[/tex] using the z score given by:
[tex] z= \frac{x -\mu}{\sigma}[/tex]
And if we use this formula we got:
[tex] P(X>33000)= P(Z> \frac{33000-27000}{7000}) =P(Z>0.857) = 1-P(Z<0.857)= 0.196[/tex]
The company from which we are more likely to get an offer of $33,000 or more is determined by comparing the probabilities of receiving such an offer from each of the two companies. This probability is determined using a concept called the z-score, with which we can convert any data point into a common scale, which makes comparison easier.
1. Compute the z-scores for obtaining this salary for both companies. The z-score formula is (X - μ) / σ where X is the data point we are interested in, μ stands for the mean amount of the starting salary, and σ is the standard deviation.
For Company A:
z = (33000 - 27000) / 3000
We deduct the average salary for Company A (μ = 27000) from the target salary (X = 33000) and divide it by the standard deviation (σ = 3000).
For Company B:
Similarly, the z score is calculated by deducting the average salary for Company B (μ = 27000) the from target salary (X = 33000) and dividing by the standard deviation (σ = 7000).
2. Now we have the z-scores for both companies. The next step is to use the z-score to calculate the probability of getting an offer of $33,000 or more from both companies. We can do this by using the cumulative distribution function (CDF) which gives us the probability that a random variable is less than or equal to a certain value. But since we want the probability that the salary is more than $33,000, we subtract the result from one (1 - CDF(z)).
3. We compare these probabilities. The company with the higher probability value will be the one from which we are more likely to get an offer of $33,000 or more.
4. Based on these calculations, you are more likely to get an offer of $33,000 or more from Company B.
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A basketball player shoots a basketball with an initial velocity of 15 ft/sec. The ball is released from an initial height of 6.5 feet.
The function ℎ()=−162+0+h0 models the height, in feet, of an object after t seconds. v0 is the initial velocity of the object, and h0 is the initial height of the object.
Part 1: Write a function that models the height of the basketball. Use your function to answer Parts 2-4.
Part 2: How long does it take for the basketball to hit the ground? Round your answer to the nearest hundredth. Show all of your work. You're welcome to use this quadratic formula calculator, but please explain your answer.
Part 3: When does the basketball reach its maximum height? Round your answer to the nearest hundredth. Show all of your work and explain your answer.
Part 4: What is the maximum height of the basketball? Round your answer to the nearest hundredth. Show all of your work and explain your answer.
Answer:
∞
Step-by-step explanation:
A recent survey found that 71% of senior adults wear glasses for driving. In a group of 20 senior adults, how like that no more than 10 wear glasses for driving? Group of answer choices 95.2% 4.8% 3.1% 1.7%
Answer:
[tex] P(X\leq 10) = 1- 0.961525= 0.0385[/tex]
The nearest answer for this case would be 3.1%
Step-by-step explanation:
Previous concepts
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Solution to the problem
Let X the random variable of interest, on this case we now that:
[tex]X \sim Binom(n=20, p=0.71)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
For this case we can begin finding the probability P(X>10). If we find the individual probabilities we got:
[tex]P(X=11)=(20C11)(0.71)^{11} (1-0.71)^{20-11}=0.0563[/tex]
[tex]P(X=12)=(20C12)(0.71)^{12} (1-0.71)^{20-12}=0.1034[/tex]
[tex]P(X=13)=(20C13)(0.71)^{13} (1-0.71)^{20-13}=0.1558[/tex]
[tex]P(X=14)=(20C14)(0.71)^{14} (1-0.71)^{20-14}=0.1907[/tex]
[tex]P(X=15)=(20C15)(0.71)^{15} (1-0.71)^{20-15}=0.1867[/tex]
[tex]P(X=16)=(20C16)(0.71)^{16} (1-0.71)^{20-16}=0.1429[/tex]
[tex]P(X=17)=(20C17)(0.71)^{17} (1-0.71)^{20-17}=0.082[/tex]
[tex]P(X=18)=(20C18)(0.71)^{18} (1-0.71)^{20-18}=0.036[/tex]
[tex]P(X=19)=(20C19)(0.71)^{19} (1-0.71)^{20-19}=0.0087[/tex]
[tex]P(X=20)=(20C20)(0.71)^{20} (1-0.71)^{20-20}=0.00106[/tex]
And if we add the values we got:
[tex] P(X>10)= P(X=11) +.... +P(X=20) = 0.961525[/tex]
And if we use the complement rule the probability that "no more than 10 wear glasses for driving" we can do this:
[tex] P(X\leq 10) = 1- 0.961525= 0.0385[/tex]
The nearest answer for this case would be 3.1%