It is known that IQ scores form a normal distribution with a mean of 100 and a standard deviation of 15. If a researcher obtains a sample of 16 students’ IQ scores from a statistics class at UT. What is the shape of this sampling distribution?

Answer:

mmmm

yesssirrreeee

Answer:

0.84 or 84%

Step-by-step explanation:

The probability of an apple tree being injured in this period, is the probability of it being injured by the first frost (0.6), added to the probability of it being injured by the second frost (0.6) minus the probability of it being injured by both frosts (0.6 x 0.6):

[tex]P = P(F_1)+P(F_2)-P(F_1\ and\ F_2)\\P=0.6+0.6-(0.6*0.6)\\P=0.84 = 84\%[/tex]

There is a 0.84 or 84% probability of an apple tree being injured in this period.

The value of probability can only be from 0 to 1. The probability of an apple tree being injured in this period is 0.84 or 84 %.

Probability means possibility. It deals with the occurrence of a random event. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.

Frost damage to apple blossoms can severely reduce apple yield in commercial orchards.

It has been determined that the probability of a late spring frost causing blossom damage to Empire apple trees in the Hudson Valley of New York State is 0.6.

The probability of it being injured by first frost (0.6), added to the probability of it being injured second frost (0.6) minus the probability of it being injured by both touches of frost (0.6 × 0.6). That can be written as mathematically,

P = P(F₁) + P(F₂) - P(F₁ and F₂)

P = 0.6 + 0.6 - (0.6 × 0.6)

P 0.84

P = 84%

Thus, the probability of an apple tree being injured in this period is 0.84 or 84 %.

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Answer:

(a)2/3

(b)2/3

(c)9/13

Step-by-step explanation:

Total Number of Cards in new Deck=39

Hearts(Red)=13

Diamonds(Red)=13

Spades(Black)=13

(a)P(drawing a red card)

Total number of red cards = 13+13=26

P(drawing a red card)=26/39=2/3

(b)Drawing a heart or a red card

Number of Hearts=13

Number of red cards=26

Number of Red Hearts = 13

Since the two events are not mutually exclusive

P(Hearts or Red) = P(Hearts) + P(Red) - P( Hearts and Red)

P(H∪R)=P(H)+P(R)-P(H∩R)

=13/39 + 26/39 - 13/39

=26/39 =2/3

(c)Drawing a jack or a red card.

Number of Jacks=3

Number of red cards=26

Number of Red Jacks = 2

Since the two events are not mutually exclusive

P(Jack or Red) = P(Jacks) + P(Red) - P( Jacks and Red)

P(J∪R)=P(J)+P(R)-P(J∩R)

=3/39 + 26/39 - 2/39

=27/39 =9/13

Answer:

The formula to the sequence

5/3, -6/9, 7/27, -8/81, 9/243, ...

is

(-1)^n. (4 + n). 3^(-n)

For n = 1, 2, 3, ...

Step-by-step explanation:

The sequence is

5/3, - 6/9, 7/27, - 8/81, 9/243, ...

We notice the following

- That the numbers are alternating between - and +

- That the numerator of a number is one greater than the numerator of the preceding number. The first number being 5.

- That the denominator of a number is 3 raised to the power of (2 minus the position of the number)

Using these observations, we can write a formula for the sequence.

(-1)^n for n = 1, 2, 3, ... takes care of the alternation between + and -

(4 + n) for n = 1, 2, 3, ... takes care of the numerators 5, 6, 7, 8, ...

3^(-n) for n = 1, 2, 3, ... takes care of the denominators 3, 9, 27, 81, 243, ...

Combining these, we have the formula to be

(-1)^n. (4 + n). 3^(-n)

For n = 1, 2, 3, ...

The final formula is [tex]a_n = ((-1)^{ (n+1)} * (n + 4)) / (3^n).[/tex]

Finding the General Term of the Sequence

The sequence given is: 5/3, -6/9, 7/27, -8/81, 9/243. To find the formula for the general term (nth term) of this sequence, we need to carefully analyze the patterns in both the numerators and the denominators separately.

Combining the results from the numerator and denominator analysis, the general term of the sequence, an, is:

[tex]a_n = ((-1)^{ (n+1)} * (n + 4)) / (3^n).[/tex]

Complete Question:- Find a formula for the general term of the sequence 5 3 , − 6 9 , 7 27 , − 8 81 , 9 243 , assuming that the pattern of the first few terms continues

Answer:

Answer is explained in the attached document

Step-by-step explanation:

Hessenberg matrix- it a special type of square matrix,there there are two subtypes of hessenberg matrix that is upper Hessenberg matrix and lower Hessenberg matrix.

upper Hessenberg matrix:- in this type of matrix  zero entries below the first subdiagonal or in another words square matrix of n\times n is said to be in upper Hessenberg form  if ai,j=0

for all i,j with i>j+1.and upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero

lower Hessenberg matrix:-  in this type of matrix  zero entries upper the first subdiagonal,square matrix of n\times n is said to be in lower Hessenberg form  if ai,j=0  for all i,j with j>i+1.and lower Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero.

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

Answer: 771,243

Step-by-step explanation:

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',

Transformation 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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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

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.

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:

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.

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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Answer:

A) Estimated standard deviation = 148cm³

B) The estimated standard deviation of 148 cm³ is less than the true standard deviation of 183.2 cm³. This estimated standard deviation is not even within 15 cm³ of the true standard deviation and thus we can say it's not accurate.

Step-by-step explanation:

From the question, the given standard deviation is 183.2 cm³

Also that the range of values is between 902 cm³ to 1494 cm³.

Range is the difference between highest and lowest values in the data set.

Thus, Range = 1494 cm³ - 902 cm³ = 592 cm³

Now, The range rule tells us that the standard deviation of a sample is approximately equal to one-fourth of the range of the data. In other words s = (Maximum – Minimum)/4

Thus, estimated standard deviation = 592/4 = 148 cm³

Answer:

[tex]z=1.28<\frac{a-239}{4.6}[/tex]

And if we solve for a we got

[tex]a=239 +1.28*4.6=244.89[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 244.89.  

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".  

Solution to the problem:

Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(239,23)[/tex]  

Where [tex]\mu=239[/tex] and [tex]\sigma=23[/tex]

Since the distribution for X is normal then we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And we are interested on a value a such that:

[tex]P(\bar X>a)=0.10[/tex]   (a)

[tex]P(\bar X<a)=0.90[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.90 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1

If we use condition (b) from previous we have this:

[tex]P(\bar X<a)=P(\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{a-\mu}{\frac{\sigma}{\sqrt{n}}})=0.9[/tex]  

[tex]P(z<\frac{a-\mu}{\frac{\sigma}{\sqrt{n}}})=0.9[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=1.28<\frac{a-239}{4.6}[/tex]

And if we solve for a we got

[tex]a=239 +1.28*4.6=244.89[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 244.89.  

Answer:

[tex]\sum_{i=1}^n x_i =459[/tex]

[tex]\sum_{i=1}^n y_i =1227[/tex]

[tex]\sum_{i=1}^n x^2_i =24059[/tex]

[tex]\sum_{i=1}^n y^2_i =168843[/tex]

[tex]\sum_{i=1}^n x_i y_i =63544[/tex]

With these we can find the sums:

[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=24059-\frac{459^2}{9}=650[/tex]

[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=63544-\frac{459*1227}{9}=967[/tex]

And the slope would be:

[tex]m=\frac{967}{650}=1.488[/tex]

Nowe we can find the means for x and y like this:

[tex]\bar x= \frac{\sum x_i}{n}=\frac{459}{9}=51[/tex]

[tex]\bar y= \frac{\sum y_i}{n}=\frac{1227}{9}=136.33[/tex]

And we can find the intercept using this:

[tex]b=\bar y -m \bar x=136.33-(1.488*51)=60.442[/tex]

So the line would be given by:

[tex]y=1.488 x +60.442[/tex]

And then the best predicted value of y for x = 41 is:

[tex]y=1.488*41 +60.442 =121.45[/tex]

Step-by-step explanation:

For this case we assume the following dataset given:

x: 38,41,45,48,51,53,57,61,65

y: 116,120,123,131,142,145,148,150,152

For this case we need to calculate the slope with the following formula:

[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]

Where:

[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]

[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]

So we can find the sums like this:

[tex]\sum_{i=1}^n x_i =459[/tex]

[tex]\sum_{i=1}^n y_i =1227[/tex]

[tex]\sum_{i=1}^n x^2_i =24059[/tex]

[tex]\sum_{i=1}^n y^2_i =168843[/tex]

[tex]\sum_{i=1}^n x_i y_i =63544[/tex]

With these we can find the sums:

[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=24059-\frac{459^2}{9}=650[/tex]

[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=63544-\frac{459*1227}{9}=967[/tex]

And the slope would be:

[tex]m=\frac{967}{650}=1.488[/tex]

Nowe we can find the means for x and y like this:

[tex]\bar x= \frac{\sum x_i}{n}=\frac{459}{9}=51[/tex]

[tex]\bar y= \frac{\sum y_i}{n}=\frac{1227}{9}=136.33[/tex]

And we can find the intercept using this:

[tex]b=\bar y -m \bar x=136.33-(1.488*51)=60.442[/tex]

So the line would be given by:

[tex]y=1.488 x +60.442[/tex]

And then the best predicted value of y for x = 41 is:

[tex]y=1.488*41 +60.442 =121.45[/tex]

Using linear regression and the provided dataset, we can predict that an individual's systolic blood pressure (SBP) at the age of 41 is approximately 142.91 millimeters of mercury, rounded to two decimal places.

here are the steps to predict systolic blood pressure (SBP) at age 41 using linear regression with the provided dataset:

Calculate the Means:

Calculate the mean (average) of ages (x) and SBP (y) from the dataset.

Mean(x) = (43 + 53 + 42 + 48 + 52 + 39 + 40 + 47 + 51) / 9 ≈ 46.33 (rounded to two decimal places)

Mean(y) = (139 + 146 + 139 + 153 + 159 + 138 + 135 + 144 + 154) / 9 ≈ 146 (rounded to the nearest whole number)

Calculate the Slope (b):

Use the formula for the slope (b) of the regression line:

b = Σ[(x - Mean(x))(y - Mean(y))] / Σ[(x - Mean(x))^2]

Calculate b using the values from the dataset and the means calculated earlier.

Calculate the Intercept (a):

Use the formula for the intercept (a) of the regression line:

a = Mean(y) - b * Mean(x)

Calculate a using the previously calculated means and the value of b.

Formulate the Regression Equation:

The regression equation is now established as:

y = a + b * x

Substituting the values of a and b, we have:

y = 118.31 + 0.6 * x

Predict SBP at Age 41:

Substitute x = 41 into the regression equation:

y = 118.31 + 0.6 * 41

Calculate y:

y ≈ 118.31 + 24.6 ≈ 142.91

So, based on these steps, the best-predicted SBP for an individual aged 41 is approximately 142.91 millimeters of mercury, rounded to two decimal places.

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complete question should be:

Using linear regression and the provided dataset, how can we predict the systolic blood pressure (y) for an individual with an age (x) of 41, assuming a significant correlation between age and systolic blood pressure? The dataset includes the following information:

Ages (x): 43, 53, 42, 48, 52, 39, 40, 47, 51.

Systolic Blood Pressures (y): 139, 146, 139, 153, 159, 138, 135, 144, 154.

After calculating, the best-predicted systolic blood pressure for an age of 41 is approximately 154.08 millimeters of mercury, rounded to two decimal places.

Answer:

a) 0.5 per minutes

b) 5 arrivals expected in 10 minutes

c) P ( x = 0 ) = 0.00673 , P ( x = 1 ) =  0.03368 , P ( x = 2 ) = 0.08422 ,P ( x = 3 ) = 0.14037                  

d)  P ( X >= 4 ) = 0.735                

Step-by-step explanation:

Given:

- The number of customer arriving at window is modeled by Poisson distribution. The distribution is given by:

                         P(x) = ( λ^x ) (e^-λ) / x!            x = 0 , 1 , 2 , 3 , ......

- Average rate λ = 30 / hr

Find:

a. How many customers arrive per minute, on average?

b. How many customers would you expect to arrive in a 10-minute interval?c. Use equation 13.1 to determine the probability of exactly 0, 1, 2, and 3 arrivals in a 10-minute interval.

d. What is the probability of more than three arrivals occurring in a 10-minute interval?

Solution:

- The average rate λ in number of customers that arrive in a minute is given by:

                              λ1 = 30 / 60 = 0.5 arrival per minutes

- The average number of customer that are expected to arrive in 10-minutes window is:

                              λ2 = 10*λ1 = 10*0.5 = 5 arrivals expected in 10 minutes

- The probability of exactly 0,1 , 2 , and 3 arrivals in 10 minute windows:

                   P ( x = 0 ) = ( 5^0 ) (e^-5) / 0! = 0.00673  

                   P ( x = 1 ) = ( 5^1 ) (e^-5) / 1! = 0.03368

                   P ( x = 2 ) = ( 5^2 ) (e^-5) / 2! = 0.08422

                   P ( x = 3 ) = ( 5^3 ) (e^-5) / 3! = 0.14037

- The probability of more than three arrivals occuring in 10-minute interval is:

                  P ( X >= 4 ) = 1 - P ( X =< 3 )

                  P ( X >= 4 ) = 1 - [ P ( x = 0 ) + P ( x = 1 ) +  P ( x = 2 ) + P ( x = 3 ) ]

                  P ( X >= 4 ) = 1 - [ 0.00673 + 0.03368 + 0.08422 + 0.14037 ]

                  P ( X >= 4 ) = 1 - [ 0.265 ]

                  P ( X >= 4 ) = 0.735

Using the Poisson distribution, it is found that:

a) 0.5 customers per minute.

b) 5 customers are expected to arrive.

c)

0.0068 = 0.68% probability of exactly 0 arrivals in a 10-minute interval.

0.0337 = 3.37% probability of exactly 1 arrivals in a 10-minute interval.

0.0842 = 8.42% probability of exactly 2 arrivals in a 10-minute interval.

0.1404 = 14.04% probability of exactly 3 arrivals in a 10-minute interval.

d) 0.7349 = 73.49% probability of more than three arrivals occurring in a 10-minute interval.

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

The parameters are:

Item a:

30 in one-hour(60 minutes), hence 0.5 customers per minute.

Item b:

0.5 customers per minute, hence, in a 10 minute interval, 5 customers are expected to arrive.

Item c:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-5}(5)^{0}}{(0)!} = 0.0068[/tex]

[tex]P(X = 1) = \frac{e^{-5}(5)^{1}}{(1)!} = 0.0337[/tex]

[tex]P(X = 2) = \frac{e^{-5}(5)^{2}}{(2)!} = 0.0842[/tex]

[tex]P(X = 3) = \frac{e^{-5}(5)^{3}}{(3)!} = 0.1404[/tex]

0.0068 = 0.68% probability of exactly 0 arrivals in a 10-minute interval.

0.0337 = 3.37% probability of exactly 1 arrivals in a 10-minute interval.

0.0842 = 8.42% probability of exactly 2 arrivals in a 10-minute interval.

0.1404 = 14.04% probability of exactly 3 arrivals in a 10-minute interval.

Item d:

This probability is:

[tex]P(X > 3) = 1 - P(X \leq 3)[/tex]

In which:

[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]

From item c:

[tex]P(X \leq 3) = 0.0068 + 0.0337 + 0.0842 + 0.1404 = 0.2651[/tex]

Then:

[tex]P(X > 3) = 1 - P(X \leq 3) = 1 - 0.2651 = 0.7349[/tex]

0.7349 = 73.49% probability of more than three arrivals occurring in a 10-minute interval.

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Answer:

200,000

Step-by-step explanation:

add 128,000+42,000+30,000

The implicit cost of her business is $56000

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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Answer:

D. All of these

Step-by-step explanation:

Requirements determination is the process of transforming a system's request into more detailed business statement that is clear and precise. It is the beginning sub phase of analysis, so all the given options in the question can be included.

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.

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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initial volume of balloon =

[tex] \frac{4}{3} \times \frac{22}{7} \times r^{3} \\ \frac{4}{3} \times \frac{22}{7} \times {3}^{3} \\ 113.14[/tex]

so intial volume is 113.14 cubic inches

volume after one hour will be 113.14 - 2 = 111.14 inches

new radius

[tex]111.14 = \frac{4}{3} \times \frac{22}{7} \times {x}^{3} \\ \frac{2333.94}{88} = {x}^{3} \\ 26.52 = {x}^{3} \\ x = 2.98[/tex]

Rate change of radius is

[tex] \frac{2.98}{3} \times 100 \\ \frac{298}{3} \\ 99.93\%[/tex]

Rate change is 0.7% per hour the radius is decreasing

Answer:

35.03% probability that fewer than 7 will be carrying backpacks

Step-by-step explanation:

For each student, there are only two possible outcomes. Either they are carrying a backpack, or they are not. The probability of a student carrying a backpack is independent from other students. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

The probability that an Oxnard University student is carrying a backpack is .70.

This means that [tex]p = 0.7[/tex]

If 10 students are observed at random, what is the probability that fewer than 7 will be carrying backpacks

This is [tex]P(X < 7)[/tex] when [tex]n = 10[/tex]. So

[tex]P(X < 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)[/tex]

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{10,0}.(0.7)^{0}.(0.3)^{10} = 0.000006[/tex]

[tex]P(X = 1) = C_{10,1}.(0.7)^{1}.(0.3)^{9} = 0.0001[/tex]

[tex]P(X = 2) = C_{10,2}.(0.7)^{2}.(0.3)^{8} = 0.0014[/tex]

[tex]P(X = 3) = C_{10,3}.(0.7)^{3}.(0.3)^{7} = 0.0090[/tex]

[tex]P(X = 4) = C_{10,4}.(0.7)^{4}.(0.3)^{6} = 0.0368[/tex]

[tex]P(X = 5) = C_{10,5}.(0.7)^{5}.(0.3)^{5} = 0.1029[/tex]

[tex]P(X = 6) = C_{10,6}.(0.7)^{6}.(0.3)^{4} = 0.2001[/tex]

[tex]P(X < 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) = 0.000006 + 0.0001 + 0.0014 + 0.0090 + 0.0368 + 0.1029 + 0.2001 = 0.3503[/tex]

35.03% probability that fewer than 7 will be carrying backpacks

Final answer:

To find the probability that fewer than 7 students will be carrying backpacks, use the binomial probability formula. The final probability is 0.9143, or 91.43%.

Explanation:

To find the probability that fewer than 7 students will be carrying backpacks, we can use the binomial probability formula. In this case, the probability of success (carrying a backpack) is 0.70. The number of trials is 10. We want to find the probability of getting fewer than 7 successes.

We can calculate this by finding the sum of the probabilities of getting 0, 1, 2, 3, 4, 5, and 6 successes, using the binomial probability formula for each value. Then, we subtract this sum from 1 to get the probability of getting fewer than 7 successes.

The final probability for this scenario is 0.9143, or 91.43%.

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%.

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

Answer:

D. 3 and 116

Step-by-step explanation:

d.f.N = k - 1 (numerator degrees of freedom) = 4 - 1 = 3

N = 4 × 30 = 120

d.f.D = N - k (denominator degrees of freedom) = 120 - 4 =116

Final answer:

In ANOVA, the degrees of freedom for the numerator is the number of groups minus one, and for the denominator, it is the total number of observations minus the number of groups. Thus, the correct answer is 3 and 116 for the numerator and denominator degrees of freedom.

Explanation:

The ANOVA procedure is used to compare means across multiple populations to see if there's a significant difference. With four populations and samples of 30 observations each, we are working with an F distribution in the framework of an ANOVA analysis.

The degrees of freedom for the numerator in ANOVA is calculated as the number of groups minus one. Therefore, for four populations, it is 4 - 1 = 3. The degrees of freedom for the denominator is the total number of observations minus the number of groups. Thus, with four samples of 30, the total number of observations is 4 × 30 = 120, minus the number of groups gives us 120 - 4 = 116. So, the answer is 3 for the numerator and 116 for the denominator.

Hence, the correct choice is D. 3 and 116 for the numerator and denominator degrees of freedom, respectively.

Answer:

1.732

Step-by-step explanation:

You are given that claims are reported according to a homogeneous Poisson process

LetX be the waiting time from 0 to second claim

X is Poisson with averageof 3 hours.

We know in a Poisson distribution the mean = variance

Hence average waiting time = mean = 3

This will also be equal to var(x)

Var(x) = mean of Poisson distribution= 3

Hence standard deviation = square root of variance

=[tex]\sqrt{3} \\=1.732[/tex]

Answer:

The probability of rolling a brown on the next toss is 0.2303.

Step-by-step explanation:

The data recorded is:

Brown = 35

Purple = 50

Green = 44

Yellow = 23

TOTAL = 152 tosses

The probability of an event E is computed by dividing the favorable number of outcomes by the total number of outcomes.

[tex]P(E)=\frac{Favorable\ outcomes}{Total\ no.\ of\ outcomes}[/tex]

Using this formula compute the probability of rolling a brown on the next toss as follows:

[tex]P(Brown)=\frac{35}{152}= 0.2303[/tex]

Thus, the probability of rolling a brown on the next toss is 0.2303.

the correct choice is:

E. 2-proportion Z-test

To determine if there is evidence that breathing extra oxygen can help test-takers think more clearly, we should use the 2-proportion Z-test.

This test is appropriate because we are comparing two proportions (the proportion of students who improved their SAT scores among those who breathed oxygen and those who did not) from two independent groups (students who received oxygen and those who did not).

Therefore, the correct choice is:

E. 2-proportion Z-test

The probable question maybe:

At one SAT test site students taking the test for a second time volunteered to inhale supplemental oxygen for 10 minutes before the test. In fact, some received oxygen but others (randomly assigned) were given just normal air. Test results showed that 42 of 66 students who breathed oxygen improved their SAT scores, compared to only 35 of 63 students who did not get the oxygen Which procedure should we use to see if there is evidence that breathing extra oxygen can help test-takers think more clearly?

A. 1-proportion 2-test

B matched pairs t-test

C 2-sample t-test

D. 1-sample t-test

E. 2-proportion Z-test

A chi-square test for independence should be used to analyze the effect of breathing extra oxygen on SAT score improvements, by comparing observed frequencies of score improvements with expected frequencies under the null hypothesis.

To determine if there is evidence that breathing extra oxygen can help test-takers think more clearly, a statistical test of significance is appropriate. In this scenario, you would typically use a chi-square test for independence to see if there is a significant association between the treatment (oxygen vs. normal air) and the outcome (improvement in SAT scores). The chi-square test compares the observed frequencies of events (here, the number of students who improved) with the frequencies we would expect to see if there were no association between the treatment and the outcome.

The procedure involves calculating a chi-square statistic, which reflects how far the observed frequencies are from the expected frequencies assuming the null hypothesis is true (no effect of breathing extra oxygen). If the resulting p-value is less than the chosen significance level (commonly 0.05), we can reject the null hypothesis and conclude that there is evidence to suggest a relationship between breathing extra oxygen and improved SAT scores.

Answer:

a) 94.06% probability that a water specimen collected from the river has an alkalinity level exceeding 45 milligrams per liter.

b) 94.06% probability that a water specimen collected from the river has an alkalinity level below 55 milligrams per liter.

c) 50.98% probability that a water specimen collected from the river has an alkalinity level between 48 and 52 milligrams per liter.

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 = 50, \sigma = 3.2[/tex]

a. exceeding 45 milligrams per liter.

This probability is 1 subtracted by the pvalue of Z when X = 45. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{45 - 50}{3.2}[/tex]

[tex]Z = -1.56[/tex]

[tex]Z = -1.56[/tex] has a pvalue of 0.0594.

1 - 0.0594 = 0.9406

94.06% probability that a water specimen collected from the river has an alkalinity level exceeding 45 milligrams per liter.

b. below 55 milligrams per liter.

This probability is the pvalue of Z when X = 55.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{55 - 50}{3.2}[/tex]

[tex]Z = 1.56[/tex]

[tex]Z = 1.56[/tex] has a pvalue of 0.9604.

94.06% probability that a water specimen collected from the river has an alkalinity level below 55 milligrams per liter.

c. between 48 and 52 milligrams per liter.

This is the pvalue of Z when X = 52 subtracted by the pvalue of Z when X = 48. So

X = 52

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{52 - 50}{3.2}[/tex]

[tex]Z = 0.69[/tex]

[tex]Z = 0.69[/tex] has a pvalue of 0.7549

X = 48

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{48 - 50}{3.2}[/tex]

[tex]Z = -0.69[/tex]

[tex]Z = -0.69[/tex] has a pvalue of 0.2451

0.7549 - 0.2451 = 0.5098

50.98% probability that a water specimen collected from the river has an alkalinity level between 48 and 52 milligrams per liter.

The probabilities are calculated using the normal distribution properties: exceeding 45 mg/L is approximately 0.9406, below 55 mg/L is approximately 0.9406, and between 48 and 52 mg/L is approximately 0.4678.

a) To find the probability that the alkalinity level exceeds 45 milligrams per liter, we can use the standard normal distribution with the given mean (50 mg/L) and standard deviation (3.2 mg/L). First, we need to calculate the z-score for 45 mg/L:

z = 45 - 50/3.2 = -1.5625

Then, we find the probability of the alkalinity level exceeding 45 mg/L by finding the area to the right of this z-score in the standard normal distribution. Using a standard normal table or calculator, we find this probability to be approximately 0.9406.

b) Similarly, to find the probability that the alkalinity level is below 55 milligrams per liter, we calculate the z-score for 55 mg/L:

[tex]\[ z = \frac{55 - 50}{3.2} = 1.5625 \][/tex]

Then, we find the probability of the alkalinity level being below 55 mg/L by finding the area to the left of this z-score in the standard normal distribution. Using a standard normal table or calculator, we find this probability to be approximately 0.9406.

c) To find the probability that the alkalinity level is between 48 and 52 milligrams per liter, we first calculate the z-scores for these values:

z₁ = 48-50/3.2 = -0.625

z₂ = 52-50/3.2 = 0.625

Then, we find the area between these two z-scores in the standard normal distribution, which represents the probability of the alkalinity level being between 48 and 52 mg/L. Using a standard normal table or calculator, we find this probability to be approximately 0.3146.

In conclusion:

Answer:

Answer is attached

The power-reducing formulas are used twice to rewrite 19sin^4(x) without trigonometric powers greater than 1, resulting in 19(3/8 - (1/2)cos(2x) + (1/8)cos(4x)).

The problem requires using power-reducing formulas to rewrite the expression 19 sine to the power of 4 of x (19 sin4x) as an equivalent expression that does not contain powers of trigonometric functions greater than 1. The power-reducing formula for sin2a is sin2a = (1 - cos(2a)) / 2. We must apply this formula twice because we have sin4x.

First step:

Second step:

Therefore, the final expression without powers greater than 1 is 19 multiplied by (3/8 - (1/2)cos(2x) + (1/8)cos(4x)), or

19sin4x = 19(3/8 - (1/2)cos(2x) + (1/8)cos(4x))

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.

Answer:

Test statistics = 1.87

Step-by-step explanation:

We are given that a political study took a sample of 1000 voters in the town and found that 56% of the residents favored annexation.

And, a political strategist wants to test the claim that the percentage of residents who favor annexation is more than 53%, i.e;

Null Hypothesis, [tex]H_0[/tex] : p = 0.53 {means that the percentage of residents who favor annexation is 53%}

Alternate Hypothesis, [tex]H_1[/tex] : p > 0.53 {means that the percentage of residents who favor annexation is more than 53%}

The test statistics we will use here is;

                T.S. = [tex]\frac{\hat p -p}{\sqrt{\frac{\hat p(1- \hat p)}{n} } }[/tex] ~ N(0,1)

where, p = actual percentage of residents who favor annexation = 0.53

            [tex]\hat p[/tex] = percentage of residents who favor annexation in a sample of

                  1000 voters = 0.56

            n = sample of voters = 1000

So, Test statistics = [tex]\frac{0.56 -0.53}{\sqrt{\frac{0.56(1- 0.56)}{1000} } }[/tex]

                              = 1.87

Therefore, the value of test statistics is 1.87 .

Answer:

Step-by-step explanation:

13. The set M4,6 of all 4x6 matrices is a vector space as it is closed under vector addition as also scalar multiplication. Also the 4x6 zero matrix is in M4,6.

14. The set M1,1 is a singleton, i.e. a 1x1 matrix. It is a vector space for the same reasons as in 13. above.

15. The degree of the zero polynomial is undefined and it is usually treated as a constant (of degree 0). If we treat 0 as a polynomial of degree 0, then the set of all 3rd degree polynomials is not a vector space despite being closed under vector addition and scalar multiplication as it does not contain the zero polynomial.

16. The set of all 5th degree polynomials is not a vector space despite being closed under vector addition and scalar multiplication as it does not contain the zero polynomial.

17. The set of all first degree polynomials ax (a≠0) is not a vector space. If p(x) = ax and q(x) = -ax , then p(x)+q(x) = 0*x which is not in the given set. Hence the set is not closed under vector addition and, therefore, it is not a vector space.

18. The set of all first degree polynomials ax +b (a,b ≠0) is not a vector space. If p(x) = ax +b and q(x) = -ax -b , then p(x)+q(x) = 0*x +0 which is not in the given set. Hence the set is not closed under vector addition and, therefore, it is not a vector space.

19. The set P4 of all polynomials of degree 4 or less isa vector space. If p(x) = a1x4+a2 x3+a3x2+a4x+a5 and q(x) = b1x4+b2 x3+b3x2+b4x+b5 are 2 arbitrary elements of P4, then p(x)+q(x)is in P4. Similarly, αp(x) is in P4 for any arbitrat scalar α. Hence, P4 is closed under vector addition and scalar multiplication. Also, the 0 polynomial is in P4. Hence P4 is a vector space.

A set forms a vector space if it meets all ten vector space axioms. For instance, a set of first-degree polynomial functions ax + b, where a, and b are non-zero and whose graphs do not pass through the origin, fails to meet vector space axiom for closure under addition. On the other hand, the set of all third-degree polynomials is a vector space as it satisfies all ten vector space axioms.

To determine whether a set combined with standard operations forms a vector space, it must satisfy ten vector space axioms. These axioms include requirements regarding the addition and scalar multiplication of vectors.

To illustrate, let's look at the set of all first-degree polynomial functions ax + b, with a, b ≠ 0, whose graphs do not pass through the origin. By definition, vectors in a vector space should be closed under addition, i.e., if you add any two vectors together, you should get another vector in the set. However, if we add two such functions together ax + b + cx + d = (a+c)x + (b+d), the resulting function will pass through the origin only if (b+d)=0. But given that b and d are non-zero, the sum will not yield a first-degree polynomial that passes through the origin. Therefore, this set fails to meet the vector space axiom for closure under addition.

On the other hand, the set of all third-degree polynomials will be a vector space as it adheres to all the ten axioms. For example, if two polynomials in the set are added together or multiplied by a scalar, the resultant is still a third-degree polynomial which means it still belongs to the set.

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