Deck 11: Statistical Inferences for Population Variances

Question 1

The exact shape of the chi-square distribution depends on the degrees of freedom.

Answer 1

True

Question 2

In order to make statistical inferences about σ² that are valid using a chi-square distribution,the assumption is that the sampled population is also a chi-square distribution.

Answer 2

False

Question 3

In testing the equality of population variances,two assumptions are required: independent samples and normally distributed populations.

Answer 3

True

Question 4

When comparing the variances of two normally distributed populations using independent random samples,the correct test statistic to use is __________.

A)z
B)t
C)F
D)Chi-square
E)None of these

Question 5

When comparing the variances of two normally distributed populations using independent random samples,if $s _ { 1 } ^ { 2 } = s _ { 2 } ^ { 2 }$
the calculated value of F will always be equal to one.

Question 6

Parameters of the F distribution include:

A)n₁
B)Degrees of freedom
C)n₂
D)n₁ and n₂
E)None of these

Question 7

To compute a 95 percent confidence interval for σ²,we use n - 1 degrees of freedom and the chi-square points on the distribution curve of χ_α/2² and of χ_1-(α/2)².

Question 8

The value of F_α in a particular situation depends on the size of the right-hand tail area and on the numerator degrees of freedom.

Question 9

When evaluating a new process,using the square root of the upper end of the confidence interval for σ² gives an estimate of the smallest that σ for the new process might reasonably be.

Question 10

The chi-square distribution is a continuous probability distribution that is skewed to the left.

Question 11

When computing a 95 percent confidence interval for σ² with a sample of n = 30,we would use the following values of χ² in the calculations:

A)45.7222 and 16.0471
B)46.9792 and 16.7908
C)42.5569 and 17.7083
D)43.7729 and 18.4926

Question 12

In general,the shape of the F distribution is _________.

A)skewed right
B)skewed left
C)normal
D)binomial

Question 13

The exact shape of the curve of the F distribution depends on two parameters,df₁ and df₂.

Question 14

The F statistic can assume either a positive or a negative value.

Question 15

When comparing two independent population variances,the correct test statistic to use is __________.

A)z
B)t
C)F
D)t²

Question 16

In testing the difference between two population variances,it is a common practice to compute the F statistic so that its value is always greater than or equal to one.

Question 17

Using a χ² test statistic to test the null hypothesis that the variance of a new process is equal to the variance of the current process and rejecting at p-value less than α,we can conclude that the new process is more consistent than the current process.

Question 18

In testing the equality of population variance,what assumption(s)should be considered?

A)Independent samples
B)Equal sample sizes
C)Normal distribution of the populations
D)Independent samples and Equal sample sizes
E)Independent samples and Normal distribution of the populations

Question 19

The value of χ²_α in a particular situation depends on:

A)The left-hand tail area α.
B)The number of degrees of freedom.
C)The right-hand tail area α.
D)The left-hand tail area α.and The number of degrees of freedom.
E)The number of degrees of freedom.and The right-hand tail area α.

Question 20

χ²_α is the point on the vertical axis under the curve of the chi-square distribution that gives a righthand tail area equal to α.

Question 21

In a sample of n = 25 selected from a normally distributed population,we find a population variance of s² = 150.What is the value of χ² if we are testing H₀: σ² = 100?

Question 22

In a sample of n = 16 selected from a normally distributed population,we find a population standard deviation of s = 10.What are the degrees of freedom for the hypothesis test?

Question 23

Given the following information about a hypothesis test of the difference between two variances based on independent random samples,what is the critical value of the test statistic at a significance level of .05? Assume that the samples are obtained from normally distributed populations. $H _ { A } : \sigma ^ { 2 } A > \sigma ^ { 2 } B , \bar { X } _ { 1 } = 12 , \bar { X } _ { 2 } = 9 , s _ { 1 } = 5 , s _ { 2 } = 3 , n _ { 1 } = 13 , n _ { 2 } = 10$

A)3.87
B)3.44
C)3.07
D)2.8
E)2.38

Question 24

Two independent samples selected from two normally distributed populations have variances of σ₁² and σ₂² with n₁ = 10 and n₂ = 15.The degrees of freedom for the F distribution when testing the equality of the two population variances are:

A)10,15
B)11,16
C)9,14
D)8,13

Question 25

A manufacturer of an automobile part has a process that is designed to produce the part with a target of 2.5 inches in length.In the past,the standard deviation of the length has been 0.035 inches.In an effort to reduce the variation in the process,the manufacturer has redesigned the process.A sample of 25 parts produced under the new process shows a sample standard deviation of 0.025 inches.Test the claim that the new process standard deviation has improved from the current process at α = .05.

Question 26

Testing H₀: σ₁² = σ₁²,H_A: σ₁² > σ₂² at α = .01,where n₁ = 5,n₂ = 6,s₁² = 15,750,and s₂² = 10,920,can we reject the null hypothesis?

Question 27

A manufacturer of an automobile part has a process that is designed to produce the part with a target of 2.5 inches in length.In the past,the standard deviation of the length has been 0.035 inches.In an effort to reduce the variation in the process,the manufacturer has redesigned the process.A sample of 25 parts produced under the new process shows a sample standard deviation of 0.025 inches.Calculate the test statistic for testing whether the new process standard deviation has improved from the current process.

Question 28

In a sample of n = 16 selected from a normally distributed population,we find a population standard deviation of s = 10.What is the value of χ² if we are testing H₀: σ² = 144?

Question 29

A baker must monitor the temperature at which cookies are baked.Too much variation will cause inconsistency in the texture of the cookies.Past records show that the variance of the temperatures has been 1.44°.A random sample of 30 batches of cookies is selected,and the sample variance of the temperature is 4.41°.What is the 95 percent confidence interval for σ² at α = .05?

Question 30

Consider an engine parts supplier,and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are,respectively,2.5 inches and .00075.To reduce this variance,a new machine is designed.A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches,a variance of s² = .0002 (normal distribution).In order for a cylindrical engine part to give an engine long life,the outside diameter must be between 2.43 and 2.57 inches.Using the upper end of the 95 percent confidence interval for σ and assuming that μ = 2.5,determine whether 99.73 percent of the outside diameters produced by the new machine are within the specification limits.

Question 31

What is the value of the computed F statistic for testing equality of population variances where s₁² = .004 and s₂² = .002? Consider H_A: σ₁² > σ₂²

Question 32

When testing H₀: σ₁² = σ₁²,H_A: σ₁² > σ₂² at α = .01,where n₁ = 5,n₂ = 6,s₁² = 15,750,and s₂² = 10,920,what critical value do we use?

Question 33

When testing H₀: σ₁² ≤ σ₂² and H_A: σ₁² > σ₂²,where s₁² = .004,s₂² = .002,n₁ = 4,and n₂ = 7 at α = .05,what is the decision on H₀?

Question 34

When testing H₀: σ₁² ≤ σ₂² and H_A: σ₁² > σ₂²,where s₁² = .004,s₂² = .002,n₁ = 4,and n₂ = 7 at α = .05,what critical value do we use?

Question 35

Consider an engine parts supplier,and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are,respectively,2.5 inches and .00075.To reduce this variance,a new machine is designed.A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches and a variance of s² = .0002 (normal distribution).In order for a cylindrical engine part to give an engine long life,the outside diameter must be between 2.43 and 2.57 inches.Assuming normality,determine whether 99.73 percent of the outside diameters produced by the current machine are within specification limits.

Question 36

A baker must monitor the temperature at which cookies are baked.Too much variation will cause inconsistency in the texture of the cookies.Past records show that the variance of the temperatures has been 1.44°.A random sample of 30 batches of cookies is selected,and the sample variance of the temperature is 4.41°.What is the test statistic for testing the null hypothesis that the population variance has increased above 1.44°?

Question 37

Consider an engine parts supplier,and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are,respectively,2.5 inches and .00075.To reduce this variance,a new machine is designed.A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches,a variance of s² = .0002 (normal distribution).In order for a cylindrical engine part to give an engine long life,the outside diameter must be between 2.43 and 2.57 inches.Find the 95 percent confidence intervals for σ² and σ.

Question 38

Consider an engine parts supplier,and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are,respectively,2.5 inches and .00075.To reduce this variance,a new machine is designed.A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches and a variance of s² = .0002 (normal distribution).In order for a cylindrical engine part to give an engine long life,the outside diameter must be between 2.43 and 2.57 inches.If σ² denotes the variance of the population of all outside diameters that would be produced by the new machine,test H₀: σ² = .00075 versus H_a: σ² < .00075 by setting α = .05.

Question 39

What is the value of the F statistic for H₀: σ₁² ≤ σ₁²,H_A: σ₁² > σ₁²,where s₁ = 3.3 and s₂ = 2.1?

Question 40

A baker must monitor the temperature at which cookies are baked.Too much variation will cause inconsistency in the texture of the cookies.Past records show that the variance of the temperatures has been 1.44°.A random sample of 30 batches of cookies is selected,and the sample variance of the temperature is 4.41°.Test the hypothesis that the temperature variance has increased above 1.44° at α = .05.

Question 41

What is the F statistic for testing H₀: σ₁² ≤ σ₂²,H_A: σ₂² > σ₁² at α = .05,where n₁ = 16,n₂ = 19,s₁² = .03,and s₂² = .02?

Question 42

Testing H₀: σ₁² ≤ σ₂²,H_A: σ₁² > σ₂² at α = .05,where n₁ = 16,n₂ = 19,s₁² = .03,and s₂² = .02,can we reject the null hypothesis?

Question 43

We use the following data for a test of the equality of variances for two populations at α = .10.Sample 1 is randomly selected from population 1 and sample 2 is randomly selected from population 2.Can we reject H₀ at α = .10?