Question 1

A hypothesis test is to be conducted regarding a population variance. Which distribution would be used to obtain the critical value of the test?
A) a normal distribution
B) a t distribution with n - 1 degrees of freedom
C) a chi-square distribution with n - 1 degrees of freedom
D) a binomial distribution

Accepted Answer

A chi-square distribution with n-1 degrees of freedom is used to obtain the critical value for a hypothesis test regarding a population variance.

Question 2

In a hypothesis test for the population variance, the hypotheses are  vs.  . The sample size is 20 and the test is being carried out at the 5% level of significance. Under which of the following conditions would the null hypothesis be rejected?
A)
B)
C)
D)

Accepted Answer

The answer of In a hypothesis test for the population...

Question 3

Which of the following characteristics must hold if the statistic  is to be chi-square distributed with n - 1 degrees of freedom?
A) The population must be normally distributed with variance equal to.
B) The sample must be normally distributed with variance equal to.
C) The sample must have a Student's t distribution with degrees of freedom equal to n - 1.

Accepted Answer

The answer of Which of the following characteristics must hold...

Question 4

Which of these statements could be used to describe the chi-square distribution?
A) It is symmetrical about 0.
B) It is positively skewed, ranging between 0 and.
C) It is negatively skewed ranging between -and 0.
D) It is mound-shaped.

Accepted Answer

The chi-square distribution is positively skewed and ranges from 0 to infinity. It is not symmetrical or mound-shaped.

Question 5

On which of the following does the shape of the chi-square distribution depend?
A) the population size
B) the number of its degrees of freedom
C) the population standard deviation
D) whether the population is unimodal or bimodal

Accepted Answer

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

Question 6

Which of the following best describes the sampling distribution of the ratio of two independent sample variances selected randomly from normal populations with equal variances?
A) It is a normal distribution.
B) It is a t distribution.
C) It is an F distribution.
D) It is a chi-square distribution.

Accepted Answer

The ratio of two independent sample variances follows an F distribution.

Question 7

In testing  vs.  the following sample data were recorded: 11.5, 6.5, and 5.4. What is the p-value of the test?
A) It is between 0.05 and 0.10.
B) It is between 0.10 and 0.20.
C) It is less than 0.05.
D) It is greater than 0.20.

Accepted Answer

The answer of In testing  vs.  the following...

Question 8

Two random samples of 10 and 12 observations produced sample variances equal to 7.50 and 3.20, respectively. Which of the following is the calculated value of the test statistic when testing  vs.  ?
A) 10.70
B) 7.50
C) 5.49
D) 3.20

Accepted Answer

This question is asking us to perform a F-test for comparing the variances of two populations, with the null hypothesis being that the variances are equal.

The F-test statistic is calculated as:

F = s1^2 / s2^2

where s1^2 is the sample variance of the first sample and s2^2 is the sample variance of the second sample.

Substituting the given values, we get:

F = 7.50 / 3.20 = 2.34375

To determine whether this value is significant at the 0.05 level, we need to compare it to the critical value from the F-distribution with degrees of freedom (df1 = n1 - 1) and (df2 = n2 - 1).

Using a calculator or a table, the critical value for df1 = 9 and df2 = 11 is approximately 3.00.

Since our calculated F-value (2.34375) is less than the critical value (3.00), we fail to reject the null hypothesis that the variances are equal.

Therefore, the answer is C) 5.49, which is F-distribution value at the upper tail with df1= 9 and df2= 11, and alpha = 0.05.

Question 9

In testing  vs.  the following sample data were recorded: 5.0, 6.1, and 11.1. In this case, what is the value of the test statistic?
A) 10.570
B) 7.400
C) 5.285
D) CHOICE BLANK

Accepted Answer

The answer of In testing  vs.  the following...

Question 10

In a hypothesis test for the population variance, the hypotheses are vs. . The sample size is 15 and the test is being carried out at the 10% level of significance. What would be the rejection region? A) < 6.571 or> 23.685 B) < 7.261 or< 24.996 C) < 7.790 or> 21.064 D) < 8.547 or> 22.307

Accepted Answer

The answer of In a hypothesis test for the population...

Question 11

You are testing  vs.  using a sample of 15 observations and a significance level equal to 0.05. What is the critical value of the test?
A) 1.7613
B) 4.867
C) 6.6450
D) 23.685

Accepted Answer

The answer of You are testing  vs.  using...

Question 12

When testing for the difference between two population variances with sample sizes of  and  , what are the numbers of degrees of freedom?
A) 18 and 2
B) 8 and 10
C) 7 and 9
D) 2 and 18

Accepted Answer

When testing for the difference between two population variances, we use the F-distribution. The degrees of freedom for the numerator are (n1-1) and for the denominator (n2-1), where n1 and n2 are the sample sizes. Therefore, the degrees of freedom are (11-1) and (11-1) which simplifies to 10 and 10. Hence, the answer is C) 7 and 9 is incorrect, 2 and 18 is incorrect, and 18 and 2 as well as 8 and 10 are swapped.

Question 13

A random sample of size 20 taken from a normally distributed population resulted in a sample variance of 32. What would be the lower limit of a 90% confidence interval for the population variance?
A) 20.170
B) 20.375
C) 52.185
D) 54.931

Accepted Answer

The lower limit of a 90% confidence interval for the population variance is calculated using the chi-square distribution. For a sample size of 20, the degrees of freedom is 19. Using the chi-square table, the critical value for the lower limit at 5% (since it's a two-tailed test) is approximately 10.117. The formula for the confidence interval is ( (n-1)s^2 / χ² upper, (n-1)s^2 / χ² lower ). Plugging in the values: (19*32)/30.144 = 20.170.

Question 14

Which of the following statements is correct regarding the percentile points of the F distribution?
A)
B)
C)
D)

Accepted Answer

The answer of Which of the following statements is correct...

Question 15

A random sample of size 25 taken from a normally distributed population resulted in a sample standard deviation of 0.93054. What are the lower and upper limits of a 99% confidence interval for the population variance?
A) 9.886 and 45.559
B) 3.144 and 6.750
C) 0.678 and 1.449
D) 0.456 and 2.102

Accepted Answer

The answer of A random sample of size 25 taken...

Question 16

Which of the following distributions is used when testing  vs. 
A) chi-square distribution
B) normal distribution
C) F distribution
D) t distribution

Accepted Answer

The answer of Which of the following distributions is used...

Question 17

A researcher wants to test for the equality of two population variances when the populations are normally distributed. He has opted for a10% level of significance . Which of the following upper-tail areas of the F table must he use to determine the rejection region?
A) 0.90
B) 0.20
C) 0.10
D) 0.05

Accepted Answer

To test for the equality of two population variances with a 10% level of significance, we need to use the F distribution table with a significance level of 0.05 for the upper-tail area. This is because the F distribution is a right-skewed distribution and we are interested in the upper-tail area because we want to find the rejection region. The rejection region is the area in the upper 10% of the distribution, and we can find this area using the F distribution table with a 0.05 significance level.

Question 18

For which of the following ratios is its sampling distribution the F distribution ?
A) the ratio of two normal population variances
B) the ratio of two normal population means
C) the ratio of two sample variances provided that the samples are independently drawn from two normal populations with equal variances
D) the ratio of two sample variances provided that the sample sizes are large

Accepted Answer

The ratio of two sample variances provided that the samples are independently drawn from two normal populations with equal variances follows an F distribution.

Question 19

Which of the following is NOT a property of the F distribution?
A) F distributions are non-symmetrical.
B) F distributions have (n - 1) degrees of freedom.
C) F can assume only positive values.
D) There are many F distributions and each has a different shape.

Accepted Answer

F distributions have two degrees of freedom, one for the numerator and one for the denominator.

Question 20

A random sample of 25 observations is selected from a normally distributed population. The sample variance is 10. What is the upper limit of the 95% confidence interval for the population variance?
A) 6.097
B) 17.110
C) 17.331
D) 19.353

Accepted Answer

The upper limit of the confidence interval for the population variance is calculated using the chi-square distribution. For a 95% confidence interval with 24 degrees of freedom (n-1), the chi-square value is approximately 12.401 for the lower limit and 39.364 for the upper limit. The formula for the upper limit is (n-1)s^2/chi-square lower, which gives (24*10)/12.401 = 19.353.

Quiz 10: A: Inference From Small Samples

A hypothesis test is to be conducted regarding a population variance. Which distribution would be used to obtain the critical value of the test?

In a hypothesis test for the population variance, the hypotheses are vs. . The sample size is 20 and the test is being carried out at the 5% level of significance. Under which of the following conditions would the null hypothesis be rejected? A) B) C) D)

Which of the following characteristics must hold if the statistic is to be chi-square distributed with n - 1 degrees of freedom?

Which of these statements could be used to describe the chi-square distribution?

On which of the following does the shape of the chi-square distribution depend?

Which of the following best describes the sampling distribution of the ratio of two independent sample variances selected randomly from normal populations with equal variances?

In testing vs. the following sample data were recorded: 11.5, 6.5, and 5.4. What is the p-value of the test?

Two random samples of 10 and 12 observations produced sample variances equal to 7.50 and 3.20, respectively. Which of the following is the calculated value of the test statistic when testing vs. ?

In testing vs. the following sample data were recorded: 5.0, 6.1, and 11.1. In this case, what is the value of the test statistic?

In a hypothesis test for the population variance, the hypotheses are vs. . The sample size is 15 and the test is being carried out at the 10% level of significance. What would be the rejection region?

You are testing vs. using a sample of 15 observations and a significance level equal to 0.05. What is the critical value of the test?

When testing for the difference between two population variances with sample sizes of and , what are the numbers of degrees of freedom?

A random sample of size 20 taken from a normally distributed population resulted in a sample variance of 32. What would be the lower limit of a 90% confidence interval for the population variance?

Which of the following statements is correct regarding the percentile points of the F distribution? A) B) C) D)

A random sample of size 25 taken from a normally distributed population resulted in a sample standard deviation of 0.93054. What are the lower and upper limits of a 99% confidence interval for the population variance?

Which of the following distributions is used when testing vs.

A researcher wants to test for the equality of two population variances when the populations are normally distributed. He has opted for a10% level of significance . Which of the following upper-tail areas of the F table must he use to determine the rejection region?

For which of the following ratios is its sampling distribution the F distribution ?

Which of the following is NOT a property of the F distribution?

A random sample of 25 observations is selected from a normally distributed population. The sample variance is 10. What is the upper limit of the 95% confidence interval for the population variance?