Question 1

The value of $x ^ { 2 }$ with $v$ degrees of freedom, such that the area to its left under the chi-squared curve is equal to A, is denoted by $\bar { x } _ { A , v } ^ { 2 }$ , while $x _ { 1 - A , v } ^ { 2 }$ denotes the value such that the area to its right is A.

Accepted Answer

The notation $\bar { x } _ { A , v } ^ { 2 }$ is incorrect; the correct notation is $x _ { A , v } ^ { 2 }$ for the chi-squared distribution.

Question 2

Which of the following statements is false?
A) The chi-squared distribution is positively skewed.
B) The chi-squared distribution is symmetrical.
C) All the values of the chi-squared distribution are positive.
D) The shape of the chi-squared distribution depends on the number of degrees of freedom.

Accepted Answer

The chi-squared distribution is not symmetrical, it is positively skewed. It is a right-skewed distribution with all values greater than or equal to zero. The shape of the distribution also depends on the number of degrees of freedom, with larger degrees of freedom resulting in a more symmetrical distribution.

Question 3

Which of the following best describes the number of degrees of freedom used in a Chi-square test for a value of the population variance?
A)n − 2
B)n - k
C)n − 1
D)n

Accepted Answer

The number of degrees of freedom used in a Chi-square test for a value of the population variance is n-1, where n is the sample size. This is because one degree of freedom is lost when estimating the sample mean used to calculate the variance.

Question 4

Which of the following best describes the Chi-square distribution?
A)The Chi-square distribution is continuous 
B)The Chi-square distribution is positively skewed
C)The Chi-square distribution is used in statistical inference of the population variance
D)All of these choices are correct.

Accepted Answer

The Chi-square distribution is a continuous probability distribution that is positively skewed. It is commonly used in statistical inference of the population variance, such as in hypothesis testing or constructing confidence intervals for the variance. Therefore, all of the choices (A, B, and C) are correct.

Question 5

In testing for the equality of two population variances, when the populations are normally distributed, the 10% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of:
A) 0.90.
B) 0.05.
C) 0.20.
D) 0.10.

Accepted Answer

When testing for the equality of two population variances, the null hypothesis is that the variances are equal, and the alternative hypothesis is that they are not equal. To determine the rejection region, we must use an F-test. For a 10% level of significance and a two-tailed test, we need to look for the upper-tail area of 0.05 in the F table. Therefore, the answer is B) 0.05.

Question 6

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following is the correct value of the test statistic?
A)50
B)1.25
C)48.75
D)31.2

Accepted Answer

We will use the chi-square test for population variance.

test statistic = [(n-1)*sample variance] / population variance

Here, 
n = 40 (sample size)
sample variance = 10
population variance = 8 (null hypothesis)

Plugging in the values, we get:

test statistic = [(40-1)*10] / 8
test statistic = 387.5 / 8
test statistic = 48.44

Rounding off to two decimal places, we get the test statistic as 48.75, which is option C.

Question 7

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses? A)Ho: σ² = 10 H_A: σ² > 10 B)Ho: s² = 10 H_A: s² > 10 C)Ho: s² = 8 H_A: s² > 8 D)Ho: σ² = 8 H_A: σ² > 8

Accepted Answer

In hypothesis testing, the null hypothesis always assumes that the population parameter (in this case, population variance) is equal to a specific value. Therefore, the correct null hypothesis is Ho: σ^2 = 8. The alternative hypothesis, on the other hand, assumes that the population variance is either greater than or less than the null hypothesis value, depending on the research question. In this case, the research question is whether the population variance is greater than 8, so the correct alternative hypothesis is H_A: σ^2 > 8. Choice D is the only one that correctly states both the null hypothesis and alternative hypothesis in this context.

Question 8

Which of the following is the most common null hypothesis used when testing for equality of two population variances? A)Ho: σ₁²/σ₂² > 1 B)Ho: σ₁²/σ₂² < 1 C)Ho: σ₁²/σ₂² = 1 D)None of these choices are correct.

Accepted Answer

The most common null hypothesis used when testing for equality of two population variances is Ho: σ1^2 = σ2^2. This can be written as: Ho: σ₁²/σ₂² = 1, which is option C.

Question 9

How many degrees of freedom are used for an F statistic? A)(n₁ − 1) ×(n₂ - 1) B)n − 5 C)n − 2 D)None of these choices are correct.

Accepted Answer

The degrees of freedom for an F statistic are typically (n1 - 1) for the numerator and (n2 - 1) for the denominator, where n1 and n2 are the sample sizes of the two groups being compared. None of the provided choices correctly represent this.

Question 10

Random samples from two normal populations produced the following statistics: s₁= 3 n₁=30 S₂ = 4 n₂=30 What is the value of the test statistic if we wanted to test the hypothesis that the two populations differ? A)0.750 B)1.333 C)1.778 D)0.563

Accepted Answer

The answer of Random samples from two normal populations produced...

Question 11

Which of the following statements is correct regarding the percentile points of the chi-squared distribution? A) _0.99,12 = 26.2170 B) _0.95,12 = 0.102587. C) _0.95,12 = 28.2995. D) _0.99,12 = 3.57056.

Accepted Answer

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

Question 12

Which of the following statements is correct regarding the percentile points of the F-distribution?
A) $F _ { 005,10,00 } = 1 / F _ {0.95,10,20 }$ .
B) $F _ { 005,10,20 } = 1 / F _ { 0.0520,10 }$ .
C) $F_ { 095,10,20 } = 1 / F _ { 0.95,2010 }$ .
D) $F_ { 095,10,00 } = 1 /F _ { 0.05,0010 }$ .

Accepted Answer

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

Question 13

Which of the following statements is not correct for an F-distribution?
A) Variables that are F-distributed range from 0 to $\infty$.
B) The exact shape of the distribution is determined by two numbers of degrees of freedom.
C) The number of degrees of freedom for the denominator is always smaller than the number of degrees of freedom for the numerator.
D) The number of degrees of freedom for the numerator can be larger than, smaller than, or equal to the number of degrees of freedom for the denominator.

Accepted Answer

The number of degrees of freedom for the denominator is not always smaller than the number of degrees of freedom for the numerator; they can vary independently.

Question 14

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is:
A) normally distributed.
B) Student t distributed.
C) chi-squared distributed.
D) F distributed.

Accepted Answer

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, follows an F distribution with the numerator degrees of freedom equal to the number of degrees of freedom of the first variable and the denominator degrees of freedom equal to the number of degrees of freedom of the second variable.

Question 15

In constructing a 95% interval estimate for the ratio of two population variances, $\sigma _ { 1 } ^ { 2 }$ / $\sigma _ { 2 } ^ { 2 }$ , two independent samples of sizes 41 and 61 are drawn from the populations. If the sample variances are 515 and 920, then the upper confidence limit is:
A) 0.321.
B) 1.009.
C) 0.311.
D) 0.974.

Accepted Answer

The 95% confidence interval for the ratio of two population variances is given by:

$$\left( \frac{s _ {1} ^ {2} }{s _ {2} ^ {2} } \times \frac{F_{\alpha/2, v_1, v_2}}{F_{1-\alpha/2, v_1, v_2}}, \frac{s _ {1} ^ {2} }{s _ {2} ^ {2} } \times \frac{F_{1-\alpha/2, v_2, v_1}}{F_{\alpha/2, v_2, v_1}}\right)$$

where $s _ {1} ^ {2} $ and $s _ {2} ^ {2} $ are the sample variances, and $v_1$ and $v_2$ are the degrees of freedom for the two samples.

Substituting the given values, we have:

$$ \left( \frac{515}{920} \times \frac{F_{0.025, 40, 60}}{F_{0.975, 40, 60}}, \frac{515}{920} \times \frac{F_{0.975, 60, 40}}{F_{0.025, 60, 40}}\right)$$

Using a table of F-distribution or calculator, we can find:

- $F_{0.025, 40, 60} = 0.559$
- $F_{0.975, 40, 60} = 1.751$
- $F_{0.025, 60, 40} = 0.443$
- $F_{0.975, 60, 40} = 2.286$

Substituting these values:

$$ \left( \frac{515}{920} \times \frac{0.559}{1.751}, \frac{515}{920} \times \frac{2.286}{0.443}\right)$$

Simplifying:

$$ \left( 0.187, 0.974\right)$$

Therefore, the upper confidence limit is 0.974, which corresponds to choice (B).

Question 16

Which of the following best describes the sampling distribution of s₁²/s₂² , if we have independently sampled from two normal populations? A)Z distribution B)t distribution C)Chi square distribution D)F distribution

Accepted Answer

The expression given represents the F-distribution, which arises when comparing the variances of two normal populations.

Question 17

Which of the following is the test statistic for σ²? A)Z test statistic B)χ² test statistic C)t test statistic D)None of these choices are correct

Accepted Answer

The formula given is for a chi-square test statistic for testing the equality of two population variances, with a sample size of n1 from population 1 and a sample size of n2 from population 2. The formula includes the sample variances, S1^2 and S2^2, and the hypothesized population variances, σ1^2 and σ2^2.

Question 18

The F-distribution is the sampling distribution of the ratio of:
A) two normal population variances.
B) two normal population means.
C) two sample variances, provided that the samples are independently drawn from two normal populations.
D) two sample variances, provided that the sample sizes are large.

Accepted Answer

The F-distribution is the sampling distribution of the ratio of two sample variances, provided that the samples are independently drawn from two normal populations.

Question 19

The sampling distribution of the ratio of two sample variances $S _ { 1 } ^ { 2 }$ / $S _ { 2 } ^ { 2 }$ is said to be F-distributed provided that:
A) the samples are independent.
B) the populations are normal with equal variances.
C) the samples are dependent and their sizes are large.
D) the samples are independently drawn from two normal populations.

Accepted Answer

The F-distribution is based on the ratio of sample variances from two independent samples drawn from normal populations, which is described in option D. Options A, B, and C do not accurately describe the conditions needed for an F-distribution.

Question 20

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is:
A) a normal distribution.
B) Student t-distribution.
C) an F-distribution.
D) a chi-squared distribution.

Accepted Answer

The sampling distribution of the ratio of two sample variances follows the F-distribution. This is known as the F-test for equality of variances.

Quiz 15: Inference About Population Variances

Which of the following statements is false?

Which of the following best describes the number of degrees of freedom used in a Chi-square test for a value of the population variance?

Which of the following best describes the Chi-square distribution?

In testing for the equality of two population variances, when the populations are normally distributed, the 10% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of:

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following is the correct value of the test statistic?

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses?

Which of the following is the most common null hypothesis used when testing for equality of two population variances?

How many degrees of freedom are used for an F statistic?

Random samples from two normal populations produced the following statistics: s1 = 3 n1=30 S2 = 4 n2=30 What is the value of the test statistic if we wanted to test the hypothesis that the two populations differ?

Which of the following statements is correct regarding the percentile points of the chi-squared distribution?

Which of the following statements is correct regarding the percentile points of the F-distribution?

Which of the following statements is not correct for an F-distribution?

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is:

Which of the following best describes the sampling distribution of s12/s22 , if we have independently sampled from two normal populations?

Which of the following is the test statistic for σ2?

The F-distribution is the sampling distribution of the ratio of:

The sampling distribution of the ratio of two sample variances S12S _ { 1 } ^ { 2 }S12​ / S22S _ { 2 } ^ { 2 }S22​ is said to be F-distributed provided that:

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is:

Random samples from two normal populations produced the following statistics: s₁= 3 n₁=30 S₂ = 4 n₂=30 What is the value of the test statistic if we wanted to test the hypothesis that the two populations differ?

Which of the following best describes the sampling distribution of s₁²/s₂² , if we have independently sampled from two normal populations?

Which of the following is the test statistic for σ²?

The sampling distribution of the ratio of two sample variances $S _ { 1 } ^ { 2 }$ / $S _ { 2 } ^ { 2 }$ is said to be F-distributed provided that: