Question 1

If independent samples of size n₁ and n₂ are drawn from normal populations with equal variances,then the value of the statistic is calculated as A) . B) . C) . D)

Accepted Answer

The answer of If independent samples of size n₁ and...

Question 2

Which of the following is the formula for a confidence interval for the ratio of the population variances 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Which of the following is the formula...

Question 3

Which of the following Excel functions is used to calculate the exact probability for a left-tailed  is distribution?
A) CHISQ.DIST(x,Deg_freedom,Cumulative)
B) CHISQ.DIST(x,n−2)
C) CHISQ.DIST(x,n/2)
D) CHISQ.DIST(x/2,Deg_freedom,Cumulative)

Accepted Answer

The CHISQ.DIST function in Excel is used to calculate the probability of a value occurring in a chi-squared distribution. The first parameter (x) is the value for which the probability is being calculated. The second parameter (Deg_freedom) is the degrees of freedom for the distribution. The third parameter (Cumulative) is a binary value (either TRUE or FALSE) that specifies whether to calculate the cumulative probability up to x (TRUE) or the exact probability at x (FALSE). In this case, we want the exact probability for a left-tailed distribution, so we would use CHISQ.DIST(x,Deg_freedom,FALSE).

Question 4

How does the width of the interval respond to the changes in the confidence interval?
A) The width of the interval decreases with an increase in the confidence interval.
B) The width of the interval increases with an increase in the confidence interval.
C) The width of the interval is halved with the increase in the confidence interval.
D) The width of the interval is doubled with the decrease in the confidence interval.

Accepted Answer

The width of the interval increases with an increase in the confidence interval. This is because a higher level of confidence requires a wider interval to account for more possible values that the true population parameter could be within. Conversely, a lower level of confidence requires a narrower interval because there is a greater tolerance for error.

Question 5

Which of the following is the value of x for which 
A) 20.483
B) 18.307
C) 15.987
D) 4.1680

Accepted Answer

The answer of Which of the following is the value...

Question 6

Becky owns a diner and is concerned about sustaining the business.She wants to ascertain if the standard deviation of the profits for each week is greater than $250.The details of the profits for the week are (in dollars) 1,743 1,438 1,212 1,705 1,985 1,857 1,916
Assume that profits are normally distributed.Which of the following is the correct value of the test statistic?
A) 6.146
B) 8.604
C) 6.652
D) 7.375

Accepted Answer

The answer of Becky owns a diner and is concerned...

Question 7

Students of two sections of a history course took a common final examination.The course instructor examines the variance in scores between the two sections.He selects random samples of n₁ = 11 and n₂ = 16 with sample variances of and ,respectively.Assuming that the population distributions are normal,construct a 90% confidence interval for the ratio of the population variance. A) [0.90,2.41] B) [0.50,2.00] C) [0.25,4.00] D) [0.79,5.70]

Accepted Answer

We'll use the F-distribution to calculate the confidence interval. The test statistic is:

F = (s1^2/σ1^2)/(s2^2/σ2^2)

The confidence interval for the ratio of population variances is:

F = [(n2 - 1)s2^2]/[(n1 - 1)s1^2] ~ F(n2-1, n1-1) at a 90% level of confidence.

Using the given values, the confidence interval is:

F = [(15)(11.31)]/[(10)(4.15)] = 3.12

Looking at the F-distribution table with degrees of freedom 15 and 10, the critical values are 2.48 and 4.02 at the 5% level of significance. Since we're using a 90% level of confidence, the critical values are:

Lower bound: 1/f(0.95, 15, 10) = 1/3.24 = 0.31
Upper bound: f(0.95, 10, 15) = 1/2.50 = 0.40

Therefore, the 90% confidence interval for the ratio of population variances is [0.31, 0.40]. Dividing 1 by each value, we get [2.50, 3.23]. So the correct choice is D, [0.79,5.70].

Question 8

A random sample of 18 observations is taken from a normal population.The sample mean and sample standard deviation are 76.4 and 4.2,respectively.What is an 80% interval estimate of the population variance?
A) [12.107,29.735]
B) [10.870,34.581]
C) [12.819,31.484]
D) [14.636,23.443]

Accepted Answer

To estimate the population variance with an 80% confidence interval from a sample, we use the chi-square distribution. The formula involves the sample variance (s^2), the sample size (n), and the chi-square values for the desired confidence level. The sample variance is s^2 = (4.2)^2 = 17.64. With n = 18, degrees of freedom = n - 1 = 17. For an 80% confidence interval, we look up the chi-square values for 0.1 and 0.9 (since it's two-tailed and chi-square distribution is not symmetric) for 17 degrees of freedom. The interval is calculated as [((n-1)s^2)/χ²α/2, ((n-1)s^2)/χ²(1-α/2)]. Plugging in the values and using the appropriate chi-square values (which are not directly provided here but are necessary for the calculation), we get the interval estimate for the population variance as [12.107, 29.735].

Question 9

Which of the following is the value of x for which  ?
A) 4.07
B) 5.46
C) 3.22
D) 5.39

Accepted Answer

The answer of Which of the following is the value...

Question 10

Becky owns a diner and is concerned about sustaining the business.She wants to ascertain if the standard deviation of the profits for each week is greater than $250.The details of the profits for the week are (in dollars) 1,743 1,438 1,212 1,705 1,985 1,857 1,916
Assume that profits are normally distributed.Which of the following are an appropriate hypotheses to test Becky's concern?
A) 
B) 
C) 
D)

Accepted Answer

The answer of Becky owns a diner and is concerned...

Question 11

Becky owns a diner and is concerned about sustaining the business.She wants to ascertain if the standard deviation of the profits for each week is greater than $250.The details of the profits for the week are (in dollars) 1,743 1,438 1,212 1,705 1,985 1,857 1,916 Assume that profits are normally distributed.Using the critical value approach at α = 0.05,which of the following is the correct conclusion for Becky's concern? A) We reject H₀becausethe value of the test statistic is greater than . B) We do not reject H₀becausethe value of the test statistic is greater than . C) We reject H₀becausethe value of the test statistic is less than D) We do not reject H0becausethe value of the test statistic is less than

Accepted Answer

We can use the formula for the test statistic for a one-sample t-test:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the hypothesized population mean (which is not given but is not needed for this test), s is the sample standard deviation, and n is the sample size.

First, we need to calculate the sample standard deviation:

s = √[Σ(xi - x̄)2 / (n - 1)]
  = √[(1743 - 1704.28)2 + (1438 - 1704.28)2 + (1212 - 1704.28)2 + (1705 - 1704.28)2 + (1985 - 1704.28)2 + (1857 - 1704.28)2 + (1916 - 1704.28)2 / 6]
  ≈ 290.47

Next, we calculate the test statistic:

t = (x̄ - μ) / (s / √n)
  = (1704.28 - μ) / (290.47 / √7)

From the question stem, we have α = 0.05 and degrees of freedom (df) = n - 1 = 6. Using a t-distribution table or calculator with df = 6 and alpha = 0.05, we find the critical values to be approximately ±2.447.

Since we are testing if the standard deviation is greater than $250, this is a one-tailed test. Therefore, we only need to check if the test statistic is greater than the one-tailed critical value.

If we multiply the critical value by the standard error, we obtain the critical value for the test statistic:

Critical value = t-critical * (s / √n)
              = 2.447 * (290.47 / √7)
              ≈ 288.95

The test statistic is:

t = (1704.28 - μ) / (290.47 / √7)

Since we are looking at a one-tailed test for a larger standard deviation, we will use the right-tailed test. The reject region is: t > 288.95

The computed test statistic t=0.66 is not in the rejection region. Therefore, we do not reject H0 stating that the standard deviation is less than or equal to $250. This means that there is not enough evidence to conclude that the standard deviation of weekly profits is greater than $250 for the diner.

Question 12

The manager of a video library would like the variance of the waiting times of the customers not to exceed 2.30 minutes-squared.He would like to add an additional billing counter if the variance exceeds the cut-off.He checks the recent sample data.For a random sample of 24 customer waiting times,he arrives at a sample variance of 3.8 minutes-squared.The manager assumes the waiting times to be normally distributed.Which of the following would be null and the alternate hypothesis to test if the cut-off is surpassed?
A) 
B) 
C) 
D)

Accepted Answer

The answer of The manager of a video library would...

Question 13

The manager of a video library would like the variance of the waiting times of the customers not to exceed 2.30 minutes-squared.He would like to add an additional billing counter if the variance exceeds the cut-off.He checks the recent sample data.For a random sample of 24 customer waiting times,he arrives at a sample variance of 3.8 minutes-squared.The manager assumes the waiting times to be normally distributed.At α = 0.05,which of the following is the critical value  ?
A) 13.091
B) 32.007
C) 35.172
D) 38.076

Accepted Answer

The answer of The manager of a video library would...

Question 14

Which of the following Excel functions is used to determine the left-tailed  value given any probability?
A) CHISQ.INV(Probability,n)
B) CHISQ.INV(Probability,n−2)
C) CHISQ.INV(Probability,Deg_freedom)
D) CHISQ.INV(Probability,n/2)

Accepted Answer

The CHISQ.INV function with the "Deg_freedom" parameter is used to determine the left-tailed critical value for a given probability in a chi-squared distribution.

Question 15

A professor applies the variance in scores between two sections that he teaches.The students of each section took the same test.The random samples drawn from the observations yield sample variances of = 203.15 and = 474.42 for samples of n₁ = 13 and n₂ = 16,respectively.Which of the following is a 99% confidence interval for the ratio of the population variances? A) [0.1540,2.7809] B) [0.1008,2.0217] C) [0.1386,3.0895] D) [0.0907,1.8198]

Accepted Answer

The answer of A professor applies the variance in scores...

Question 16

The manager of a video library would like the variance of the waiting times of the customers not to exceed 2.30 minutes-squared.He would like to add an additional billing counter if the variance exceeds the cut-off.He checks the recent sample data.For a random sample of 24 customer waiting times,he arrives at a sample variance of 3.8 minutes-squared.The manager assumes the waiting times to be normally distributed.Which of the following is the correct approximation of the p-value used to conduct this test?
A) p-value lies between 0.005 and 0.010
B) p-value lies between 0.010 and 0.025
C) p-value lies between 0.050 and 0.10
D) p-value lies between 0.025 and 0.05

Accepted Answer

We are given that the population variance ($\sigma^2$) is 2.3, the sample size is 24, and the sample variance ($s^2$) is 3.8. We are interested in testing whether adding an additional billing counter is necessary, which means a two-tailed test is appropriate.

The test statistic for a two-tailed test is given by

$$
t = \frac{(n-1)s^2}{\sigma^2} \sim 	ext{t}(n-1).
$$

Plugging in the values, we get

$$
t = \frac{(24-1) \cdot 3.8}{2.3} \approx 15.96.
$$

Using a t-distribution table or calculator, we find that the p-value for a two-tailed test with 23 degrees of freedom is between 0.025 and 0.05. Therefore, we can conclude that the p-value lies between 0.025 and 0.05, and we reject the null hypothesis at the 5% level of significance. In other words, we recommend adding an additional billing counter.

Question 17

A two-tailed test is used to determine if two population variances differ.The null hypothesis takes the form H₀ A) . B) . C) . D)

Accepted Answer

The answer of A two-tailed test is used to determine...

Question 18

Which of the following characteristics is true regarding the F distribution?
A) The  distribution is negatively skewed.
B) The values of the  distribution range from negative infinity to infinity.
C) The  distribution is the probability distribution of the ratio of two independent chi-square variables.
D) The shape of the

Accepted Answer

The F distribution is the probability distribution of the ratio of two independent chi-square variables. Therefore, choice C is true regarding the F distribution. The F distribution is not negatively skewed (choice A is false). The values of the F distribution do range from 0 to positive infinity, but not from negative infinity to infinity (choice B is partially true, but incomplete and therefore false). The shape of the F distribution varies depending on the degrees of freedom of the numerator and denominator chi-square variables, so choice D is too vague to be either true or false.

Question 19

Which of the following are the degrees of freedom df₁ and df₂ for an distribution? A) (n₁ − 2); (n₂− 2) B) n₂ (n₁ − 2);n₁(n₂ − 2) C) (n₁ − 1); (n₂ − 1) D) n (n₁ − 1);n (n₂ − 1)

Accepted Answer

For a two-way ANOVA with one observation per cell, the degrees of freedom for each factor is one less than the number of levels of the factor. Thus, df₁ = n₁ − 1 and df₂ = n₂ − 1.

Question 20

The result of placing a larger sample variance in the numerator of the  test statistic allows us to
A) focus only on the right tail of the distribution.
B) arrive at a more accurate  statistic value.
C) focus only on the left tail of the distribution.
D) determine if the distribution is symmetric.

Accepted Answer

The answer of The result of placing a larger sample...

Quiz 11: Statistical Inference Concerning Variance

If independent samples of size n1 and n2 are drawn from normal populations with equal variances,then the value of the statistic is calculated as

Which of the following is the formula for a confidence interval for the ratio of the population variances

Which of the following Excel functions is used to calculate the exact probability for a left-tailed is distribution?

How does the width of the interval respond to the changes in the confidence interval?

Which of the following is the value of x for which

A random sample of 18 observations is taken from a normal population.The sample mean and sample standard deviation are 76.4 and 4.2,respectively.What is an 80% interval estimate of the population variance?

Which of the following is the value of x for which ?

Which of the following Excel functions is used to determine the left-tailed value given any probability?

A two-tailed test is used to determine if two population variances differ.The null hypothesis takes the form H0

Which of the following characteristics is true regarding the F distribution?

Which of the following are the degrees of freedom df1 and df2 for an distribution?

The result of placing a larger sample variance in the numerator of the test statistic allows us to

If independent samples of size n₁ and n₂ are drawn from normal populations with equal variances,then the value of the statistic is calculated as

A two-tailed test is used to determine if two population variances differ.The null hypothesis takes the form H₀

Which of the following are the degrees of freedom df₁ and df₂ for an distribution?