Question 1

We are interested in testing to see if the variance of a population is less than 7. The correct null hypothesis is _____. A) σ < 7 B) σ² ≥ 49 C) s < 49 D) s > 49

Accepted Answer

The correct null hypothesis for testing if the variance of a population is less than a specific value involves setting the null hypothesis to the opposite (i.e., greater than or equal to). Since variance is denoted by σ^2, and we're testing if it's less than 7, the null hypothesis should be that the variance is greater than or equal to 7^2, which is 49. Hence, the correct formulation is σ^2 ≥ 49.

Question 2

A sample of 20 cans of tomato juice showed a standard deviation of 0.4 ounce. A 95% confidence interval estimate of the variance for the population is _____.
A) .2313 to .8533
B) .2224 to .7924
C) .0889 to .3169
D) .0925 to .3413

Accepted Answer

We use the chi-square distribution to construct the confidence interval for the population variance. The formula for the confidence interval is:

[(n-1)*S^2)/X^2_u, (n-1)*S^2/X^2_l]

where n is the sample size, S is the sample standard deviation, X^2_u and X^2_l are the upper and lower critical values of the chi-square distribution with degrees of freedom (n-1) and the desired level of confidence. In this case, n = 20, S = 0.4, and we want a 95% confidence interval, which corresponds to alpha/2 = 0.025 and degrees of freedom = 19. Looking up the critical values in a chi-square distribution table, we find X^2_u = 32.852 and X^2_l = 10.117. Plugging in the values, we get:

[(19*0.4^2)/32.852, 19*0.4^2/10.117] = [0.0925, 0.3413]

Therefore, the 95% confidence interval estimate of the variance for the population is 0.0925 to 0.3413. The answer is D.

Question 3

The symbol used for the variance of the sample is _____. A) σ B) σ² C) s D) s²

Accepted Answer

D

Question 4

The χ² values (for interval estimation) for a sample size of 10 at 95% confidence are _____. A) 3.32511 and 16.9190 B) 2.70039 and 19.0228 C) 4.16816 and 14.6837 D) 3.24697 and 20.4831

Accepted Answer

Using a chi-square distribution table with 9 degrees of freedom (10-1), we find the values to be 2.70039 and 19.0228 at 95% confidence.

Question 5

The χ² value for a one-tailed (upper tail) hypothesis test at 95% confidence and a sample size of 25 is _____. A) 33.1963 B) 36.4151 C) 39.3641 D) 37.6525

Accepted Answer

Using a chi-square distribution table with 24 degrees of freedom (25-1), and a 95% confidence level for an upper-tailed test, the critical value of chi-square is 36.4151. Therefore, the answer is B.

Question 6

The symbol used for the variance of the population is _____. A) σ B) σ² C) s D) s²

Accepted Answer

The symbol for the variance of the population is represented by σ^2, which is option B (σ²) in the given choices. σ represents the standard deviation of the population, and when squared (σ^2), it denotes the variance of the population.

Question 7

The bottler of a certain soft drink claims its equipment to be accurate and that the variance of all filled bottles is .05 or less. The null hypothesis in a test to confirm the claim would be written as _____. A) H₀: σ² ≥ .05 B) H₀: σ² > .05 C) H₀: σ² < .05 D) H₀: σ² ≤ .05

Accepted Answer

The null hypothesis should state that the variance is equal to or greater than the claimed value, which is .05. Therefore, the correct choice is D as it states that the variance is less than or equal to .05.

Question 8

The producer of a certain bottling equipment claims that the variance of all its filled bottles is .027 or less. A sample of 30 bottles showed a standard deviation of .2. The p-value for the test is _____.
A) between .025 and .05
B) between .05 and .01
C) .05
D) .025

Accepted Answer

The p-value indicates the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. In this case, the null hypothesis would be that the variance is .027 or less. Given the sample size of 30 and a sample standard deviation of .2, which is significantly higher than the claimed variance, the p-value would likely indicate that it is relatively unlikely to observe such a deviation if the true variance were indeed .027 or less. The option "between .025 and .05" suggests a low but not extremely low probability, fitting the scenario where the observed data somewhat strongly contradicts the claim, but not to an extreme extent.

Question 9

The value of F_.05 with 8 numerator and 19 denominator degrees of freedom is _____. A) 2.48 B) 2.58 C) 3.63 D) 2.96

Accepted Answer

The answer of The value of F_.05 with 8 numerator...

Question 10

A sample of 41 observations yielded a sample standard deviation of 5. If we want to test H₀: σ² = 20, the test statistic is _____. A) 100 B) 10 C) 51.25 D) 50

Accepted Answer

The test statistic for testing the variance is calculated using the formula: (n-1)s^2/σ^2. Here, n=41, s=5, and σ^2=20. So, (41-1)(5^2)/20 = 50.

Question 11

A random sample of 31 charge sales showed a sample standard deviation of $50. A 90% confidence interval estimate of the population standard deviation is _____.
A) 1,715.10 to 4,055.68
B) 1,596.45 to 4,466.73
C) 39.96 to 66.83
D) 41.39 to 63.68

Accepted Answer

To find the confidence interval for the population standard deviation, we use the chi-squared distribution. For a 90% confidence interval with 30 degrees of freedom (n-1), the critical values from the chi-squared distribution are 41.39 and 63.68. The formula is:

(sample standard deviation * sqrt(n-1)) / sqrt(chi-squared upper, alpha/2 df) < population standard deviation < (sample standard deviation * sqrt(n-1)) / sqrt(chi-squared lower, alpha/2 df)

Plugging in the values, we get:

(50 * sqrt(31-1)) / sqrt(63.68) < population standard deviation < (50 * sqrt(31-1)) / sqrt(41.39)
39.96 < population standard deviation < 66.83

Therefore, the 90% confidence interval estimate of the population standard deviation is 41.39 to 63.68.

Question 12

The manager of the service department of a local car dealership has noted that the service times of a sample of 15 new automobiles has a standard deviation of 4 minutes. A 95% confidence interval estimate for the variance of service times for all its new automobiles is _____.
A) 8.58 to 39.79
B) 4 to 16
C) 4 to 15
D) 1.64 to 1.96

Accepted Answer

The correct confidence interval for the variance is calculated using the chi-square distribution with the formula $\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}$ to $\frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}$, where $n$ is the sample size, $s^2$ is the sample variance, and $\chi^2$ values are based on the degrees of freedom ($n-1$) and the confidence level ($1-\alpha$). Given $n=15$ and $s=4$, and aiming for a 95% confidence interval, the calculation leads to the interval provided in option A, which correctly accounts for the variability in the sample to estimate the population variance.

Question 13

A sample of 28 elements is selected to estimate a 95% confidence interval for the variance of the population. The χ²values to be used for this interval estimation are _____. A) -1.96 and 1.96 B) 14.573 and 43.195 C) 16.151 and 40.113 D) 15.308 and 44.461

Accepted Answer

The correct χ^2 values for a 95% confidence interval for the variance when the sample size is 28 (degrees of freedom = 27) are found in the χ^2 distribution table. The values 14.573 and 43.195 correspond to the lower and upper critical values for 27 degrees of freedom at a 95% confidence level.

Question 14

A sample of n observations is taken from a population. When performing statistical inference about a population variance, the appropriate χ²distribution has _____. A) n degrees of freedom B) n - 1 degrees of freedom C) n - 2 degrees of freedom D) n - 3 degrees of freedom

Accepted Answer

When performing statistical inference about a population variance, the appropriate distribution is the chi-square distribution with n - 1 degrees of freedom. This is because the sample variance is calculated with n - 1 degrees of freedom (dividing by n - 1 instead of n) in order to account for the fact that estimating the population variance requires using the sample mean, which introduces an additional source of variability. Therefore, when using the sample variance to make inferences about the population variance, we must use the chi-square distribution with n - 1 degrees of freedom.

Question 15

The manager of the service department of a local car dealership has noted that the service times of a sample of 30 new automobiles has a standard deviation of 6 minutes. A 95% confidence interval estimate for the standard deviation of the service times for all its new automobiles is _____.
A) 16.05 to 45.72
B) 4.78 to 8.07
C) 2.93 to 6.31
D) 22.83 to 65.06

Accepted Answer

To estimate the confidence interval for the standard deviation of a population based on a sample, we use the Chi-Square distribution. The formula involves the sample standard deviation, the sample size, and the Chi-Square values for the desired confidence level. The correct interval is calculated using the Chi-Square distribution values for 29 degrees of freedom (n-1) at the 95% confidence level, which results in the interval 4.78 to 8.07.

Question 16

We are interested in testing whether the variance of a population is significantly less than 1.44. The null hypothesis for this test is _____. A) H₀: σ² < 1.44 B) H₀: s² ≥ 1.44 C) H₀: σ ≤ 1.20 D) H₀: σ² ≥ 1.44

Accepted Answer

The null hypothesis for this test is that the variance of the population is not less than 1.44. Using the symbol σ² to represent the population variance, the null hypothesis can be stated as:
H0: σ² ≥ 1.44
Option D is the only one that matches this null hypothesis statement. Option A reverses the inequality and is therefore incorrect. Option B uses the sample variance (s) instead of the population variance (σ²) and also reverses the inequality. Option C sets a specific value for σ, which is not what we are testing.

Question 17

The random variable for a chi-square distribution may assume any _____.
A) value between -1 and 1
B) value between -infinity and +infinity
C) negative value
D) value greater than 0

Accepted Answer

The chi-square distribution is defined only for non-negative values, meaning its random variable can assume any value greater than 0.

Question 18

For an F distribution, the number of degrees of freedom for the numerator _____.
A) must be larger than the number of degrees for the denominator
B) must be smaller than the number of degrees of freedom for the denominator
C) must be equal to the number of degrees of freedom for the denominator
D) can be larger, smaller, or equal to the number of degrees of freedom for the denominator

Accepted Answer

There is no requirement or restriction on the relationship between the numerator and denominator degrees of freedom for an F distribution. The numerator and denominator degrees of freedom can be any positive integers.

Question 19

The producer of a certain medicine claims that its bottling equipment is very accurate and that the standard deviation of all its filled bottles is 0.1 ounce or less. A sample of 20 bottles showed a standard deviation of .11. The test statistic to test the claim is _____.
A) 400
B) 22.99
C) 4.85
D) 20

Accepted Answer

The test statistic for testing a claim about a population standard deviation (σ) when the sample standard deviation (s) is known, is calculated using the formula: $\chi^2 = \frac{(n-1)s^2}{\sigma^2}$, where $n$ is the sample size, $s$ is the sample standard deviation, and $\sigma$ is the claimed population standard deviation. Here, $n = 20$, $s = 0.11$, and $\sigma = 0.1$. Plugging these values into the formula gives: $\chi^2 = \frac{(20-1)(0.11)^2}{(0.1)^2} = 22.99$.

Question 20

To avoid the problem of not having access to tables of F distribution with values given for the lower tail, the numerator of the test statistic should be the one with the _____.
A) larger sample size
B) smaller sample size
C) larger sample variance
D) smaller sample variance

Accepted Answer

The numerator of the test statistic should be the one with the larger sample variance to ensure the F-ratio is always greater than or equal to 1. This approach simplifies the use of F distribution tables, which typically only provide critical values for the upper tail of the distribution.

Quiz 11: Inferences About Population Variances

We are interested in testing to see if the variance of a population is less than 7. The correct null hypothesis is _____.

A sample of 20 cans of tomato juice showed a standard deviation of 0.4 ounce. A 95% confidence interval estimate of the variance for the population is _____.

The symbol used for the variance of the sample is _____.

The χ2 values (for interval estimation) for a sample size of 10 at 95% confidence are _____.

The χ2 value for a one-tailed (upper tail) hypothesis test at 95% confidence and a sample size of 25 is _____.

The symbol used for the variance of the population is _____.

The bottler of a certain soft drink claims its equipment to be accurate and that the variance of all filled bottles is .05 or less. The null hypothesis in a test to confirm the claim would be written as _____.

The producer of a certain bottling equipment claims that the variance of all its filled bottles is .027 or less. A sample of 30 bottles showed a standard deviation of .2. The p-value for the test is _____.

The value of F.05 with 8 numerator and 19 denominator degrees of freedom is _____.

A sample of 41 observations yielded a sample standard deviation of 5. If we want to test H0: σ2 = 20, the test statistic is _____.

A random sample of 31 charge sales showed a sample standard deviation of $50. A 90% confidence interval estimate of the population standard deviation is _____.

The manager of the service department of a local car dealership has noted that the service times of a sample of 15 new automobiles has a standard deviation of 4 minutes. A 95% confidence interval estimate for the variance of service times for all its new automobiles is _____.

A sample of 28 elements is selected to estimate a 95% confidence interval for the variance of the population. The χ2 values to be used for this interval estimation are _____.

A sample of n observations is taken from a population. When performing statistical inference about a population variance, the appropriate χ2 distribution has _____.

The manager of the service department of a local car dealership has noted that the service times of a sample of 30 new automobiles has a standard deviation of 6 minutes. A 95% confidence interval estimate for the standard deviation of the service times for all its new automobiles is _____.

We are interested in testing whether the variance of a population is significantly less than 1.44. The null hypothesis for this test is _____.

The random variable for a chi-square distribution may assume any _____.

For an F distribution, the number of degrees of freedom for the numerator _____.

The producer of a certain medicine claims that its bottling equipment is very accurate and that the standard deviation of all its filled bottles is 0.1 ounce or less. A sample of 20 bottles showed a standard deviation of .11. The test statistic to test the claim is _____.

To avoid the problem of not having access to tables of F distribution with values given for the lower tail, the numerator of the test statistic should be the one with the _____.

The χ² values (for interval estimation) for a sample size of 10 at 95% confidence are _____.

The χ² value for a one-tailed (upper tail) hypothesis test at 95% confidence and a sample size of 25 is _____.

The value of F_.05 with 8 numerator and 19 denominator degrees of freedom is _____.

A sample of 41 observations yielded a sample standard deviation of 5. If we want to test H₀: σ² = 20, the test statistic is _____.

A sample of 28 elements is selected to estimate a 95% confidence interval for the variance of the population. The χ²values to be used for this interval estimation are _____.

A sample of n observations is taken from a population. When performing statistical inference about a population variance, the appropriate χ²distribution has _____.