Question 1

A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 51 students and asked each to provide the amount of time they spent traveling to campus. The sample results found that the sample mean was 23.243 minutes and the sample standard deviation was 20.40 minutes. Find the rejection region for determining if the population standard deviation exceeds 20 minutes. Use α = 0.05. A) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { z } > 1.645$
B) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 67.5048$
C) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 71.4202$
D) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 34.7642$ 2 Perform Test of Hypothesis for Population Variance

Accepted Answer

The answer of A large university is interested in learning...

Question 2

A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than .07 to ensure proper inoculation. A random sample of 25 injections was measured. Suppose the p-value for the test is p = .0024. State the proper conclusion using α = .01.

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Question 3

A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 51 students and asked each to provide the amount of time they spent traveling to campus. The sample results found that the sample mean was 23.243 minutes and the sample standard deviation was 20.40 minutes. It is desired to determine if the population standard deviation exceeds 20 minutes. Calculate the test statistic for this test of hypothesis. A) $\chi ^ { 2 } = 51$
B) $\chi ^ { 2 } = 53.06$
C) $\chi ^ { 2 } = 52.02$
D) $\chi ^ { 2 } = 58.11$

Accepted Answer

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Question 4

It is desired to test H0: μ = 50 against HA: μ ≠ 50 using α = 0.10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 45, what is the power of the test?

Accepted Answer

The power of the test is calculated as 1 minus the probability of a Type II error (β). Given the sample size, standard deviation, and significance level, the power is approximately 0.7544 when μ is actually 45.

Question 5

Under the assumption that ? = ?a, where ?a is the alternative mean, the distribution of $x$ is mound shaped and symmetric about ?a.

Accepted Answer

Under the assumption that the mean is the alternative mean (?a), the distribution of $x$ is symmetric and mound-shaped around this mean, which is a characteristic of a normal distribution.

Question 6

It is desired to test H0: μ = 50 against HA: μ ≠ 50 using α = 0.10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 48, what is the probability that the hypothesis test would lead the investigator to commit a Type II error?

Accepted Answer

The probability of committing a Type II error (β) is calculated using the non-centrality parameter and the critical value. Here, β is 0.7567.

Question 7

A random sample of $n$ observations, selected from a normal population, is used to test the null hypothesis $H _ { 0 }$ : $\sigma ^ { 2 } = 155$. Specify the appropriate rejection region. $$H _ { \mathrm { a } } : \sigma ^ { 2 } \approx 155 , n = 10 , \alpha = .05$$ A) $\chi ^ { 2 } < 2.70039$ or $\chi ^ { 2 } > 19.0228$ B) $2.70039 < \chi ^ { 2 } < 19.0228$ C) $\chi ^ { 2 } < 3.32511$ or $\chi ^ { 2 } > 16.9190$ D) $\chi ^ { 2 } < 3.24697$ or $\chi ^ { 2 } > 20.4831$

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Question 8

It is desired to test H0: μ = 55 against Ha: μ < 55 using α = .10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 50, what is the power of this test?

Accepted Answer

The power of a test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. Here, we calculate the power by finding the probability that the test statistic falls in the critical region when μ = 50. The critical value for a one-tailed test at α = 0.10 is found using the standard normal distribution. The test statistic is calculated using the sample mean, standard deviation, and sample size. The power is then 1 minus the probability of a Type II error (failing to reject H0 when Ha is true), which is 0.1469, so the power is 0.8531.

Question 9

An educational testing service designed an achievement test so that the range in student scores would be greater than 420 points. To determine whether the objective was achieved, the testing service gave the test to a random sample of 30 students and found that the sample mean and variance were 759 and 1943, respectively. Conduct the test for $$H _ { 0 } : \sigma ^ { 2 } = 4900 \text { vs. } H _ { a } : \sigma ^ { 2 } > 4900 \text { using } \alpha = .025 \text {. Assume the range is } 6 \sigma \text {. }$$

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Question 10

The value of $\beta$ is the area under the bell curve for the distribution of $\bar { x }$ centered at $\mu _ { \mathrm { a } }$ for values of $\bar { x }$ that fall within the acceptance region of the distribution of $\bar { x }$ centered at $\mu _ { 0 }$.

Accepted Answer

The acceptance region is the range of values for $\bar { x }$ that are considered acceptable. The area under the bell curve for the distribution of $\bar { x }$ centered at $\mu _ { \mathrm { a } }$ for values of $\bar { x }$ that fall within the acceptance region is the probability that $\bar { x }$ will fall within the acceptance region. This probability is the value of $\beta$.

Question 11

It has been estimated that the G-car obtains a mean of 40 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects 64 G-cars and records the mileage obtained for each car over a driving course similar to that used to obtain the estimate. The following data resulted: x = 41.5 miles per gallon, s = 8 miles per gallon. Calculate the value of β if the true value of the mean is 42 miles per gallon. Use α = .025. 2 Find and Interpret Power of Test

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Question 12

A random sample of n observations, selected from a normal population, is used to test the null hypothesis H0: σ2 = 155. Specify the appropriate rejection region. Ha: σ2 > 155, n = 25, α = .10

Accepted Answer

The rejection region for a right-tailed test of the variance of a normal population is given by χ2 > χ2α,n-1, where χ2α,n-1 is the upper α-quantile of the chi-squared distribution with n-1 degrees of freedom. For n = 25 and α = .10, χ2α,n-1 = 33.1963. Therefore, the rejection region is χ2 > 33.1963.

Question 13

$$\text { Let } \chi _ { 0 } ^ { 2 } \text { be a particular value of } x ^ { 2 } \text {. Find the value of } \chi _ { 0 } ^ { 2 } \text { such that } \mathrm { P } \left( x ^ { 2 } > \chi _ { 0 } ^ { 2 } \right) = .10 \text { for } n = 10 \text {. }$$

Accepted Answer

The value of $\chi_0^2$ such that $P(x^2 > \chi_0^2) = 0.10$ for $n = 10$ degrees of freedom is found using the chi-square distribution table or calculator. The critical value for the 90th percentile is approximately 14.6837.

Question 14

Type I errors and Type II errors are complementary events so that α = P(Type I error) = 1 - P(Type II error) = 1 - β.

Accepted Answer

Type I and Type II errors are not complementary events. α is the probability of a Type I error, and β is the probability of a Type II error. They are related but not complementary, as they depend on different hypotheses.

Question 15

A random sample of $n$ observations, selected from a normal population, is used to test the null hypothesis $H _ { 0 }$ : $\sigma ^ { 2 } = 155$. Specify the appropriate rejection region.
$$H _ { \mathrm { a } } : \sigma ^ { 2 } < 155 , n = 14 , \alpha = .01$$
A) $\chi ^ { 2 } < 4.10691$
B) $\chi ^ { 2 } < 27.6883$
C) $x ^ { 2 } < 4.66043$
D) $\chi ^ { 2 } < 29.1413$

Accepted Answer

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Question 16

A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than .05 to ensure proper inoculation. A random sample of 31 injections resulted in a variance of .103. Specify the rejection region for the test. Use α = .10.

Accepted Answer

The answer of A new apparatus has been devised to...

Question 17

A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than .08 to ensure proper inoculation. A random sample of 25 injections resulted in a variance of .103. Calculate the test statistic for the test of interest.

Accepted Answer

The answer of A new apparatus has been devised to...

Question 18

An educational testing service designed an achievement test so that the range in student scores would be greater than 300 points. To determine whether the objective was achieved, the testing service gave the test to a random sample of 41 students and found that the sample mean and variance were 687 and 2605, respectively. Specify the null and alternative hypotheses for determining whether the test achieved the desired dispersion in scores. Assume that range = 6σ.

Accepted Answer

The answer of An educational testing service designed an achievement...

Question 19

It has been estimated that the G-car obtains a mean of 30 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects 36 G-cars and records the mileage obtained for each car over a driving course similar to that used to obtain the estimate. The following data resulted: x = 31.8 miles per gallon, s = 6 miles per gallon. Calculate the power of the test if the true value of the mean is 31 miles per gallon. Use a value of α = .025.

Accepted Answer

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Question 20

The null distribution is the distribution of the test statistic assuming the null hypothesis is true; it mound shaped and symmetric about the null mean μ0.

Accepted Answer

The null distribution represents the expected distribution of the test statistic under the assumption that the null hypothesis is true, often resembling a normal distribution which is symmetric and centered around the null mean μ0.

Quiz 8: Inferences Based on a Single Sample: Tests of Hypothesis

It is desired to test H0: μ = 50 against HA: μ ≠ 50 using α = 0.10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 45, what is the power of the test?

Under the assumption that ? = ?a, where ?a is the alternative mean, the distribution of $x$ is mound shaped and symmetric about ?a.

It is desired to test H0: μ = 55 against Ha: μ < 55 using α = .10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 50, what is the power of this test?

The value of $\beta$ is the area under the bell curve for the distribution of $\bar { x }$ centered at $\mu _ { \mathrm { a } }$ for values of $\bar { x }$ that fall within the acceptance region of the distribution of $\bar { x }$ centered at $\mu _ { 0 }$ .

A random sample of n observations, selected from a normal population, is used to test the null hypothesis H0: σ2 = 155. Specify the appropriate rejection region. Ha: σ2 > 155, n = 25, α = .10

$\text { Let } \chi _ { 0 } ^ { 2 } \text { be a particular value of } x ^ { 2 } \text {. Find the value of } \chi _ { 0 } ^ { 2 } \text { such that } \mathrm { P } \left( x ^ { 2 } > \chi _ { 0 } ^ { 2 } \right) = .10 \text { for } n = 10 \text {. }$

Type I errors and Type II errors are complementary events so that α = P(Type I error) = 1 - P(Type II error) = 1 - β.

The null distribution is the distribution of the test statistic assuming the null hypothesis is true; it mound shaped and symmetric about the null mean μ0.

Quiz 8: Inferences Based on a Single Sample: Tests of Hypothesis

It is desired to test H0: μ = 50 against HA: μ ≠ 50 using α = 0.10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 45, what is the power of the test?

Under the assumption that ? = ?a, where ?a is the alternative mean, the distribution of xxx is mound shaped and symmetric about ?a.

It is desired to test H0: μ = 55 against Ha: μ < 55 using α = .10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 50, what is the power of this test?

The value of β\betaβ is the area under the bell curve for the distribution of xˉ\bar { x }xˉ centered at μa\mu _ { \mathrm { a } }μa​ for values of xˉ\bar { x }xˉ that fall within the acceptance region of the distribution of xˉ\bar { x }xˉ centered at μ0\mu _ { 0 }μ0​ .

A random sample of n observations, selected from a normal population, is used to test the null hypothesis H0: σ2 = 155. Specify the appropriate rejection region. Ha: σ2 > 155, n = 25, α = .10

Type I errors and Type II errors are complementary events so that α = P(Type I error) = 1 - P(Type II error) = 1 - β.

The null distribution is the distribution of the test statistic assuming the null hypothesis is true; it mound shaped and symmetric about the null mean μ0.

Under the assumption that ? = ?a, where ?a is the alternative mean, the distribution of $x$ is mound shaped and symmetric about ?a.

The value of $\beta$ is the area under the bell curve for the distribution of $\bar { x }$ centered at $\mu _ { \mathrm { a } }$ for values of $\bar { x }$ that fall within the acceptance region of the distribution of $\bar { x }$ centered at $\mu _ { 0 }$ .