#### HW #17 – Hypothesis Tests I – Means

- A survey organization sampled 56 households in a community and found that the mean number of TV sets per household was 2.49. The standard deviation is 1.4. Can you conclude that the mean number of TV sets per household is greater than 2? Use the � = 0.01 level.
- State the null and alternative hypotheses.

�0: ? p μ ? < > = ≠

�1: ? p μ ? < > = ≠

- Find the critical value and sketch the rejection region.

Critical value(s):

(If there are two critical values, list both values separated by a comma)

- Find the test statistic.

- Find the p-value.

- Decide: Select an answer Accept Reject Don’t Reject �0

- Conclusion: At � = 0.01 significance level, there Select an answer is not is enough evidence to conclude that the mean number of TV sets per household is greater than 2.

- The Russell-2000 is a group of small-company stocks. A random sample of 39 of these stocks had a mean price of $33.88 with a standard deivation of $6.54. A stock market analyst predicted that the mean price of all Russell-2000 stocks would be $32. Can you conclude that the mean price differs from $32? Use the � = 0.07 level.

- State the null and alternative hypotheses.

�0: ? p μ ? < > = ≠

�1: ? p μ ? < > = ≠

- Find the critical value and sketch the rejection region.

Critical value(s):

(If there are two critical values, list both values separated by a comma)

- Find the test statistic.

- Find the p-value.

- Decide: Select an answer Accept Reject Don’t Reject �0

- Conclusion: At � = 0.07 significance level, there Select an answer is is not enough evidence to conclude that mean price differs from $32.

- For the past several years, the mean number of people in a household has been declining. A social scientist believes that in a certain city, the mean number of people per household is less than 3.5. To investigate this, she takes a random sample of 171 households in the city and finds that the mean number of people is 3.34 with a standard deviation of 1.3. Can you conclude that the mean number of people per household is less than 3.5? Use the � = 0.06 level.

- State the null and alternative hypotheses.

�0: ? p μ ? < > = ≠

�1: ? p μ ? < > = ≠

Critical value(s):

(If there are two critical values, list both values separated by a comma)

- Find the test statistic.

- Find the p-value.

- Decide: Select an answer Reject Don’t Reject Accept �0

- Conclusion: At � = 0.06 significance level, there Select an answer is not is enough evidence to conclude that the mean number of people per household is less than 3.5.

- A housing official in a certain city claims that the mean monthly rent for apartments in the city is less than $1000. To verify this claim, a simple random sample of 40 renters in the city was taken, and the mean rent paid was $950 with a standard deviation of $223. Can you conclude that the mean monthly rent in the city is less than $1000? Use the � = 0.09 level.

- State the null and alternative hypotheses.

�0: ? p μ p₁ μ₁ ? < > = ≠ Select an answer μ₂ p₂ 223 1000 950

�1: ? p μ p₁ μ₁ ? < > = ≠ Select an answer μ₂ p₂ 223 950 1000

Critical value(s):

(If there are two critical values, list both values separated by a comma)

- Find the test statistic.

- Find the p-value.

- Decide: Select an answer Accept Reject Don’t Reject �0

- Conclusion: At � = 0.09 significance level, there Select an answer is not is enough evidence to conclude that the mean monthly rent in the city is less than $1000.

- Type of Error: A Select an answer Type II Type IV Type III Type I error could occur in this problem. The probablity of this error occurring is Select an answer 0.08 0.09 1000 950 β 0.1 . We can reduce the probability of this error occurring by Select an answer being more careful decreasing the sample size lowering the significance level α increasing the sample size .

- Credit scores are used by banks and other lenders to determine whether someone is a good credit risk. Scores range from 300 to 850, with a score of 720 or more indicating that a person is a good credit risk. An economist finds that a simple random sample of 164 people had a mean credit score of 738 with a standard deviation of 82. Can the economist conclude that the mean credit score is more than 720? Use the � = 0.01 level.
- State the null and alternative hypotheses.

�0: ? p μ p₁ μ₁ ? < > = ≠ ? 738 p₂ μ₂ 82 720

�1: ? p μ p₁ μ₁ ? < > = ≠ ? μ₂ 720 p₂ 738 82

Critical value(s):

(If there are two critical values, list both values separated by a comma)

- Find the test statistic.

- Find the p-value.

- Decide: Select an answer Reject Accept Don’t Reject �0

- Conclusion: At � = 0.01 significance level, there Select an answer is is not enough evidence to conclude that the mean credit score is more than 720.

- Type of Error: A Select an answer Type II Type IV Type I Type III error could occur in this problem. The probablity of this error occurring is Select an answer β 738 0.02 720 0.01 0 . We can reduce the probability of this error occurring by Select an answer lowering the significance level α being more careful decreasing the sample size increasing the sample size .

- Answer the following True or False:

Suppose a sample size of 103 was used to conduct a left tailed hypothesis test for a single population mean. The null hypothesis was �0:�=22 and the p-value was 0.04. Then there is a 4% chance that the population mean is equal to 22. - Answer the following True or False:

A researcher hypothesizes that the average student spends less than 20% of their total study time reading the textbook. The appropriate hypothesis test is a left tailed test for a population mean.

- true
- Answer the following True or False:

Suppose a sample size of 87 was used to conduct a right tailed hypothesis test for a single population mean. The null hypothesis was H_{0}: � = 45 and the sample mean was 50 and the p-value was 0.03. Then there is a 3% chance that 87 randomly selected individuals from a population with a mean of 45 and the same standard deviation would have a sample mean greater than 50.

- true
- false
- Answer the following True or False:

If the p-value is 0.003 and the level of significance , � , is 0.05 in a hypothesis test for a mean, then we accept the alternative hypothesis.

- true
- false
- Answer the following True or False:

Suppose a hypothesis test was performed with a level of significance of 0.05. Then if the null hypothesis is actually true, then there is a 5% chance that the researcher will end up accepting the alternative hypothesis in error.

- false
- true
- Answer the following True or False:

If the p-value is 0.13 and the level of significance , � , is 0.05 in a hypothesis test for a mean, then we accept the null hypothesis.

- true
- false

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