Q: A sample of 350 cell phone users finds that 238 have 4G contracts. Last year, the proportion was $$p = 0.62. $$Test $$H_0$$: p \leq 0.62$$ vs. H_1$: p \u003e 0.62 at$$ $$\alpha = 0.05. $$What is the test statistic?

$$z = \frac{0.68 - 0.62}{\sqrt{0.62 \times 0.38 / 350}} = 2.31$$

Question 1

Which of the following statements best describes the null hypothesis ($$H_0) in a $$hypothesis test for a population mean?

Accepted Answer

It is the claim that the population mean is equal to, less than, or greater than a specific value, representing the status quo.

Question 2

A company claims that the average lifetime of its light bulbs is at least 8,000 hours. An independent lab tests a sample and finds a mean of 7,850 hours with a known population standard deviation of 200 hours and a sample size of 36. Which is the correct set of hypotheses for this test?

Accepted Answer

$$H_0$$: \mu \geq 8,000$$; H_1$: \mu \u003c 8,000$$

Question 3

Which formula should be used to calculate the test statistic for a one-sample $$z-$$test when the population standard deviation is known?

Accepted Answer

$$z = \frac{\bar{x} - \mu_0$}{\sigma / \sqrt{n}}$$

Question 4

A researcher wants to test if the average monthly utility bill has decreased from last year’s mean of $$\186$. A sample of 40 customers has a mean of $$\$$178.10$$ and a sample standard deviation of $$\22.40$. Which test should be used?

Accepted Answer

One-sample $$t-$$test

Question 5

If the significance level $$\alpha is $$decreased from $$0.05 to$$ $$0.01$$, what is the effect on the probability of a Type II error ($$\beta$$), assuming the sample size remains the same?

Accepted Answer

Type II error probability ($$\beta) $$increases.

Question 6

A two-tailed hypothesis test is performed at $$\alpha = 0.05$$ with a calculated $$z-$$statistic of $$2.10. $$What is the correct decision rule and conclusion?

Accepted Answer

Reject $$H_0 if$$ $$|z| \u003e 1.96$$; since $$2.10 \u003e 1.96$$, reject $$H_0.$$

Question 7

Which of the following is a correct interpretation of a p-value in hypothesis testing?

Accepted Answer

The probability of obtaining a test statistic as extreme as the observed value, assuming $$H_0 is $$true.

Question 8

A sample of 350 cell phone users finds that 238 have 4G contracts. Last year, the proportion was $$p = 0.62. $$Test $$H_0$$: p \leq 0.62$$ vs. H_1$: p \u003e 0.62 at$$ $$\alpha = 0.05. $$What is the test statistic?

Accepted Answer

$$z = \frac{0.68 - 0.62}{\sqrt{0.62 	imes 0.38 / 350}} = 2.31$$

Question 9

Which of the following conditions must be met to use the normal approximation for a hypothesis test about a population proportion?

Accepted Answer

$$np \geq 5$$ and $$n(1-p) \geq 5$$

Question 10

A Type I error in hypothesis testing is:

Accepted Answer

Rejecting $$H_0$$ when $$H_0 is $$actually true.

Question 11

A sample of 25 hotels in Chicago has a mean nightly rate of $$\177.50$ and a sample standard deviation of $$\$$25.40. $$The population mean is claimed to be $$\188$. What is the test statistic for H_0$: \mu \geq 188 vs$$. $$H_1$$: \mu \u003c 188$?

Accepted Answer

$$t = \frac{177.5 - 188}{25.4 / \sqrt{25}} = -2.07$$

Question 12

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean approaches normality as the sample size increases, regardless of the population distribution.

Question 13

If a hypothesis test for a mean with $$\sigma$$ unknown yields a test statistic $$|t_{\bar{x}}| = 0.98$$ and the critical value is $$|t_{\alpha/2}| = 2.821$$, what is the correct conclusion?

Accepted Answer

Do not reject $$H_0$$ because $$|t_{\bar{x}}| \u003c |t_{\alpha/2}|.$$

Question 14

Which of the following is a correct decision rule for a right-tailed $$z-$$test for the mean?

Accepted Answer

Reject $$H_0 if$$ $$z_{\bar{x}} \u003e z_{\alpha}$$

Question 15

A business wants to know if the average class size has increased from 35 students. A sample of 38 classes has a mean of 36.9 and a standard deviation of 8. At $$\alpha = 0.05$$, what is the $$z-$$statistic?

Accepted Answer

$$z = \frac{36.9 - 35}{8 / \sqrt{38}} = 1.46$$

Question 16

Which of the following is true about the relationship between significance level ($$\alpha) $$and the power of a test?

Accepted Answer

Increasing $$\alpha$$ increases the power of the test, all else equal.

Chapter 9

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