Question 1

Which is a valid null hypothesis?

Accepted Answer

A valid null hypothesis is always an equality statement, therefore option B is the only valid null hypothesis as it states that the population mean is equal to 18. Options A, C and D are alternative hypotheses that make a claim about the population mean being different, greater or lesser than 18, respectively.

Question 2

A two-tailed hypothesis test for H₀: π = .30 at α = .05 is analogous to:

Accepted Answer

A two-tailed hypothesis test at a significance level of α = .05 is analogous to asking if the 95 percent confidence interval for π contains .30. This is because the confidence level (95%) plus the significance level (5%) sums to 100%, and the confidence interval approach and the hypothesis testing approach are conceptually linked in this way.

Question 3

The Melodic Kortholt Company will change its current health plan if at least half the employees are dissatisfied with it. A trial sample of 25 employees shows that 16 are dissatisfied. In this problem:

Accepted Answer

The sample size (25) is large enough to assume normality of the sample proportion (np=25*0.5=12.5 and nq=25*0.5=12.5 are both greater than 10). Therefore, we can use the normal distribution to make inferences about the population proportion.

Question 4

If sample size increases from 25 to 100 and the level of significance stays the same, then:

Accepted Answer

Increasing the sample size generally leads to a decrease in the risk of a Type II error, as a larger sample size provides more information about the population, making it easier to detect a true effect if one exists. The level of significance (alpha) directly controls the risk of a Type I error, and it remains unchanged regardless of sample size.

Question 5

In a two-tailed test, a statistician got a z test statistic of 1.47. What is the p-value?

Accepted Answer

In a two-tailed test, the p-value is calculated by finding the probability of getting a z score as extreme as the test statistic in both tails of the distribution. For a z score of 1.47, the one-tailed probability is approximately 0.0708. Since it's a two-tailed test, this value is doubled, resulting in a p-value of approximately 0.1416.

Question 6

The Melodic Kortholt Company will change its current health plan if at least half the employees are dissatisfied with it. A trial sample of 25 employees shows that 16 are dissatisfied. For a right-tailed test, the test statistic would be:

Accepted Answer

The answer of The Melodic Kortholt Company will change its...

Question 7

The hypotheses H₀: π ≥ .40, H₁: π < .40 would require:

Accepted Answer

The hypotheses state that the true proportion (π) is either greater than or equal to 0.40 (H₀: π ≥ .40) or less than 0.40 (H₁: π < .40). Since the alternative hypothesis (H₁) is less than the null hypothesis value (.40), this implies a left-tailed test. Therefore, the best choice is A) a left-tailed test.

Question 8

High power in a hypothesis test about one sample mean is likely to be associated with:

Accepted Answer

High power in a hypothesis test is associated with a smaller standard deviation (σ), as this reduces the variability in the data, making it easier to detect a true effect if one exists.

Question 9

Which of the following statements is true?

Accepted Answer

The power of a test is the probability of correctly rejecting a false null hypothesis. The distance between the true mean and the hypothesized mean is a factor that affects the power of the test, so if the true mean is farther from the hypothesized mean, the power of the test increases. Option A is incorrect because decreasing type I error rate (α) will decrease the power of the test. Option B is incorrect because doubling the sample size will increase the power of the test by a factor of four, not by two. Option C is incorrect because a higher standard deviation will decrease the power of the test.

Question 10

Which of the following decisions could result in a Type II error for a test?

Accepted Answer

Type II error refers to the incorrect conclusion of retaining the null hypothesis when the alternative hypothesis is true. Therefore, failing to reject the null hypothesis (option C) could lead to a type II error.

Question 11

"Currently, only 20 percent of arrested drug pushers are convicted," cried candidate Courageous Calvin in a campaign speech. "Elect me and you'll see a big increase in convictions." A year after his election a random sample of 144 case files of arrested drug pushers showed 36 convictions. For a right-tailed test, the p-value is approximately:

Accepted Answer

We can use a one-sample proportion test to test if the proportion of convicted drug pushers has increased from the previous rate of 20%.

Under the null hypothesis that the conviction rate is still 20%, the expected number of convictions out of 144 cases would be:

E = 0.2 * 144 = 28.8

The standard deviation of the sampling distribution can be calculated as:

sqrt(var) = sqrt((0.2 * 0.8) / 144) = 0.0354

Using the observed number of convictions, we can calculate the test statistic:

z = (36 - 28.8) / 0.0354 = 204.5

The p-value for a right-tailed test is the probability of observing a z-score greater than or equal to 204.5 under the standard normal distribution. This is an extremely small value, well below any of the choices provided.

However, since the p-value is less than 0.5, we can instead report the complement of the p-value as:

p-value = P(z > 204.5) = 1 - P(z <= 204.5) < 1 - P(z <= 3.34) = 0.0004

Rounding to four decimal places, the closest choice is B: 0.0668.

Question 12

In a test of a mean, the reported p-value is .025. Using α =.05 the conclusion would be to:

Accepted Answer

The p-value is less than the chosen significance level of α=0.05, so we can reject the null hypothesis at the 5% level of significance. Therefore, we have evidence of a significant difference between the sample mean and the hypothesized population mean.

Question 13

The power of a test is the probability of:

Accepted Answer

The power of a test is the probability of rejecting the null hypothesis when it is false.

Question 14

Which is an invalid alternative hypothesis?

Accepted Answer

An alternative hypothesis cannot be a statement of equality (i.e. "μ = 18"), as it does not provide a direction of difference between the null and alternative hypotheses. The correct alternative hypotheses should reference a direction of difference (e.g. "μ > 18" or "μ < 18") or simply state that there is a difference without specifying a direction (e.g. "μ ≠ 18").

Question 15

At α = .05, the critical value to test the hypotheses H₀: π ≥ .40, H₁: π < .40 would be:

Accepted Answer

For a one-tailed test at α = .05, the critical value for testing a hypothesis about a population proportion (π) when the alternative hypothesis (H₁) is π < .40 (indicating a left-tailed test) is -1.645. This value corresponds to the z-score that cuts off the lower 5% of the standard normal distribution.

Question 16

In a left-tailed test, a statistician got a z test statistic of -1.720. What is the p-value?

Accepted Answer

For a left-tailed test, we need to find the area to the left of the test statistic (-1.720) on the standard normal distribution. We can look this up on a standard normal distribution table or use technology to find the p-value. Using a standard normal distribution table, we find that the area to the left of -1.72 is 0.0427. Therefore, the p-value is 0.0427. So, the answer is (C).

Question 17

The Melodic Kortholt Company will change its current health plan if at least half the employees are dissatisfied with it. A trial sample of 25 employees shows that 16 are dissatisfied. The p-value for a right-tailed test is:

Accepted Answer

The p-value of .0808 suggests that there is an 8.08% chance of observing a sample with 16 or more dissatisfied employees out of 25 if the true proportion of dissatisfied employees in the company does not exceed half. This relatively low p-value indicates evidence against the null hypothesis (that half or fewer are dissatisfied), supporting the consideration for changing the health plan.

Question 18

Which is not a step in hypothesis testing?

Accepted Answer

Finding the test statistic from a table is not a step in hypothesis testing. The test statistic is either computed using statistical software or calculated by hand using the formula for the appropriate test. The steps in hypothesis testing are as follows: 
1. Formulate the hypotheses. 
2. Specify the desired Type I error. 
3. Choose the appropriate test and calculate the test statistic. 
4. Determine the p-value or critical value. 
5. Formulate a decision rule. 
6. Make a decision and interpret the results.

Question 19

In the nation of Gondor, the EPA requires that half the new cars sold will meet a certain particulate emission standard a year later. A sample of 64 one-year-old cars revealed that only 24 met the particulate emission standard. The test statistic to see whether the proportion is below the requirement is:

Accepted Answer

The test statistic is calculated using the formula for a proportion test: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size. Here, $\hat{p} = \frac{24}{64} = 0.375$, $p_0 = 0.5$, and $n = 64$. Plugging these values into the formula gives $z = \frac{0.375 - 0.5}{\sqrt{\frac{0.5(0.5)}{64}}} = -2.000$.

Question 20

In a right-tailed test, a statistician got a z test statistic of 1.47. What is the p-value?

Accepted Answer

The p-value in a right-tailed test corresponds to the area to the right of the z test statistic on the standard normal distribution. For a z test statistic of 1.47, the area to the left is approximately .9292, so the area to the right (the p-value) is 1 - .9292 = .0708.

Quiz 9: One-Sample Hypothesis Tests