Question 1

When using the p-value method for a two-tailed hypothesis, the p-value is found by finding the area in the tail beyond the test statistic, then doubling it.

Accepted Answer

In a two-tailed hypothesis test, the p-value is calculated by finding the area in one tail (beyond the test statistic) of the probability distribution and then doubling this area to account for both tails of the distribution. This reflects the probability of observing a test statistic as extreme as, or more extreme than, the one observed under the null hypothesis.

Question 2

When using the p-value method, the null hypothesis is rejected when the calculated p-value > α.

Accepted Answer

The null hypothesis is rejected when the calculated p-value is less than or equal to the significance level (α), not greater.

Question 3

A two-tailed hypothesis test is used when the null hypothesis looks like the following: H0 : = 100.

Accepted Answer

A two-tailed hypothesis test is used when we are interested in determining whether the parameter is not equal to the value specified in the null hypothesis (e.g., H0: μ ≠ 100), rather than testing if it is specifically greater than or less than that value.

Question 4

A company that makes and markets a device that is aimed at helping people quit smoking claims that at least 70 percent of the people who have used the product have quit smoking. To test this, a random sample of n = 100 product users was selected. Of these, 65 people were found to have quit smoking. Given these results, the test statistic value is z = -1.0911.

Accepted Answer

The test statistic is calculated using the formula for a proportion: z = (p̂ - p0) / √(p0(1-p0)/n), where p̂ = 0.65, p0 = 0.70, and n = 100. This results in z ≈ -1.0911.

Question 5

One claim states the IRS conducts audits for not more than 5 percent of total tax returns each year. In order to test this claim statistically, the appropriate null and alternative hypotheses are: H0 : μ ≤ 0.05 Ha : μ > 0.05

Accepted Answer

The correct null hypothesis should state the condition of equality or the status quo, which in this case is that the IRS audits at most 5 percent of total tax returns each year, so H0 should be μ = 0.05. The alternative hypothesis should reflect the claim being tested, which is that more than 5 percent of tax returns are audited, so Ha : μ > 0.05 is correct. However, the null hypothesis as presented (H0 : μ ≤ 0.05) incorrectly includes conditions both equal to and less than 5 percent, which is not appropriate for testing the specific claim that more than 5 percent are audited.

Question 6

A major package delivery company claims that at least 95 percent of the packages it delivers reach the destination on time. As part of the evidence in a lawsuit against the package company, a random sample of n = 200 packages was selected. A total of 188 of these packages were delivered on time. Using a significance level of .05, the test statistic for this test is approximately z = -0.65.

Accepted Answer

The test statistic is calculated as (188 - 190) / sqrt(200 * 0.05 * 0.95) = -0.65.

Question 7

When testing a hypothesis involving population proportions, an increase in sample size will result in a smaller chance of making a Type I statistical error.

Accepted Answer

Type I error (α) is predetermined by the significance level chosen for the test, not affected by sample size. Increasing sample size can reduce Type II error and increase test power.

Question 8

When deciding the null and alternative hypotheses, the rule of thumb is that if the claim contains the equality (e.g., at least, at most, no different from, etc.), the claim becomes the null hypothesis. If the claim does not contain the equality (e.g., less than, more than, different from), the claim is the alternative hypothesis.

Accepted Answer

The null hypothesis typically represents a theory that has been put forward, either because it is believed to be true or because it is to be used as a basis for argument, but has not been proved. It usually reflects the theory that one wishes to test and is formulated as a statement of no effect or no difference (i.e., equality). The alternative hypothesis, on the other hand, is the hypothesis that reflects the substantiation of the null hypothesis to be false or incorrect, often representing a new theory or an effect (i.e., it usually contains the inequality).

Question 9

When the hypothesized proportion is close to 0.50, the spread in the sampling distribution of is greater than when the hypothesized proportion is close to 0.0 or 1.0.

Accepted Answer

The spread in the sampling distribution of a proportion is determined by the formula for the standard error of a proportion, which is sqrt[p(1-p)/n], where p is the hypothesized proportion and n is the sample size. When p is close to 0.50, the product p(1-p) reaches its maximum value, resulting in a greater spread in the sampling distribution compared to when p is close to 0.0 or 1.0, where the product p(1-p) is smaller.

Question 10

A cell phone company believes that 90 percent of their customers are satisfied. They survey a sample of n = 100 customers and find that 82 say they are satisfied. In calculating the standard error of the sampling distribution (σp) the proportion to use is 0.82.

Accepted Answer

The standard error of the sampling distribution should be calculated using the hypothesized population proportion (0.90), not the sample proportion (0.82).

Question 11

Aceco has a contract with a supplier to ship parts that contain no more than three percent defects. When a large shipment of parts comes in, Aceco samples n = 150. Based on the results of the sample, they either accept the shipment or reject it. If Aceco wants no more than a 0.10 chance of rejecting a good shipment, the cut-off between accepting and rejecting should be 0.0478 or 4.78 percent of the sample.

Accepted Answer

The cut-off of 4.78% is calculated using statistical methods to ensure that the probability of rejecting a shipment with a true defect rate of 3% (considered a "good" shipment) does not exceed 10%. This involves using the binomial distribution and normal approximation to set an appropriate threshold.

Question 12

The executive director of the United Way believes that more than 24 percent of the employees in the high-tech industry have made voluntary contributions to the United Way. In order to test this statistically, the appropriate null and alternative hypotheses are: H0 : ≤ .24 HA : > .24

Accepted Answer

The correct null hypothesis (H0) should state there is no effect or no difference, so it would be H0: p = .24, and the alternative hypothesis (HA) should reflect the belief or claim being tested, so it should be HA: p > .24, indicating that the proportion is greater than 24%.

Question 13

Lube-Tech is a major chain whose primary business is performing lube and oil changes for passenger vehicles. The national operations manager has stated in an industry newsletter that the mean number of miles between oil changes for all passenger cars exceeds 4,200 miles. To test this, an industry group has selected a random sample of 100 vehicles that have come into a lube shop and determined the number of miles since the last oil change and lube. The sample mean was 4,278 and the standard deviation was known to be 780 miles. Based on this information, the p-value for the hypothesis test is less than 0.10.

Accepted Answer

The answer of Lube-Tech is a major chain whose primary...

Question 14

Lube-Tech is a major chain whose primary business is performing lube and oil changes for passenger vehicles. The national operations manager has stated in an industry newsletter that the mean number of miles between oil changes for all passenger cars exceeds 4,200 miles. To test this, an industry group has selected a random sample of 100 vehicles that have come into a lube shop and determined the number of miles since the last oil change and lube. The sample mean was 4,278 and the sample standard deviation was 780 miles. Based on this information, the test statistic is approximately t = 1.000.

Accepted Answer

The test statistic is calculated using the formula $t = \frac{\bar{x} - \mu}{(s/\sqrt{n})}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean under the null hypothesis, $s$ is the sample standard deviation, and $n$ is the sample size. Plugging in the given values ($\bar{x} = 4278$, $\mu = 4200$, $s = 780$, $n = 100$), the calculation yields a test statistic close to 1.

Question 15

Generally, it is possible to appropriately test a null and alternative hypotheses using the test statistic approach and reach a different conclusion than would be reached if the p-value approach were used.

Accepted Answer

Both the test statistic approach and the p-value approach are methods used to determine whether to reject the null hypothesis, and they should lead to the same conclusion when applied correctly to the same dataset and hypothesis test.

Question 16

A company that makes and markets a device that is aimed at helping people quit smoking claims that at least 70 percent of the people who have used the product have quit smoking. To test this, a random sample of n = 100 product users was selected. The critical value for the hypothesis test using a significance level of 0.05 would be approximately -1.645.

Accepted Answer

The critical value for a one-tailed test with a significance level of 0.05 is approximately -1.645.

Question 17

Lube-Tech is a major chain whose primary business is performing lube and oil changes for passenger vehicles. The national operations manager has stated in an industry newsletter that the mean number of miles between oil changes for all passenger cars exceeds 4,200 miles. To test this, an industry group has selected a random sample of 100 vehicles that have come into a lube shop and determined the number of miles since the last oil change and lube. The sample mean was 4,278 and the standard deviation was known to be 780 miles. Based on a significance level of 0.10, the critical value for the test is approximately z = 1.28.

Accepted Answer

Given the sample mean, known standard deviation, and the significance level, the calculation for the z-score aligns with standard statistical methods for hypothesis testing, making the critical value of approximately z = 1.28 reasonable for a one-tailed test at the 0.10 significance level.

Question 18

A major package delivery company claims that at least 95 percent of the packages it delivers reach the destination on time. As part of the evidence in a lawsuit against the package company, a random sample of n = 200 packages was selected. A total of 188 of these packages were delivered on time. Using a significance level of 0.05, the critical value for this hypothesis test is approximately 0.90.

Accepted Answer

The critical value in hypothesis testing is typically found from a statistical distribution (e.g., z-distribution, t-distribution) and represents the threshold for rejecting the null hypothesis. It is not calculated as a percentage like 0.90 but rather as a value from these distributions corresponding to the chosen significance level (e.g., z = 1.96 for a two-tailed test with α = 0.05). The statement mixes up concepts related to hypothesis testing.

Question 19

A cell phone company believes that 90 percent of its customers are satisfied with their service. They survey n = 30 customers. Based on this, it is acceptable to assume the sample distribution is normally distributed.

Accepted Answer

The sample size of 30 is too small to assume a normal distribution for a proportion, especially when the expected number of successes (np) and failures (n(1-p)) are not both greater than 5.

Question 20

For testing a research hypothesis, the burden of proof that a new product is no better than the original is placed on the new product, and the research hypothesis is formulated as the null hypothesis.

Accepted Answer

The null hypothesis typically assumes no difference or effect, so it would state that the new product is no better than the original. The research hypothesis (alternative hypothesis) posits that there is a difference, suggesting the new product is better. The burden of proof is on showing the research hypothesis is true, not the null.

Quiz 9: Introduction to Hypothesis Testing