Exam 10: Hypothesis Testing

We are testing H₀: μ ≥ 2.5; versus H_A: μ < 2.5. When = 2.46, s = .05, and n = 26, at α = .10 we reject the null hypothesis. Assume that the population from which the sample is selected is normally distributed.

Using either the critical value rule or the p-value rule, if a one-sided null hypothesis is rejected at a given significance level, then the corresponding two-sided null hypothesis (i.e., the same sample size, the same standard deviation, and the same mean) will ________ be rejected at the same significance level.

When we test H₀: p = .2; versus H_A: p ≠.2, with = .26 and n = 100, at alpha = .05, we reject the null hypothesis.

The manager of the quality department for a tire manufacturing company wants to know the average tensile strength of rubber used in making a certain brand of radial tire. The population is normally distributed and the population standard deviation is known. She uses a z test to test the null hypothesis that the mean tensile strength is 800 pounds per square inch. The calculated z test statistic is a positive value that leads to a p-value of .045 for the test. If the significance level (α) for this test is .05, the null hypothesis would be rejected.

Consider an engine parts supplier and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are, respectively, 2.5 inches and .00075. To reduce this variance, a new machine is designed. A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches and a variance of s² = .0002 (normal distribution). In order for a cylindrical engine part to give an engine long life, the outside diameter must be between 2.43 and 2.57 inches. Using the upper end of the 95 percent confidence interval for σ and assuming that μ = 2.5, determine whether 99.73 percent of the outside diameters produced by the new machine are within the specification limits.

A Type I error is rejecting a true null hypothesis.

Based on a random sample of 25 units of product X, the average weight is 102 lb and the sample standard deviation is 10 lb. We would like to decide if there is enough evidence to establish that the average weight for the population of product X is greater than 100 lb. Assume the population is normally distributed. What is the critical value to test the claim at α = .01?

The local pharmacy prides itself on the accuracy of the number of tablets that are dispensed in a 60-count prescription. The new manager feels that the pharmacy assistants might have become careless in counting due to an increase in the volume of prescriptions. To test her theory, she randomly selects 40 prescriptions requiring 60 tablets and recounts the number in each bottle. She finds a sample mean of 61.35. Assume a population standard deviation of 4.45. If we have preset the probability of a Type I error equal to .05, compute the probability of a Type II error. (Assume an alternative value of the population mean of 61.) The claim is that the tablet count is different from 60.

When testing a hypothesis about a single mean, if the sample size is 51 and the population standard deviation is known, the correct test statistic to use is ________.

A manufacturer of an automobile part has a process that is designed to produce the part with a target of 2.5 inches in length. In the past, the standard deviation of the length has been .035 inches. In an effort to reduce the variation in the process, the manufacturer has redesigned the process. A sample of 25 parts produced under the new process shows a sample standard deviation of .025 inches. Test the claim that the new process standard deviation has improved from the current process at α = .05.

Last year, during an investigation of the time spent reading emails on a daily basis, researchers found that on Monday the average time was 50 minutes. Office workers claim that with the increased spam and junk mail, this time has now increased. To conduct a test, a sample of 25 employees is selected, with the following results: sample mean = 51.05 minutes and sample standard deviation = 14.2 minutes. What is the critical value for α = .05 to test the hypotheses and apply the critical value rule to determine if the null hypothesis can be rejected.

A major airline company is concerned that its proportion of late arrivals has substantially increased in the past month. Historical data shows that on average 18 percent of the company airplanes have arrived late. In a random sample of 1,240 airplanes, 310 airplanes have arrived late. If we are conducting a hypothesis test using the critical value rule to determine if the proportion of late arrivals has increased, what is the value of the calculated test statistic?

A recent study conducted by the state government attempts to determine whether the voting public supports a further increase in cigarette taxes. The opinion poll recently sampled 1,500 voting age citizens. 1,020 of the sampled citizens were in favor of an increase in cigarette taxes. The state government would like to decide if there is enough evidence to establish whether the proportion of citizens supporting an increase in cigarette taxes is significantly greater than .66. What is the alternative hypothesis?

In a bottling process, a manufacturer will lose money if the bottles contain either more or less than is claimed on the label. Suppose a quality manager for a steak sauce company is interested in testing whether the mean number of ounces of steak sauce per restaurant-size bottle differs from the labeled amount of 20 ounces. The manager samples nine bottles, measures their contents, and finds the sample mean is 19.7 ounces and the sample standard deviation is .3 ounces. Test the hypotheses at α = .05 using the critical value rule.

Using a χ² test statistic to test the null hypothesis that the variance of a new process is equal to the variance of the current process and when applying the p-value rule, we are rejecting at a p-value less than α, we can conclude that the new process is more consistent than the current process.

The local pharmacy prides itself on the accuracy of the number of tablets that are dispensed in a 60-count prescription. The new manager feels that the pharmacy assistants might have become careless in counting due to an increase in the volume of prescriptions. To test her theory, she randomly selects 40 prescriptions requiring 60 tablets and recounts the number in each bottle. She finds a sample mean of 61.35. Assume a population standard deviation of 4.45. If we want the probability of a Type I error and Type II error to be equal to .05, what is the sample size needed to make both the probability of a Type I error and the probability of a Type II error as small as possible. (Assume an alternative value of the population mean of 61.) The claim is that the tablet count is different from 60.

Using the p-value rule, if a null hypothesis is rejected at a significance level of .01, it will ________ be rejected at a significance level of .05.

If you live in California, the decision to buy earthquake insurance is an important one. A survey revealed that only 133 of 337 randomly selected residences in one California county were protected by earthquake insurance. Calculate the appropriate test statistic to test the claim that less than 40 percent of the residents in the county are protected by earthquake insurance.

For a fixed sample size, the lower we set α, the higher is the ________.

Rejecting a true null hypothesis is called a ________ error.