Exam 10: Hypothesis Testing

A cereal manufacturer is concerned that the boxes of cereal not be underfilled or overfilled. Each box of cereal is supposed to contain 13 ounces of cereal. A random sample of 31 boxes is tested. The average weight is 12.58 ounces and the standard deviation is .25 ounces. What is the critical value for testing the hypotheses that the mean cereal weight is different from 13 ounces at α = .001.

When applying either the critical value rule or the p-value rule about a single mean, reducing the level of significance (α) will increase the size of the rejection region.

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 whether 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. Therefore, one way of expressing the null hypothesis is H₀: μ = 100.

You cannot make a Type II error when the null hypothesis is true.

A major car manufacturer wants to test a new catalytic converter to determine whether it meets new air pollution standards. The mean emission of all converters of this type must be less than 20 parts per million of carbon monoxide. Ten converters are manufactured for testing purposes and their emission levels are measured, with a mean of 17.17 and a standard deviation of 2.98. Test the hypotheses at α = .01 using the critical value rule.

A null hypothesis is not rejected at a given level of significance. As the assumed value of the mean gets further away from the true population mean, the Type II error will ________.

Using the p-value rule, if a null hypothesis is not rejected at a significance level of .05, it will ________ be rejected at a significance level of .01.

To compute a 95 percent confidence interval for σ², we use n − 1 degrees of freedom and the chi-square points on the distribution curve of χ²_α/2 and of χ²_1−(α/2).

A baker must monitor the temperature at which cookies are baked. Too much variation will cause inconsistency in the texture of the cookies. Past records show that the variance of the temperatures has been 1.44 degrees. A random sample of 30 batches of cookies is selected, and the sample variance of the temperature is 4.41 degrees. What is the test statistic for testing the null hypothesis that the population variance has increased above 1.44 degrees?

It has been hypothesized that, on average, employees spend one hour a day playing video games at work. To test this at her company, a manager takes a random sample of 35 employees, who showed a mean time of 55 minutes per day, with an assumed population standard deviation of 5 minutes. Calculate the test statistic.

According to a national survey, the average commuting time for people living in a city with a population of 1 to 3 million is 19.0 minutes. Suppose a researcher lives in a city with a population of 2.4 million and wants to test this claim for her city. Taking a random sample of 20 commuters, she calculates a mean time of 19.346 minutes and a standard deviation of 2.842 minutes. Test the hypotheses that the commuting time in this city is different from the average 19 minutes at α = .10 using the critical value rule.

In a study of distances traveled by buses before the first major engine failure, a sample of 191 buses results in a mean of 96,700 miles and a population standard deviation of 37,500 miles. Calculate the appropriate test statistic to test the claim that the mean distance traveled before a major engine failure is more than 90,000 miles.

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 whether 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 value of the test statistic to test the claim?

A mail-order business prides itself in its ability to fill customers' orders in less than six calendar days, on average. Periodically, the operations manager selects a random sample of customer orders and determines the number of days required to fill the orders. On one occasion when a sample of 39 orders was selected, the average number of days was 6.65 with a population standard deviation of 1.5 days. Calculate the appropriate test statistic to test the hypotheses.

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 = 4.42 minutes. Calculate the appropriate test statistic to apply the critical value rule on the hypothesis that the time has increased from last year.

When applying either the critical value rule or the p-value rule about a single mean, other relevant factors held constant, increasing the level of significance, α, from .05 to .10 will reduce the probability of a Type I error.

A Type II error is defined as ________ H₀, when it should ________.

A cereal manufacturer is concerned that the boxes of cereal not be underfilled or overfilled. Each box of cereal is supposed to contain 13 ounces of cereal. A random sample of 31 boxes is tested. The average weight is 12.58 ounces and the standard deviation is .25 ounces. Calculate a confidence interval to test the hypotheses that the mean cereal weight is different from 13 ounces at α = .10 and interpret.

The HR department tested how long employees stay with the company in their current positions. A random sample of 50 employees yielded a mean of 2.79 years and σ = .76. The sample evidence indicates that the average time is less than 3 years and is significant at α = .01.

The manager of a grocery store wants to determine whether the amount of water contained in 1-gallon bottles purchased from a nationally known manufacturer actually averages 1 gallon. It is known from the specifications that the standard deviation of the amount of water is equal to .02 gallon. A random sample of 32 bottles is selected, and the mean amount of water per 1-gallon bottle is found to be .995 gallon. Calculate the test statistic.