Exam 10: Hypothesis Testing

Standard X-ray machines should give radiation dosages below 5.00 milliroentgens (mR). To test a certain X-ray machine, a sample of 36 observations is taken, with a mean of 4.13 mR and a population standard deviation of 1.91 mR. Calculate the appropriate test statistic to test the hypotheses using the critical value rule.

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. Calculate the appropriate test statistic to test the hypotheses using the critical value rule.

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 95 percent confidence interval for σ² at α = .05?

We are testing H₀: μ = 32; versus H_A: μ > 32. If = 36, s = 1.6, and n = 30, at α = .05, we should reject H₀. Assume that the population from which the sample is selected is normally distributed.

Failure to meet payments on student loans guaranteed by the US government has been a major problem for both banks and the government. Approximately 50 percent of all student loans guaranteed by the government are in default. A random sample of 350 loans to college students in one region of the United States indicates that 147 are in default. Calculate the p-value associated with the test statistic, and, using the p-value rule, test the hypothesis that the default rate for the selected region is lower than the national percentage at α = .01.

χ²_α is the point on the vertical axis under the curve of the chi-square distribution that gives a right-hand tail area equal to α.

As the level of significance, α, increases, we are more likely to reject the null hypothesis.

A manufacturer of salad dressings uses machines to dispense liquid ingredients into bottles that move along a filling line. The machine that dispenses dressing is working properly when 8 ounces are dispensed. The standard deviation of the process is .15 ounces. A sample of 48 bottles is selected periodically, and the filling line is stopped if there is evidence that the mean amount dispensed is different from 8 ounces. Suppose that the mean amount dispensed in a particular sample of 48 bottles is 7.983 ounces. Calculate the p-value and determine if the process is working properly when testing at α = .10 using the p-value rule.

As the Type II error, β, of a statistical test increases, the power of the test ________.

The ________ hypothesis is not rejected unless there is sufficient sample evidence to do so.

A company has developed a new ink-jet cartridge for its printer that it believes has a longer lifetime on average than the one currently being produced. To investigate its length of life, 40 of the new cartridges were tested by counting the number of high-quality printed pages each was able to produce. The sample mean and standard deviation were determined to be 1511.4 pages and 35.7 pages, respectively. The historical average lifetime for cartridges produced by the current process is 1502.5 pages. Calculate the appropriate test statistic to test the hypotheses using the critical value rule.

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. Assuming normality, determine whether 99.73 percent of the outside diameters produced by the current machine are within specification limits.

After testing H₀: p = .33; versus H_A: p < .33; at α = .05, with = .20 and n = 100, we do not reject H₀.

When computing a 95 percent confidence interval for σ² with a sample of n = 30, what values of χ² would we use in the calculations?

A random sample of 80 companies who announced corrections to their balance sheets took a mean time of 8.1 days for the time between balance sheet construction and the complete audit. The population standard deviation is assumed to be 1.3 days. If the claim is that the population mean is greater than 7.5 days, calculate the appropriate test statistic to test the hypotheses.

A manufacturer of salad dressings uses machines to dispense liquid ingredients into bottles that move along a filling line. The machine that dispenses dressing is working properly when 8 ounces are dispensed. The standard deviation of the process is .15 ounces. A sample of 48 bottles is selected periodically, and the filling line is stopped if there is evidence that the mean amount dispensed is different from 8 ounces. Suppose that the mean amount dispensed in a particular sample of 48 bottles is 7.983 ounces. Calculate a confidence interval to test the hypotheses at α = .05 and determine if the process should be stopped.

Last year, TV station KAAA had a share of the 6 p.m. news audience approximately equal to, but not greater than, 25 percent. The advertising department for the station believes the current audience share is higher than the 25 percent share they had last year. In an attempt to substantiate this belief, the station surveyed a random sample of 400 viewers of 6 p.m. news and found that 146 watched KAAA. Calculate the appropriate test statistic to test the hypotheses using the critical value rule.

A company has developed a new ink-jet cartridge for its printer that it believes has a longer lifetime on average than the one currently being produced. To investigate its length of life, 40 of the new cartridges were tested by counting the number of high-quality printed pages each was able to produce. The sample mean and standard deviation were determined to be 1511.4 pages and 35.7 pages, respectively. The historical average lifetime for cartridges produced by the current process is 1502.5 pages. At α = .05, test the claim that the new cartridge has a longer lifetime using the critical value rule.

We are testing H₀: p ≥ .7; versus H_A: p < .7. With = .63 and n = 100, at α = .01, we do not reject the null hypothesis.

A mail-order business prides itself in its ability to fill 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. What is the critical value for α = .10 when using the critical value rule to test the hypothesis that the time to fill orders is less than 6 days?