# Testing Hypotheses: One Sample Tests

How many standard errors around the hypothesized value should we use to be 99.44 percent certain that we accept the hypothesis when it is true?

An automobile manufacturer claims that a particular model gets 28 miles to the gallon. The Environmental Protection Agency, using a sample of 49 automobiles of this model, finds the sample mean to be 26.8 miles per gallon. From previous studies, the population standard deviation is known to be 5 miles per gallon. Could we reasonably expect (within 2 standard errors) that we could select such a sample if indeed the population mean is actually 28 miles per gallon

If we reject a hypothesized value because it differs from a sample statistic by more than 1.75 standard errors, what is the probability that we have rejected a hypothesis that is in fact true?

How many standard errors around the hypothesized value should we use to be 98 percent certain that we accept the hypothesis when it is true?

Sports and media magnate Ned Sterner is interested in purchasing the Atlanta Stalwarts if he can be reasonably certain that operating the team will not be too costly. He figures that average attendance would have to be about 28,500 fans per game to make the purchase attractive to him. Ned randomly chooses 64 home games over the past 4 years and finds from figures reported in Sporting Reviews that average attendance at these games was 26,100. A study he commissioned the last time he purchased a team showed that the population standard deviation for attendance at similar events had been quite stable for the past 10 years at about 6,000 fans. Using 2 standard errors as the decision criterion, should Ned purchase the Stalwarts? Can you think of any reason(s) why your conclusion might not be valid?

Computing World has asserted that the amount of time owners of personal computers spend on their machines averages 23.9 hours per week and has a standard deviation of 12.6 hours per week. A random sampling of 81 of its subscribers revealed a sample mean usage of 27.2 hours per week. On the basis of this sample, is it reasonable to conclude (using 2 standard errors as the decision criterion) that Computing World’s subscribers are different from average personal computer owners?

A grocery store has specially packaged oranges and has claimed a bag of oranges will yield 2.5 quarts of juice. After randomly selecting 42 bags, a stacker found the average juice production per bag to be 2.2 quarts. Historically, we know the population standard deviation is 0.2 quart. Using this sample and a decision criterion of 2.5 standard errors, could we conclude the store’s claims are correct?

For the following cases, specify which probability distribution to use in a hypothesis test: (a) H 0: μ = 27, H1: μ ≠ 27, x = 33, ˆσ = 4, n = 25. (b) H 0: μ = 98.6, H1: μ > 98.6, x = 99.1, σ = 1.5, n = 50. (c) H0: μ = 3.5, H1: μ < 3.5, x = 2.8, ˆσ = 0.6, n = 18. (d) H 0: μ = 382, H1: μ ≠ 382, x = 363, σ = 68, n = 12. (e) H 0: μ = 57, H1: μ > 57, x = 65, ˆ σ = 12, n = 42.

Martha Inman, a highway safety engineer, decides to test the load-bearing capacity of a bridge that is 20 years old. Considerable data are available from similar tests on the same type of bridge. Which is appropriate, a one-tailed or a two-tailed test? If the minimum load-bearing capacity of this bridge must be 10 tons, what are the null and alternative hypotheses?

Formulate null and alternative hypotheses to test whether the mean annual snowfall in Buffalo, New York, exceeds 45 inches.