The South Carolina Department of Natural Resources personnel enforce hunting and fishing regulations and conduct routine safety checks on...

The South Carolina Department of Natural Resources personnel enforce hunting and fishing regulations and conduct routine safety checks on recreational boats. Past experience indicates that 15% of all boats inspected have at least one safety violation. Recent accidents on lakes in South Carolina have prompted calls for more extensive inspections. A random sample of recreational boats will be obtained and inspected. If there is evidence that the true proportion of boats with safety violations, p, is more than 15%, a methodical inspection of every boat launched at popular sites will be started. Answer the following three questions: 

a. State the null and alternative hypotheses in terms of p.

Ho: p is less than 0.15; Ha: p is greater than 0.15

Ho: p is greater than 0.85; Ha: p is equal to 0.85

Ho: p is equal to 0.15; Ha: p is greater than 0.15

Ho: p is equal to 0.15; Ha: p is equal to 0.15

b. Describe a type I error and a type II error in this context.

A type I error: decide p is greater than 0.15 when the true proportion is really 0.15 (or less). Type II error: decide p = 0.15 (or less) when the true proportion is really smaller than 0.15.

A type I error: decide p is greater than 0.15 when the true proportion is really 0.15 (or less). Type II error: decide p = 0.15 (or less) when the true proportion is really greater than 0.15.

A type I error: decide p is less than 0.15 when the true proportion is really 0.15 (or more). Type II error: decide p = 0.15 (or less) when the true proportion is really greater than 0.15.

A type I error: decide p = 0.15 (or less) when the true proportion is really greater than 0.15. Type II error: decide p is greater than 0.15 when the true proportion is really 0.15 (or less).

c. What happens to the probability of a type I error as the value of p approaches 0.15 from the right (that is, 0.20, 0.19, . . .)?

The probability of a type I error becomes greater.

Type I error is not defined in this region.

The probability of a type I error becomes smaller.