# Problem 409

A county is considering the speed limit on a road because they claim that the mean speed of vehicle is greater than 50 miles per hour. A random sample of 27 vehicles has a mean speed of 53 miles per hour and a standard deviation of 4.1 miles per hour. At α = 0.01, do you have enough evidence to support the county’s claim? Complete parts (a) through (d) below.

Solution

(a)    Write the claim mathematically and identify and . A null hypothesis is a statistical hypothesis that contain a statement of equality, such as ≤ , = , or ≥. The alternative hypothesis is the complement of the null hypothesis. It is statement must be true if is false and it contain a statement of strict inequality, such as > , ≠ , or <.

The appropriate hypothesis are stated below. : ≤ 50 : > 50

(b)   Calculate the standardized test statistic.

First determine which type of test is being conduct. Since the county believes the average speed is greater than the posted limit, this is a right-tailed test.

Calculate the standardized test statistic. The standardized test statistic t is given by the formula below, where is the sample mean, µ is the hypothesis mean s is the sample standard deviation, and n is the sample size. Determine the standardized test statistic, rounding to two decimal places.   3.80

A P-value of a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data if the null hypothesis is true. While either the t-distribution table or technology could be used to determine the P-value, for this example, use technology.

Use technology to calculate the P-calculate the P-value for t 3.80, rounding to three decimal places.

P-value 0.000

(c)    Decimal whether to reject or fail to reject the null hypothesis. If  the P-value less than the level of significance, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Use the information above to decide whether to reject or fail to reject the null hypothesis.

(d)   Interpret the decision in the context of the original claim. If the null hypothesis is rejected then there is sufficient evidence to support the county’s claim that the mean speed of vehicles is greater than the posted speed limit. If the null hypothesis is not rejected then there is not sufficient evidence to support the claim that the mean vehicle speed is greater than the posted speed limit.

Use the information above to interpret the decision in the context if the original claim.