Question

Use this information to answers Questions 1 through 7. The U.S. government provides money to each state to maintain the interstate highway system in the state. The U.S. can revoke or reduce the money if the states do not safely maintain the highways. The U.S. government is particularly concerned about the speed of traffic on Kansas highways. If there is convincing evidence that the average speed of all interstate highway vehicles in Kansas exceeds the posted speed limit of 70 mph, the federal government will reduce the amount of funding it provides. If there is not convincing evidence, then the government will not reduce the funding.

Kansas Highway Patrol recorded the speed of 450 interstate vehicles and found the mean speed of 70.2 mph with standard deviation 1.6 mph. The U.S. government will use this information to conduct a hypothesis test at significance level 0.01 to decide whether or not to reduce to money sent to Kansas.

Suppose that mu is the true mean speed of all vehicles on the Kansas interstate highway system.

1. What are the null and alternative hypotheses that the U.S. government should test?
2. What is the value of the test statistic?

Answer 1

1. Null Hypothesis,
`H_0` :
`\mu \leq` 70 mph

Alternate Hypothesis,
`H_A` :
`\mu` > 70 mph

2. Value of test statistics is 2.652.

Explanation:

We are given that Kansas Highway Patrol recorded the speed of 450 interstate vehicles and found the mean speed of 70.2 mph with standard deviation 1.6 mph.

We have to conduct a hypothesis test at significance level 0.01 to decide whether or not to reduce the money sent to Kansas.

Let
`\mu` = true mean speed of all vehicles on the Kansas interstate highway system.

SO, Null Hypothesis,
`H_0` :
`\mu \leq` 70 mph {means that the federal government will not reduce the amount of funding it provides as the speed limit is less than or equal to 70 mph}

Alternate Hypothesis,
`H_A` :
`\mu` > 70 mph {means that the federal government will reduce the amount of funding it provides as the speed limit exceed 70 mph}

The test statistics that will be used here is One-sample t test statistics as we don't know about the population standard deviation;

T.S. =
`(\bar X -\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean speed limit of 450 interstate vehicles = 70.2 mph

s = sample standard deviation = 1.6 mph

n = sample of vehicles = 450

So, test statistics =
`(70.2-70)/((1.6)/(√(450) ) )` ~
`t_4_4_9`

= 2.652

Hence, the value of test statistics is 2.652.