Question

It is advertised that the average braking distance for a small car traveling at 75 miles per hour equals 124 feet. A transportation researcher wants to determine if the statement made in the advertisement is false. She randomly test drives 37 small cars at 75 miles per hour and records the braking distance. The sample average braking distance is computed as 112 feet. Assume that the population standard deviation is 22 feet.

Answer 1

The researcher is using a hypothesis test to compare the sampled average braking distance of 37 small cars at 112 feet with the advertised distance of 124 feet to determine if there is a significant difference, using a known population standard deviation of 22 feet.

Step-by-step explanation:

The transportation researcher's goal is to determine if the advertised average braking distance for a small car traveling at 75 miles per hour, which is 124 feet, is accurate. To test this, she has conducted an experiment with a sample of 37 small cars at the same speed and found a sample average braking distance of 112 feet. Given that the population standard deviation is known to be 22 feet, statistical analysis can be used to infer whether the difference between the sample mean and the advertised mean is statistically significant.

This analysis would typically involve constructing a hypothesis test using the sample data, where the null hypothesis assumes that the true mean is equal to the advertised mean (124 feet), and the alternative hypothesis suggests that the true mean is different. The researcher would calculate the test statistic using the sample mean, population standard deviation, and sample size, and then determine a p-value to assess the evidence against the null hypothesis.

Answer 2

`z=(112-124)/((22)/(√(37)))=-3.318`

The p value would be given by this probability:

`p_v =2*P(z<-3.318)=0.0009`

Since the p value is a very small value at any significance level used we can reject the null hypothesis and we can conclude that the true mean for this case is different from 124 ft

Step-by-step explanation:

Data given and notation

`\bar X=112` represent the sample mean

`\sigma =22` represent the population standard deviation

`n=37` sample size

`\mu_o =124` represent the value that we want to test

z would represent the statistic (variable of interest)

`p_v` represent the p value

Hypothesis to test

We want to check the following system of hypothesis:

Null hypothesis:
`\mu = 124`

Alternative hypothesis :
`\mu \\eq 124`

The statistic is given by:

`z=(\bar X-\mu_o)/((\sigma)/(√(n)))` (1)

Replacing the info given we got:

`z=(112-124)/((22)/(√(37)))=-3.318`

The p value would be given by this probability:

`p_v =2*P(z<-3.318)=0.0009`

Since the p value is a very small value at any significance level used we can reject the null hypothesis and we can conclude that the true mean for this case is different from 124 ft