Question

The police that patrol a heavily traveled highway claim that the average driver exceeds the 65 miles per hour speed limit by more than 10 miles per hour. Seventy-two randomly selected cars were clocked by airplane radar. The average speed of the 72 cars was 77.40 miles per hour, and the standard deviation of these speeds was 5.90 miles per hour. Test a 5% level of significance whether the average speed by all drivers is more than 75 mph. Make sure to show all of your work and include every step.

Answer 1

Solution :

The null and alternative hypotheses are as follows:

i.e. The average speed by all drivers is 75 mph.

i.e. The average speed by all drivers is greater than 75 mph.

To test the hypothesis we shall use one sample t-test. The test statistic is given as follows :

Where, x̄ is sample mean, s is sample standard deviation, μ is hypothesized value of population mean under H​​​0 and n is sample size.

We have,  x̄ = 77.40 mph, s = 5.90 mph, μ = 75 mph and n = 72

The value of the test statistic is 3.45164

Since, our test is right-tailed test, therefore we shall obtain right-tailed p-value for the test statistic. The right-tailed p-value is given as follows :

p-value = P(T > t)

p-value = P(T > 3.45164)

Using the R software we get, P(T > 3.45164) = 0.0005

Hence, p-value = 0.0005

The p-value is 0.0005.

We make decision rule as follows :

If p-value is less than the significance level, then we reject the null hypothesis (H​​​​​​​​​​​0) at given significance level.

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H​​​​​​​​​​​0) at given significance level.

We have, p-value = 0.0005

significance level = 5% = 0.05

(0.0005 < 0.05)

Since, p-value is less than the significance level of 5%, therefore we shall reject the null hypothesis (H​​​​​​​​​​​0) at 5% significance level.

Conclusion : At 5% significance level, there is sufficient evidence to conclude that the average speed by all drivers is more than 75 mph.

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