Question

The U.S. Department of Transportation, National Highway Traffic Safety Administration, reported that 77% of all fatally injured automobile drivers were intoxicated. A random sample of 27 records of automobile driver fatalities in Kit Carson County, Colorado, showed that 15 involved an intoxicated driver. Do these data indicate that the population proportion of driver fatalities related to alcohol is less than 77% in Kit Carson County? Use α = 0.01. Solve the problem using both the traditional method and the P-value method. Since the sampling distribution of p̂ is the normal distribution, you can use critical values from the standard normal distribution as shown in the table of critical values of the z distribution. (Round the test statistic and the critical value to two decimal places. Round the P-value to four decimal places.)

test statistic	=
critical value	=
P-value	=

State your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the true proportion of driver fatalities related to alcohol is less than 77%.

There is insufficient evidence at the 0.01 level to conclude that the true proportion of driver fatalities related to alcohol is less than 77%.    

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.

The conclusions obtained by using both methods are the same.    

We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.

Answer 1

Given that, n = 27 and x = 15

=> sample proportion = 15/27 = 0.5556

The null and alternative hypotheses are,

H0 : p = 0.77

Ha : p < 0.77

This hypothesis test is a left-tailed test.

Test statistic is,

=> Test statistic = Z = -2.65

critical value at significance level of 0.01 is, Zcrit = -2.33

=> Critical value = -2.33

p-value = P(Z < -2.65) = 1 - P(Z < 2.65) = 1 - 0.9960 = 0.0040

=> p-value = 0.0040

Since, test statistic = -2.65 < -2.33, we reject H0.

And since, p-value is less than 0.01 significance level, we reject H0.

Conclusion : There is sufficient evidence at the 0.01 level to conclude that the true proportion of driver fatalities related to alcohol is less than 77%.

The conclusions obtained by using both methods are the same.