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 51 records of automobile driver fatalities in a certain county showed that 33 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 = 0.10.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.77; H₁:  p ≠ 0.77

H₀: p = 0.77; H₁:  p > 0.77     

H₀: p = 0.77; H₁:  p < 0.77

H₀: p < 0.77; H₁:  p = 0.77

(b) What sampling distribution will you use?

The standard normal, since np > 5 and nq > 5.

The Student's t, since np > 5 and nq > 5.     

The standard normal, since np < 5 and nq < 5.

The Student's t, since np < 5 and nq < 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.     

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that the true proportion of driver fatalities related to alcohol in the county is less than 0.77.

There is insufficient evidence at the 0.10 level to conclude that the true proportion of driver fatalities related to alcohol in the county is less than 0.77.

Answer 1

Solution:

Given:  The U.S. Department of Transportation, National Highway Traffic Safety Administration, reported that 77% of all fatally injured automobile drivers were intoxicated.

thus p = proportion of all fatally injured automobile drivers were intoxicated = 0.77

n = 51

x = Number of fatally injured automobile drivers were  involved an intoxicated driver = 33

Part a) What is the level of significance?

The level of significance =   α = 0.10

State the null and alternate hypotheses.

We have to test if these data indicate that the population proportion of driver fatalities related to alcohol is less than 77% in Kit Carson County, thus this is left tailed test. ( < type)

thus:

H₀: p = 0.77; H₁:  p < 0.77

Part b) What sampling distribution will you use?

n*p = 51 * 0.77 =39.27 > 5

n*q = n*(1-p) = 51 * ( 1- 0.77) = 11.73 > 5

The standard normal, since np > 5 and nq > 5.

What is the value of the sample test statistic?

where

.

thus

Part c) Find the P-value of the test statistic.

For left tailed test , p-value is:

p-value = P(Z < z test statistic)

p-value = P(Z < -2.09 )

Look in z table for z = -2.0 and 0.09 and find corresponding area.

P( Z<-2.09) = 0.0183

thus

p-value = P(Z < -2.09 )

p-value = 0.0183

Sketch the sampling distribution and show the area corresponding to the P-value.

Part d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

Since p-value = 0.0183 < 0.10 level of significance , we reject H0,

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

Part (e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that the true proportion of driver fatalities related to alcohol in the county is less than 0.77.