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 53 records of automobile driver fatalities in a certain county showed that 35 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 = 0.05.

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 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.

The standard normal, 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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.   

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 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 0.05 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 0.05 level to conclude that the true proportion of driver fatalities related to alcohol in the county is less than 0.77.    

Answer 1

a) 5 % l.o.s.

The hypotheses are,

H0 : p=0.77 v/s H1 : p<0.77

b) Sampling distribution will be

The standard normal since np>5 & nq<5

> n=53
> x=35
> P=0.77
> prop.test(x,n)

   1-sample proportions test with continuity correction

data: x out of n, null probability 0.5
X-squared = 4.8302, df = 1, p-value = 0.02797
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
0.5164050 0.7811293
sample estimates:
p
0.6603774

##Hence p-value of sample test statistic is p-value = 0.02

c)

p-value = 0.02797

d) Since p-value<0.05

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

There is sufficient evidence at the 0.05 level to conclude that the true proportion of driver fatalities related to alcohol in the county is less than 0.77.