Question

A recent study shows that 78% of teenagers have used cell phones while driving. In Oct. 2010, Massachusetts enacted a law that forbids cell phone use by drivers under the age of 18. A policy analyst would like to test whether the law has lowered the proportion of drivers under the age of 18 who use a cell phone. In a random sample of 85 young drivers, 59 of them said that they used cell phones while driving.

Required:
a. State the null and the alternative hypotheses to test the policy analyst’s objective.
b. Suppose a sample of 200 drivers under the age of 18 results in 150 who still use a cell phone while driving. What is the value of the test statistic? What is the p-value?
c. At α = 0.05 has the law been effective?
d. Repeat this exercise using the critical value approach with α = 0.05.

Answer 1

a) Null hypothesis:
`p\geq 0.78`

Alternative hypothesis:
`p < 0.78`

b)
`z=\frac{0.75 -0.78}{\sqrt{(0.78(1-0.78))/(200)}}=-1.024`

We are conducting a left tailed test so then the p value would be:

`p_v =P(z<-1.024)=0.1529`

c) Since the p value is higher than the confidence level of 0.05 we don;t have enough evidence to conclude that the true proportion of people under 18 in the sample who use cell phones while driving is significantly lower than 0.78. so then the law is not effective

d) For this case we need to find a critical value in the normal distribution who accumulates 0.05 of the area on the left and we got:

`z_(crit)= -1.64`

Since the calculated value -1.024 is not less than the critical value we don't have enpugh evidence to conclude that the true proportion is less than 0.78 and the law is not effective

Explanation:

Information given

n=85 represent the random sample selected

X=59 represent the people under 18 in the sample who use cell phones while driving

`\hat p=(59)/(85)=0.694` estimated proportion of people under 18 in the sample who use cell phones while driving

`p_o=0.78` is the value that we want to analyze

Part a

We want to check if the law has lowered the proportion of drivers under the age of 18 who use a cell phone so then the system of hypothesis arE:

Null hypothesis:
`p\geq 0.78`

Alternative hypothesis:
`p < 0.78`

Part b

For this case the estimated proportion would be:

`\hat p =(X)/(n)= (150)/(200)= 0.75`

The statistic is given by this formula:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

Replacing the info given we got:

`z=\frac{0.75 -0.78}{\sqrt{(0.78(1-0.78))/(200)}}=-1.024`

We are conducting a left tailed test so then the p value would be:

`p_v =P(z<-1.024)=0.1529`

Part c

Since the p value is higher than the confidence level of 0.05 we don;t have enough evidence to conclude that the true proportion of people under 18 in the sample who use cell phones while driving is significantly lower than 0.78. so then the law is not effective

Part d

For this case we need to find a critical value in the normal distribution who accumulates 0.05 of the area on the left and we got:

`z_(crit)= -1.64`

Since the calculated value -1.024 is not less than the critical value we don't have enpugh evidence to conclude that the true proportion is less than 0.78 and the law is not effective