Question

Cell phone usage differs by gender. The role of cell phone in modern life was investigated by Pew Internet and American life project(April 2016) survey. A total of 1,286 cell phone users were interviewed in the sample. One of the objectives was to compare male and female cell phone users. For example, 32% of men admitted they sometimes don’t drive safely whole talking or texting on a cell phone compared to 25% of women. Also, 71% of men used their cell phone in an emergency compared to 77% of women. Assume that half (643) of the cell phone users in the sample were men and half were women.
A. Describe the two population of interest in the survey.
B. Give an estimate of the proportion of men and the proportion of women who sometimes do not drive safely while talking or texting on cell phone.
C. Conduct a test to determine whether the proportion of men and women who used their cell phone in an emergency differ.

Answer 1

Answer:

A) Two populations: American men and American women, as it estimated from the broad of the survey.

B) The proportion of men that sometimes do not drive safely while talking or texting on cell phone is 32%, and the proportion of women that sometimes do not drive safely while talking or texting on cell phone is 25%.

C) There is enough evidence to support the claim that the proportion of men and women who used their cell phone in an emergency differ significantly (P-value=0.014).

Step-by-step explanation:

A. We have two populations: American men and American women. This are the populations from which we want to infere the difference of means.

B. The estimation of the proportion is already shown in the question:

The proportion of men that sometimes do not drive safely while talking or texting on cell phone is 32%, and the proportion of women that sometimes do not drive safely while talking or texting on cell phone is 25%.

These are unbiased estimators of the populations proportions.

C. This is a hypothesis test for the difference between proportions.

The claim is that the proportion of men and women who used their cell phone in an emergency differ significantly.

Then, the null and alternative hypothesis are:

$H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\neq 0$

The significance level is 0.05.

The sample 1 (men), of size n1=643 has a proportion of p1=0.71.

The sample 2 (women), of size n2=643 has a proportion of p2=0.77.

 

The difference between proportions is (p1-p2)=-0.06.

$p_d=p_1-p_2=0.71-0.77=-0.06$

The pooled proportion, needed to calculate the standard error, is:

$p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{457+495.11}{643+643}=\dfrac{952}{1286}=0.7403$

The estimated standard error of the difference between means is computed using the formula:

$s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.7403*0.2597}{643}+\dfrac{0.7403*0.2597}{643}}\\\\\\s_{p1-p2}=\sqrt{0.0003+0.0003}=\sqrt{0.0006}=0.0245$

Then, we can calculate the z-statistic as:

$z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{-0.06-0}{0.0245}=\dfrac{-0.06}{0.0245}=-2.45$

This test is a two-tailed test, so the P-value for this test is calculated as (using a z-table):

$P-value=2\cdot P(z$

As the P-value (0.014) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the proportion of men and women who used their cell phone in an emergency differ significantly.