An insurance company collects data on seat-belt us

Questions: An insurance company collects data on seat-belt use among drivers in a country. Of 1400 drivers 20-29 years old, 21% said that they buckle up, whereas 309 of 1200 drivers 45-64 years old said that they did. At the 1% significance level, do the data suggest that there is a difference in seat-belt use between drivers 20-29 years old and those 45-64? Let population 1 be drivers of age 20-29 and let population 2 be drivers of age 45-64. Use the two-proportions z-test to conduct the required hypothesis test. What are the hypotheses for this test? A. H0: p1>p2, Ha: p1=p2 B. H0: p1=p2, Ha: p1 ≠ p2 C. H0: p1=p2, Ha: p1>p2 D. H0: p1<p2, Ha: p1=p2 E. H0: p1 ≠ p2, Ha: p1=p2 F. H0: p1=p2, Ha: p1<p2

Transcript text: An insurance company collects data on seat-belt use among drivers in a country. Of 1400 drivers 20-29 years old, $21 \%$ said that they buckle up, whereas 309 of 1200 drivers $45-64$ years old said that they did. At the $1 \%$ significance level, do the data suggest that there is a difference in seat-belt use between drivers 20-29 years old and those 45-64? Let population 1 be drivers of age 20-29 and let population 2 be drivers of age 45-64. Use the two-proportions z-test to conduct the required hypothesis test. What are the hypotheses for this test? A. $H_{0}: p_{1}>p_{2}, H_{a}: p_{1}=p_{2}$ B. $\mathrm{H}_{0}: \mathrm{p}_{1}=\mathrm{p}_{2}, \mathrm{H}_{\mathrm{a}}: \mathrm{p}_{1} \neq \mathrm{p}_{2}$ C. $H_{0}: p_{1}=p_{2}, H_{a}: p_{1}>p_{2}$ D. $H_{0}: p_{1}

Solution

Solution Steps

To determine if there is a significant difference in seat-belt use between the two age groups, we will perform a two-proportions z-test. The null hypothesis $ H_0 $ states that the proportions of seat-belt use in both age groups are equal, while the alternative hypothesis $ H_a $ suggests that the proportions are not equal. We will calculate the z-statistic and compare it to the critical value at the 1% significance level to decide whether to reject the null hypothesis.

Step 1: Define the Hypotheses

We define the null and alternative hypotheses for the two-proportions z-test as follows:

Null hypothesis: $ H_0: p_1 = p_2 $
Alternative hypothesis: $ H_a: p_1 \neq p_2 $

Where:

$ p_1 $ is the proportion of drivers aged 20-29 who buckle up.
$ p_2 $ is the proportion of drivers aged 45-64 who buckle up.

Step 2: Calculate the Counts and Observations

From the data provided:

For drivers aged 20-29:
- Number of successes (drivers who buckle up): $ 0.21 \times 1400 = 294 $
For drivers aged 45-64:
- Number of successes (drivers who buckle up): $ 309 $

Thus, we have: