Question

fewer young people are driving. in 1995, 63.0% of people under 20 years old who were eligible had a drivers license. the federal highway administration reported that percentage had dropped to 41.5% in 2020. suppose these results are based on a random sample of 1,200 people under 20 years old who were eligible to have a drivers license in 1995 and again in 2020. a. at 95% confidence, what is the margin of error and the interval estimate of the number of eligible people under 20 years old who had a drivers license in 1995? margin of error = (to four decimal places) interval estimate = to (to four decimal places) b. at 95% confidence, what is the margin of error and the interval estimate of the number of eligible people under 20 years old who had a drivers license in 2020? margin of error = (to four decimal places) interval estimate = to (to four decimal places) c. is the margin of error the same in parts (a) and (b)? no why, or why not?

Explanation:

Step1: Recall the formula for margin of error

For a proportion, the margin of error \(E = z_{\alpha/2}\sqrt{\frac{p(1 - p)}{n}}\). For a 95% confidence interval, \(z_{\alpha/2}=1.96\), \(n = 1200\).

Part (a)

Step1: Calculate margin of error for 1995

Given \(p = 0.63\).

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Step2: Calculate interval estimate for 1995

The point estimate is \(\hat{p}=0.63\). The lower limit \(=0.63 - 0.0272=0.6028\), the upper limit \(=0.63 + 0.0272 = 0.6572\)

Part (b)

Step1: Calculate margin of error for 2020

Given \(p = 0.415\).

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Step2: Calculate interval estimate for 2020

The point estimate is \(\hat{p}=0.415\). The lower limit \(=0.415- 0.0278=0.3872\), the upper limit \(=0.415 + 0.0278=0.4428\)

Part (c)

Step1: Explain why margin of error is different

The margin of error formula \(E = z_{\alpha/2}\sqrt{\frac{p(1 - p)}{n}}\) depends on \(p\). Since \(p_{1995}=0.63\) and \(p_{2020}=0.415\) (the values of \(p(1 - p)\) are different: \(0.63\times(1 - 0.63)=0.2331\) and \(0.415\times(1 - 0.415)=0.242775\)), the margin of error is different.

Answer:

a. Margin of error \(= 0.0272\), Interval estimate \(=0.6028\) to \(0.6572\)
b. Margin of error \(=0.0278\), Interval estimate \(=0.3872\) to \(0.4428\)
c. The margin of error formula \(E = z_{\alpha/2}\sqrt{\frac{p(1 - p)}{n}}\) depends on \(p\). Different \(p\) ( \(p = 0.63\) for 1995 and \(p=0.415\) for 2020) lead to different \(p(1 - p)\) values, so the margin of error is different.