Fewer young people are driving. In 1995, 63.9% of people under 20 years old who were eligible had a driver's license. Bloomberg reported that percentage had dropped to 41.7% in 2016. Suppose these results are based on a random sample of 1200 people under 20 years old who were eligible to have a driver's license in 1995 and again in 2016.

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 driver's license in 1995?



Answer :

Explanation:

To find the margin of error and the interval estimate of the number of eligible people under 20 years old who had a driver's license in 1995, we will follow these steps:

1. **Calculate the sample proportion (\( \hat{p} \)) for 1995:**

- Given: \( \hat{p} = 0.639 \)

- Sample size (\( n \)) = 1200

2. **Calculate the standard error (SE):**

\[

SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}

\]

3. **Find the critical value (\( z \)) for a 95% confidence interval:**

- For a 95% confidence level, \( z \approx 1.96 \)

4. **Calculate the margin of error (ME):**

\[

ME = z \times SE

\]

5. **Calculate the interval estimate:**

\[

\text{Interval estimate} = \hat{p} \pm ME

\]

Let's calculate these step-by-step.

### Step 1: Sample Proportion

\[

\hat{p} = 0.639

\]

### Step 2: Standard Error

\[

SE = \sqrt{\frac{0.639 \times (1 - 0.639)}{1200}}

\]

### Step 3: Critical Value

\[

z = 1.96

\]

### Step 4: Margin of Error

\[

ME = 1.96 \times SE

\]

### Step 5: Interval Estimate

\[

\text{Interval estimate} = \hat{p} \pm ME

\]

Let's do the calculations:

#### Standard Error

\[

SE = \sqrt{\frac{0.639 \times 0.361}{1200}}

\]

\[

SE = \sqrt{\frac{0.230679}{1200}}

\]

\[

SE = \sqrt{0.0001922325}

\]

\[

SE \approx 0.01386

\]

#### Margin of Error

\[

ME = 1.96 \times 0.01386

\]

\[

ME \approx 0.02715

\]

#### Interval Estimate

\[

\text{Lower bound} = \hat{p} - ME = 0.639 - 0.02715 = 0.61185

\]

\[

\text{Upper bound} = \hat{p} + ME = 0.639 + 0.02715 = 0.66615

\]

### Summary

- **Margin of Error (ME):** \( 0.02715 \)

- **Interval Estimate:** \( 0.61185 \) to \( 0.66615 \)

These are the 95% confidence interval estimates for the proportion of eligible people under 20 years old who had a driver's license in 1995.