An insurance company wants to estimate the percentage of drivers who talk on their mobile phones while driving. A random sample of 850 drivers results in 544 who talk on their mobile phones while driving.
Exercise
Find the point estimate of the percentage of all drivers who talk on their cell phones while driving.
Find a 95% interval estimate of the percentage of all drivers who talk on their cell phones while driving.
Here’s how we tackle both parts of this exercise using principles of proportion estimation:
(a) Point Estimate
The point estimate for the population proportion is the sample proportion:
\[ \hat{p} = \frac{544}{850} \approx 0.64 \]
So, the point estimate is 64% of drivers talk on their mobile phones while driving.
(b) 95% Confidence Interval Estimate
We’ll compute a confidence interval for a population proportion using the formula:
\[ \hat{p} \pm z \cdot \sqrt{ \frac{\hat{p}(1 - \hat{p})}{n} } \]
Where: - \(\hat{p} = 0.64\) - \(n = 850\) - \(z = 1.96\) (for 95% confidence)
Step 1: Compute the standard error (SE)
\[ SE = \sqrt{ \frac{0.64 \cdot 0.36}{850} } \approx \sqrt{ \frac{0.2304}{850} } \approx \sqrt{0.000271} \approx 0.0165 \]
Step 2: Apply the margin of error
\[ MOE = 1.96 \cdot 0.0165 \approx 0.0323 \]
Step 3: Final confidence interval
\[ 0.64 \pm 0.0323 = (0.6077,\ 0.6723) \]
So the 95% confidence interval for the percentage of all drivers who talk on their mobile phones while driving is between 60.8% and 67.2%.