Questions: Assume that police estimate that 16% of drivers do not wear their seatbelts. They set up a safety roadblock, stopping cars to check for seatbelt use. They stop 50 cars during the first hour. a. Find the mean, variance, and standard deviation of the number of drivers expected not to be wearing seatbelts. Use the fact that the mean of a geometric distribution is μ = 1/p and the variance is σ^2 = q/p^2. b. How many cars do they expect to stop before finding a driver whose seatbelt is not buckled?

Assume that police estimate that 16% of drivers do not wear their seatbelts. They set up a safety roadblock, stopping cars to check for seatbelt use. They stop 50 cars during the first hour.

a. Find the mean, variance, and standard deviation of the number of drivers expected not to be wearing seatbelts. Use the fact that the mean of a geometric distribution is μ = 1/p and the variance is σ^2 = q/p^2.

b. How many cars do they expect to stop before finding a driver whose seatbelt is not buckled?

Transcript text: Assume that police estimate that $16 \%$ of drivers do not wear their seatbelts. They set up a safety roadblock, stopping cars to check for seatbelt use. They stop 50 cars during the first hour. a. Find the mean, variance, and standard deviation of the number of drivers expected not to be wearing seatbelts. Use the fact that the mean of a geometric distribution is $\mu=\frac{1}{p}$ and the variance is $\sigma^{2}=\frac{q}{p^{2}}$. b. How many cars do they expect to stop before finding a driver whose seatbelt is not buckled?

Solution

Solution Steps

Step 1: Calculate the Mean

The mean $ \mu $ of the number of drivers expected not to be wearing seatbelts can be calculated using the formula: \[ \mu = n \cdot p \] where $ n = 50 $ (the number of cars stopped) and $ p = 0.16 $ (the probability that a driver does not wear a seatbelt). Thus, \[ \mu = 50 \cdot 0.16 = 8.0 \]

Step 2: Calculate the Variance

The variance $ \sigma^2 $ of the number of drivers not wearing seatbelts is given by: \[ \sigma^2 = n \cdot p \cdot q \] where $ q = 1 - p = 0.84 $. Therefore, \[ \sigma^2 = 50 \cdot 0.16 \cdot 0.84 = 6.72 \]

Step 3: Calculate the Standard Deviation

The standard deviation $ \sigma $ is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{6.72} \approx 2.592 \]

Step 4: Calculate the Expected Number of Cars to Stop

The expected number of cars to stop before finding a driver whose seatbelt is not buckled is given by the formula for the mean of a geometric distribution: \[ E(X) = \frac{1}{p} \] Thus, \[ E(X) = \frac{1}{0.16} = 6.25 \]