Answer to Question #284594 in Statistics and Probability for mercy

Question

Suppose speeds of vehicles on a particular stretch of roadway are normally distributed with mean 36.6 mph and standard deviation 1.7 mph. Find the probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph

Answer 1

Let $X=$ the mean speed of 20 randomly selected vehicles: $X\sim N(\mu, \sigma^2/n).$

Given $\mu=36.6mph, \sigma=1.7mph, n=20$

$$P(35<X<40)=P(X<40)-P(X\leq 35)$$

$$=P(Z<\dfrac{40-\mu}{\sigma/\sqrt{n}})-P(Z\leq \dfrac{35-\mu}{\sigma/\sqrt{n}})$$

$$=P(Z<\dfrac{40-36.6}{1.7/\sqrt{20}})-P(Z\leq \dfrac{35-36.6}{1.7/\sqrt{20}})$$

$$\approx P(Z<8.944272)-P(Z\leq -4.209069)$$

$$\approx1-0.000013\approx0.999987$$

The probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph is $0.999987.$