Question

Suppose that the speeds of cars travelling on California freeways are normally distributed with a mean of 57 hour miles ​. The highway patrol's policy is to issue tickets for cars with speeds exceeding 75 hour /miles ​. The records show that exactly 1% of the speeds exceed this limit. Find the standard deviation of the speeds of cars travelling on California freeways. Carry your intermediate computations to at least four decimal places. Round your answer to at least one decimal place.

Answer 1

To find the standard deviation of car speeds on California freeways, we can use the z-score for the 1% exceedance point, which is 2.33. Applying the z-score formula and solving for the standard deviation, we determine that it is approximately 7.7 miles per hour.

Step-by-step explanation:

In order to find the standard deviation of the speeds of cars travelling on California freeways, we can use the fact that the speeds are normally distributed, with a mean (μ) of 57 miles per hour, and that 1% of the speeds exceed 75 miles per hour. The z-score corresponding to the top 1% of a standard normal distribution is typically found in a z-table or by using a calculator, and it is approximately 2.33.

The formula for a z-score is:

z = (X - μ) / σ

Where:
X is the value from the dataset;
μ is the mean of the distribution;
σ is the standard deviation of the distribution.

Using the value of 75 miles per hour for X, and knowing that the z-score is 2.33 for the top 1%, we can setup the equation:

2.33 = (75 - 57) / σ

Solving for σ, we get:

σ = (75 - 57) / 2.33
σ ≈ 18 / 2.33
σ ≈

7.7251

After rounding to at least one decimal place, the standard deviation, σ, is approximately 7.7 miles per hour.