Question

Suppose that the speeds of cars travelling on Califomia freeways are normally distributed with a mean of 61 (miles /hour)

, The highway patrol's policy is to issue tickets for cars with speeds exceeding 75 ( miles /hour) ​
. The records show that exactly 2% of the speeds exceed this limit. Find the standard deviation of the speeds of cars traveiling on California freeways. Carry your intermeciate computatians to at least four decimat places. Round your answer to at feast one decimal place.

Answer 1

The standard deviation of the speeds of cars on California freeways is found to be approximately 6.8 miles per hour, using the known mean speed and the percentage of cars exceeding the speed limit.

Step-by-step explanation:

To find the standard deviation of the speeds of cars traveling on California freeways, given that 2% of the speeds exceed 75 miles per hour, we'll use the properties of the normal distribution. In a normal distribution, the percentage of data that falls above a certain value corresponds to a z-score. Since 2% exceed the limit, we are looking for the z-score that corresponds to the 98th percentile of a standard normal distribution. The z-score for the 98th percentile is approximately 2.05.

The z-score formula is given by:

Z = (X - µ) / σ

Where Z is the z-score, X is the value in question, µ is the mean, and σ is the standard deviation. Plugging in the numbers:

2.05 = (75 - 61) / σ

Solving for σ gives us:

σ = (75 - 61) / 2.05

σ = 14 / 2.05 ≈ 6.83

Therefore, the standard deviation (σ) of the speeds of cars on California freeways is approximately 6.8 miles per hour.