Suppose that the speeds of cars travelling on California freeways are normally distributed with a mean of 55 mph . The highway patrol's policy is to issue tickets for cars with speeds exceeding 80 mph. 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.
Question
Suppose that the speeds of cars travelling on California freeways are normally distributed with a mean of 55 mph . The highway patrol's policy is to issue tickets for cars with speeds exceeding 80 mph. 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.
Solution
To solve this problem, we need to use the properties of the normal distribution.
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First, we know that the mean speed of cars on California freeways is 55 mph.
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We also know that the top 1% of speeds are over 80 mph. In a standard normal distribution, the value that separates the top 1% from the rest is approximately 2.33 (this is the z-score that corresponds to a cumulative probability of 0.99).
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The z-score is calculated by the formula (X - μ) / σ, where X is the value from the dataset, μ is the mean, and σ is the standard deviation.
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We can set up the equation 2.33 = (80 - 55) / σ and solve for σ.
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This simplifies to 2.33σ = 25.
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Solving for σ gives us σ ≈ 25 / 2.33 ≈ 10.73 mph.
So, the standard deviation of the speeds of cars travelling on California freeways is approximately 10.7 mph.
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