What braking distance represents the first quartile?

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Step 1: Understand the problem. The first quartile (Q1) represents the value below which 25% of the data falls in a normal distribution. To find Q1, we need to use the z-score corresponding to the 25th percentile in a standard normal distribution.

Step 2: Recall that the z-score for the 25th percentile in a standard normal distribution is approximately -0.674. This value is derived from standard normal tables or statistical software.

Step 3: Use the formula for converting a z-score to a value in a normal distribution: x = μ + zσ, where μ is the mean, σ is the standard deviation, and z is the z-score.

Step 4: Substitute the given values into the formula. Here, μ = 132 ft, σ = 4.53 ft, and z = -0.674. Plug these values into the formula to calculate the braking distance corresponding to the first quartile.

Step 5: Interpret the result. The calculated value will represent the braking distance below which 25% of the data falls, i.e., the first quartile.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean (µ) and standard deviation (σ). In this context, the braking distances of a sedan follow a normal distribution with a mean of 132 feet and a standard deviation of 4.53 feet.

Quartiles

Quartiles are values that divide a dataset into four equal parts, each containing 25% of the data. The first quartile (Q1) is the value below which 25% of the data fall. To find Q1 in a normal distribution, one can use the z-score corresponding to the 25th percentile, which helps in determining the specific value of braking distance that represents this quartile.

Z-Score

A z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula z = (X - µ) / σ, where X is the value in question, µ is the mean, and σ is the standard deviation. For the first quartile, the z-score is approximately -0.674, which can be used to calculate the corresponding braking distance by rearranging the formula.

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Textbook Question

In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P85

Textbook Question