Question

1. an automobile service shop called \20 - minute oil\ offers fast oil changes. let ( t ) = the length of time it takes for a randomly selected vehicle to get an oil change at \20 - minute oil\. the distribution of ( t ) is approximately normal with a mean time of 19.7 minutes and a standard deviation of 2.3 minutes.

a. find ( p(t > 20) ). interpret this value.
b. do you think \20 - minute oil\ is an appropriate name for this oil shop? explain.
c. for some value, ( t ), ( p(t leq t)=0.05 ). find and interpret the value of ( t ).

Explanation:

Step1: Calculate z-score for T=20

First, use the z-score formula to standardize the value 20.
$$z = \frac{T - \mu}{\sigma} = \frac{20 - 19.7}{2.3} \approx 0.13$$

Step2: Find $P(T > 20)$

This is equivalent to $P(z > 0.13)$. Using standard normal tables or a calculator:
$$P(z > 0.13) = 1 - P(z \leq 0.13) = 1 - 0.5517 = 0.4483$$
Interpretation: About 44.83% of oil changes take longer than 20 minutes.

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Step3: Evaluate name appropriateness

Compare the mean and the probability from part a. The mean oil change time is 19.7 minutes (less than 20), and slightly less than half of oil changes take over 20 minutes. The name is reasonable because the average time is under 20 minutes, and a majority of jobs finish within the 20-minute window.

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Step4: Find t for $P(T \leq t)=0.05$

First, find the z-score corresponding to the 5th percentile. From standard normal tables, $z \approx -1.645$.
Then rearrange the z-score formula to solve for $t$:
$$t = \mu + z\sigma = 19.7 + (-1.645)(2.3)$$
$$t = 19.7 - 3.7835 = 15.9165$$
Interpretation: Only 5% of oil changes take 15.92 minutes or less; this is the 5th percentile of oil change times.

Answer:

a. $P(T > 20) \approx 0.448$. Interpretation: Approximately 44.8% of randomly selected vehicles take longer than 20 minutes for an oil change.
b. Yes, the name is appropriate. The mean oil change time is 19.7 minutes (under 20), and only ~44.8% of jobs exceed 20 minutes, so a majority are completed within the 20-minute frame.
c. $t \approx 15.92$ minutes. Interpretation: 5% of oil changes at this shop take 15.92 minutes or less; this is the value where 95% of oil change times are longer than it.