Questions: The shape of the distribution of the time required to get an oil change at a 20-minute oil-change facility is skewed right. However, records indicate that the mean time is 21.3 minutes, and the standard deviation is 4.8 minutes. Complete parts (a) through (c) below. (a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required? Choose the required sample size below. A. The sample size needs to be greater than 30. B. The normal model cannot be used if the shape of the distribution is skewed right. C. Any sample size could be used. D. The sample size needs to be less than 30. (b) What is the probability that a random sample of n=45 oil changes results in a sample mean time less than 20 minutes? The probability is approximately . (Round to four decimal places as needed.)

The shape of the distribution of the time required to get an oil change at a 20-minute oil-change facility is skewed right. However, records indicate that the mean time is 21.3 minutes, and the standard deviation is 4.8 minutes. Complete parts (a) through (c) below. (a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required? Choose the required sample size below. A. The sample size needs to be greater than 30. B. The normal model cannot be used if the shape of the distribution is skewed right. C. Any sample size could be used. D. The sample size needs to be less than 30. (b) What is the probability that a random sample of n=45 oil changes results in a sample mean time less than 20 minutes? The probability is approximately . (Round to four decimal places as needed.)
Transcript text: The shape of the distribution of the time required to get an oil change at a 20-minute oil-change facility is skewed right. However, records indicate that the mean time is 21.3 minutes, and the standard deviation is 4.8 minutes. Complete parts (a) through (c) below. (a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required? Choose the required sample size below. A. The sample size needs to be greater than 30. B. The normal model cannot be used if the shape of the distribution is skewed right. C. Any sample size could be used. D. The sample size needs to be less than 30. (b) What is the probability that a random sample of $n=45$ oil changes results in a sample mean time less than 20 minutes? The probability is approximately $\square$ $\square$. (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Sample Size Requirement

To compute probabilities regarding the sample mean using the normal model, the sample size must be sufficiently large. Given that the distribution of the time required to get an oil change is skewed right, the Central Limit Theorem suggests that a sample size greater than 30 is necessary for the normal approximation to be valid. Therefore, the correct choice is:

\[ \text{Sample size requirement: A. The sample size needs to be greater than 30.} \]

Step 2: Probability Calculation

We need to calculate the probability that a random sample of \( n = 45 \) oil changes results in a sample mean time less than \( 20 \) minutes.

First, we calculate the Z-score for \( 20 \) minutes using the formula:

\[ Z = \frac{X - \mu}{\sigma / \sqrt{n}} \]

Where:

  • \( X = 20 \)
  • \( \mu = 21.3 \)
  • \( \sigma = 4.8 \)
  • \( n = 45 \)

Calculating the standard error:

\[ \sigma_{\text{sample}} = \frac{\sigma}{\sqrt{n}} = \frac{4.8}{\sqrt{45}} \approx 0.7155 \]

Now, substituting into the Z-score formula:

\[ Z_{end} = \frac{20 - 21.3}{0.7155} \approx -1.8168 \]

Next, we find the probability:

\[ P(X < 20) = \Phi(Z_{end}) - \Phi(-\infty) = \Phi(-1.8168) - 0 \]

Using the standard normal distribution table, we find:

\[ P(X < 20) \approx 0.0346 \]

Final Answer

The answers to the questions are as follows:

  1. Sample size requirement: A. The sample size needs to be greater than 30.
  2. Probability that the sample mean is less than 20 minutes: \( P \approx 0.0346 \).

Thus, the final boxed answers are:

\[ \boxed{\text{Sample size requirement: A. The sample size needs to be greater than 30.}} \] \[ \boxed{P \approx 0.0346} \]

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