Questions: 3.- The distance that a Tesla model 3 can travel is normally distributed with a mean of 260 miles and a standard deviation of 25 miles. (d) Now, suppose that you pick a random sample of 9 Tesla model 3. What is the probability that the sample mean will be more than 250 miles. (e) Does the Central Limit Theorem applies in Part (d)? Explain.

3.- The distance that a Tesla model 3 can travel is normally distributed with a mean of 260 miles and a standard deviation of 25 miles. (d) Now, suppose that you pick a random sample of 9 Tesla model 3. What is the probability that the sample mean will be more than 250 miles. (e) Does the Central Limit Theorem applies in Part (d)? Explain.
Transcript text: 3.- The distance that a Tesla model 3 can travel is normally distributed with a mean of 260 miles and a standard deviation of 25 miles. (d) Now, suppose that you pick a random sample of 9 Tesla model 3. What is the probability that the sample mean will be more than 250 miles. (e) Does the Central Limit Theorem applies in Part (d)? Explain.
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Solution

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Solution Steps

Step 1: Calculate the Z-score

To find the probability that the sample mean of a random sample of 9 Tesla Model 3 vehicles is more than 250 miles, we first calculate the Z-score for the lower bound of 250 miles. The Z-score is given by:

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

Where:

  • \(X = 250\)
  • \(\mu = 260\)
  • \(\sigma = 25\)
  • \(n = 9\)

Substituting the values, we have:

\[ Z = \frac{250 - 260}{25 / \sqrt{9}} = \frac{-10}{25 / 3} = \frac{-10 \cdot 3}{25} = -1.2 \]

Step 2: Calculate the Probability

Next, we need to find the probability that the sample mean is more than 250 miles. This can be expressed as:

\[ P(\bar{X} > 250) = 1 - P(\bar{X} \leq 250) = 1 - \Phi(Z_{start}) \]

Where \(\Phi(Z)\) is the cumulative distribution function (CDF) of the standard normal distribution. From the calculations, we have:

\[ P(\bar{X} \leq 250) = \Phi(-1.2) \approx 0.1151 \]

Thus, the probability that the sample mean is more than 250 miles is:

\[ P(\bar{X} > 250) = 1 - 0.1151 = 0.8849 \]

Step 3: Central Limit Theorem Explanation

The Central Limit Theorem (CLT) applies in this scenario because the sample size \(n = 9\) is generally considered sufficient for the CLT to hold. According to the CLT, the distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution, as long as the sample size is large enough.

Final Answer

The probability that the sample mean will be more than 250 miles is approximately \(0.8849\), and the Central Limit Theorem applies in this case.

\[ \boxed{P(\bar{X} > 250) \approx 0.8849} \]

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