Questions: According to the Central Limit Theorem, if a sample size is at least sampled populations, we can conclude that the sample means are approximately normal. a. 25 b. 20 c. 30 d. 50 e. 100

According to the Central Limit Theorem, if a sample size is at least sampled populations, we can conclude that the sample means are approximately normal. a. 25 b. 20 c. 30 d. 50 e. 100
Transcript text: According to the Central Limit Theorem, if a sample size is at least sampled populations, we can conclude that the sample means are approximately normal. a. 25 b. 20 c. 30 d. 50 e. 100
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Solution

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Solution Steps

Step 1: Calculate the Mean of the Sampling Distribution

The mean of the sampling distribution is equal to the population mean, which is given by:

\[ \mu = 50 \]

Step 2: Calculate the Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution is calculated using the formula:

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{30}} \approx 1.8257 \]

Step 3: Calculate the Probability of the Sample Mean Falling Within a Range

To find the probability that the sample mean falls within the range \( [45, 55] \), we first convert the bounds to Z-scores:

\[ Z_{start} = \frac{45 - 50}{1.8257} \approx -2.7386 \] \[ Z_{end} = \frac{55 - 50}{1.8257} \approx 2.7386 \]

Using the standard normal distribution, we find:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) \approx \Phi(2.7386) - \Phi(-2.7386) \approx 0.9938 \]

Final Answer

The mean of the sampling distribution is \( \mu = 50 \), the standard deviation is \( \sigma_{\bar{x}} \approx 1.8257 \), and the probability that the sample mean falls within the range \( [45, 55] \) is approximately \( P \approx 0.9938 \).

Thus, the final answer is:

\[ \boxed{P \approx 0.9938} \]

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