Given a normal distribution with muequals101 and sigmaequals10​, and given you select a sample of nequals25​. What is the probability that Upper X overbar is less than 95​?

To solve this problem, we are asked to find the probability that the sample mean (\(\bar{X}\)) is less than 95, given the following information:

- Population mean (\(\mu\)) = 101
- Population standard deviation (\(\sigma\)) = 10
- Sample size (\(n\)) = 25
- We need to find \(P(\bar{X} < 95)\)

### Step 1: Find the distribution of the sample mean \(\bar{X}\)
The sample mean \(\bar{X}\) is normally distributed with:
- Mean: \(\mu_{\bar{X}} = \mu = 101\)
- Standard deviation (standard error): \(\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2\)

Thus, \(\bar{X}\) is normally distributed with a mean of 101 and a standard deviation (standard error) of 2.

### Step 2: Standardize the value
To find the probability \(P(\bar{X} < 95)\), we need to standardize 95 to a z-score using the formula:
\[
z = \frac{\bar{X} - \mu_{\bar{X}}}{\sigma_{\bar{X}}}
\]
Substitute the values:
\[
z = \frac{95 - 101}{2} = \frac{-6}{2} = -3
\]

### Step 3: Find the probability using the z-score
Now, we need to find the probability corresponding to \(z = -3\). Using standard normal distribution tables or a calculator, we find:

\[
P(Z < -3) = 0.00135
\]

### Conclusion:
The probability that the sample mean \(\bar{X}\) is less than 95 is approximately 0.00135, or 0.135%.

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Do you have any questions or would you like more details on this solution? 

Here are 5 related questions:
1. How does changing the sample size \(n\) affect the standard error?
2. What is the probability that \(\bar{X}\) is greater than 105 in this same distribution?
3. How would the probability change if \(\mu = 95\) and \(\sigma = 5\)?
4. Can you explain the relationship between sample size and the accuracy of sample mean estimates?
5. How would you calculate the probability if \(n = 100\) instead of 25?

**Tip**: The larger the sample size \(n\), the smaller the standard error, which means the sample mean becomes a more precise estimate of the population mean.

Given a normal distribution with μ = 101 and σ = 10, and given you select a sample of n = 25, what is the probability that X̄ is less than 95?

Solve a probability problem involving a sample mean from a normal distribution with population mean μ = 101, standard deviation σ = 10, and sample size n = 25. Learn how to calculate the probability that the sample mean X̄ is less than 95 using the Z-score formula. Suitable for undergraduate students.

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