if the true mean is 0.9050 with a standard deviation of 0.0100, within what interval will 90% of the sample of 48 means fall?

To determine the interval within which 90% of the sample means will fall, we can use the concept of the **sampling distribution of the sample mean**. Given the true mean (\(\mu\)) and standard deviation (\(\sigma\)), the formula for the standard error of the mean (SEM) is:

\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\]

where:
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.

Once we have the SEM, we can find the interval by using the z-score corresponding to the desired confidence level.

### Step 1: Calculate the Standard Error of the Mean (SEM)

Given:
- \(\mu = 0.9050\)
- \(\sigma = 0.0100\)
- \(n = 48\)

\[
\text{SEM} = \frac{0.0100}{\sqrt{48}} \approx 0.00144
\]

### Step 2: Find the z-score for 90% Confidence Level

For a 90% confidence interval, the z-score that corresponds to the middle 90% of the distribution is approximately \(z = 1.645\).

### Step 3: Calculate the Interval

The interval can be calculated as:

\[
\text{Interval} = \mu \pm (z \times \text{SEM})
\]

Substituting the values:

\[
\text{Interval} = 0.9050 \pm (1.645 \times 0.00144)
\]

\[
\text{Interval} = 0.9050 \pm 0.00237
\]

\[
\text{Interval} = [0.90263, 0.90737]
\]

### Final Answer
90% of the sample means will fall within the interval \([0.90263, 0.90737]\).

Would you like more details or have any questions?

Here are some related questions:
1. What is the interpretation of the confidence interval?
2. How does the sample size affect the standard error of the mean?
3. What happens to the interval if we increase the confidence level to 95%?
4. How would the interval change if the population standard deviation were higher?
5. Why do we use the z-score in calculating confidence intervals?

**Tip:** Remember that the standard error decreases as the sample size increases, leading to a narrower confidence interval.

Learn how to calculate the 90% confidence interval for a sample mean distribution with a true mean of 0.9050 and standard deviation of 0.0100. This problem involves using the sampling distribution concept, standard error of the mean (SEM), and z-score calculation. Suitable for advanced high school students or anyone studying statistics.

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