https://file.aiquickdraw.com/mathbot/file/17306025702735ksnq2a1.jpg

To construct a 90% confidence interval for the true population mean (\(\mu\)) of the textbook weights, we can use the following formula for a confidence interval when the population standard deviation (\(\sigma\)) is known:

\[
\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\bar{x} = 62\) (sample mean),
- \(\sigma = 2.6\) (population standard deviation),
- \(n = 25\) (sample size),
- \(z\) is the z-score corresponding to the 90% confidence level.

### Step 1: Determine the z-score for a 90% confidence interval
For a 90% confidence level, the z-score is approximately \(1.645\).

### Step 2: Calculate the margin of error
The margin of error is given by:

\[
z \cdot \frac{\sigma}{\sqrt{n}} = 1.645 \cdot \frac{2.6}{\sqrt{25}}
\]

\[
= 1.645 \cdot \frac{2.6}{5} = 1.645 \cdot 0.52 = 0.8554
\]

### Step 3: Calculate the confidence interval
Now we calculate the lower and upper bounds:

\[
\text{Lower bound} = \bar{x} - \text{margin of error} = 62 - 0.8554 = 61.14
\]
\[
\text{Upper bound} = \bar{x} + \text{margin of error} = 62 + 0.8554 = 62.86
\]

### Final Answer
The 90% confidence interval for the true population mean textbook weight is:

\[
61.14 < \mu < 62.86
\]

Would you like further details, or do you have any questions?

---

Here are 5 related questions that might enhance understanding:

1. How is the confidence level related to the z-score, and why does it vary?
2. What would happen to the confidence interval if we increased the sample size?
3. How would the interval change with a 95% confidence level instead of 90%?
4. Why do we use the population standard deviation in this calculation, rather than the sample standard deviation?
5. What assumptions are necessary to calculate a confidence interval for a population mean?

**Tip:** For confidence intervals, remember that a higher confidence level (like 95% or 99%) will widen the interval, reflecting greater uncertainty in the estimate.

You measure 25 textbooks' weights, and find they have a mean weight of 62 ounces. Assume the population standard deviation is 2.6 ounces. Based on this, construct a 90% confidence interval for the true population mean textbook weight.

Calculate a 90% confidence interval for the true population mean of textbook weights, given a sample mean of 62 ounces, a sample size of 25, and a population standard deviation of 2.6 ounces.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation