A survey is conducted to estimate the average number of text messages sent by high school girls daily. The population standard deviation (sigma) of the number of text messages sent by high school girls is known to 12 messages per day. A recent survey of 36 high school girls showed a sample average of 100 text messages sent per day.

What is the 90% confidence interval estimate (CIE) of the mean texts (m) that all high school girls send daily?

To calculate the 90% confidence interval (CIE) for the population mean \(\mu\) based on the sample, we use the formula for a confidence interval when the population standard deviation \(\sigma\) is known:

\[
CI = \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\bar{x}\) is the sample mean (100 messages/day),
- \(\sigma\) is the population standard deviation (12 messages/day),
- \(n\) is the sample size (36),
- \(z_{\alpha/2}\) is the z-value for a 90% confidence level (1.645 for 90% confidence).

### Step-by-step calculation:

1. **Sample mean**: \(\bar{x} = 100\)
2. **Population standard deviation**: \(\sigma = 12\)
3. **Sample size**: \(n = 36\)
4. **Z-value for 90% confidence level**: \(z_{\alpha/2} = 1.645\)

Now, calculate the margin of error:

\[
ME = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} = 1.645 \cdot \frac{12}{\sqrt{36}} = 1.645 \cdot 2 = 3.29
\]

### Confidence Interval:

\[
CI = 100 \pm 3.29
\]

So, the 90% confidence interval for the population mean is:

\[
CI = (100 - 3.29, 100 + 3.29) = (96.71, 103.29)
\]

### Conclusion:
The 90% confidence interval estimate of the mean number of text messages sent daily by high school girls is between **96.71 and 103.29 messages**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the confidence interval change if the confidence level was increased to 95%?
2. What happens to the confidence interval if the sample size is increased?
3. Can this method be used if the population standard deviation is unknown? Why or why not?
4. What is the effect of increasing the population standard deviation on the confidence interval?
5. How does the z-value influence the width of the confidence interval?

**Tip**: A larger sample size reduces the margin of error, providing a narrower confidence interval, which gives a more precise estimate of the population mean.

A survey is conducted to estimate the average number of text messages sent by high school girls daily. The population standard deviation (sigma) of the number of text messages sent by high school girls is known to 12 messages per day. A recent survey of 36 high school girls showed a sample average of 100 text messages sent per day. What is the 90% confidence interval estimate (CIE) of the mean texts (m) that all high school girls send daily?

Calculate the 90% confidence interval for the average number of text messages sent by high school girls daily. With a sample mean of 100 messages, a known population standard deviation of 12, and a sample size of 36, find the margin of error and the confidence interval range. Learn the step-by-step solution using the z-value of 1.645 for 90% confidence. Suitable for Grades 10-12.

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