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From the image you uploaded, it appears you're working on a statistical problem involving a random sample of 45 phones. Here's a breakdown of the given dataFrom the image you uploaded, it appears you're working on a statistical problem involving a random sample of 45 phones. Here's a breakdown of the given 

- **Number of phones**: 45
- **Sample mean**: 6.46
- **Sample standard deviation**: 2.37
- **Population standard deviation**: 2.42

The goal seems to involve calculating the **standard error**, **margin of error**, and potentially constructing a confidence interval based on the critical value provided in the table. 

### Standard Error (SE)
The formula for standard error is:

\[
SE = \frac{\sigma}{\sqrt{n}}
\]

Where:
- \( \sigma \) is the population standard deviation (2.42 in this case),
- \( n \) is the sample size (45 in this case).

\[
SE = \frac{2.42}{\sqrt{45}} \approx 0.3607
\]

### Margin of Error (ME)
The margin of error is calculated as:

\[
ME = z \times SE
\]

Where:
- \( z \) is the critical value (which appears to be 2.576 from the image),
- \( SE \) is the standard error calculated above.

\[
ME = 2.576 \times 0.3607 \approx 0.9283
\]

### Confidence Interval
To construct the confidence interval, you would add and subtract the margin of error from the sample mean:

\[
CI = (\bar{x} - ME, \bar{x} + ME)
\]

\[
CI = (6.46 - 0.9283, 6.46 + 0.9283)
\]

\[
CI = (5.53, 7.39)
\]

So the 99% confidence interval for the population mean is approximately \( (5.53, 7.39) \).

---

Would you like further details on any of these steps?

Here are five related questions:
1. How does increasing the sample size affect the standard error?
2. Why do we use the population standard deviation instead of the sample standard deviation in this calculation?
3. What does the critical value represent in terms of confidence level?
4. How would the confidence interval change if the critical value were lower?
5. What is the relationship between the confidence level and the width of the confidence interval?

**Tip**: Remember, a smaller standard error means that the sample mean is a more precise estimate of the population mean!

Given a random sample of 45 phones with a sample mean of 6.46, a sample standard deviation of 2.37, and a population standard deviation of 2.42, calculate the standard error, margin of error, and construct a confidence interval.

Learn how to calculate the standard error, margin of error, and 99% confidence interval for a sample of 45 phones with a sample mean of 6.46 and a population standard deviation of 2.42. This statistical problem involves key concepts such as standard error, margin of error, and confidence intervals.

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