To find the 99% confidence interval for the true mean chloride level of all healthy Alaska residents, we'll follow these steps:

Step 1: Identify the Given Information

• Sample mean $\bar{x} = 103$ mEq/L
• Population standard deviation $\sigma = 40$ mEq/L
• Sample size $n = 90$
• Confidence level = 99%

Step 2: Find the Critical Value

Since the population standard deviation is known and the sample size is large ($n \geq 30$), we will use the Z-distribution to find the critical value.

For a 99% confidence level, the critical value $Z_{\alpha/2}$ can be found using a Z-table or standard normal distribution calculator. The value for a 99% confidence level is approximately $Z_{\alpha/2} = 2.576$.

Step 3: Calculate the Standard Error

The standard error (SE) of the mean is calculated using the formula:

$SE = \frac{\sigma}{\sqrt{n}}$

Substitute the values:

$SE = \frac{40}{\sqrt{90}} \approx \frac{40}{9.487} \approx 4.216$

Step 4: Calculate the Margin of Error (ME)

The margin of error (ME) is calculated using the formula:

$ME = Z_{\alpha/2} \times SE$

Substitute the values:

$ME = 2.576 \times 4.216 \approx 10.858$

Step 5: Calculate the Confidence Interval

The confidence interval is calculated using the formula:

$\text{Lower limit} = \bar{x} - ME$ $\text{Upper limit} = \bar{x} + ME$

Substitute the values:

$\text{Lower limit} = 103 - 10.858 \approx 92.1 \text{ mEq/L}$ $\text{Upper limit} = 103 + 10.858 \approx 113.9 \text{ mEq/L}$

Final Answer:

• Lower limit: 92.1 mEq/L
• Upper limit: 113.9 mEq/L

These values give the 99% confidence interval for the true mean chloride level of all healthy Alaska residents.

Would you like more details or have any other questions? Here are some related questions you might find useful:

1. How does increasing the confidence level affect the width of the confidence interval?
2. What are the assumptions behind using the Z-distribution for confidence intervals?
3. How would the confidence interval change if the sample size were smaller?
4. What role does the standard deviation play in determining the confidence interval?
5. How can we interpret this confidence interval in the context of the population?

Tip: Always ensure to use a larger sample size when possible, as it narrows the confidence interval, providing a more precise estimate of the population parameter.