Questions: A group of 49 randomly selected students has a mean age of 22.4 years with a population standard deviation of 3.8. Use this sample data to construct a 98% confidence interval for the mean age of all students.

A group of 49 randomly selected students has a mean age of 22.4 years with a population standard deviation of 3.8. Use this sample data to construct a 98% confidence interval for the mean age of all students.
Transcript text: A group of 49 randomly selected students has a mean age of 22.4 years with a population standard deviation of 3.8. Use this sample data to construct a 98\% confidence interval for the mean age of all students.
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Solution

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Solution Steps

Step 1: Determine the Appropriate Distribution

Given that the population standard deviation is known and the sample size is 49, we use the Z-distribution.

Step 2: Calculate the Standard Error (SE)

The standard error (SE) is calculated as SE = \(\frac{std_dev}{\sqrt{sample_size}}\) = 0.543.

Step 3: Find the Critical Value

The critical value for a 98% confidence level with Z-distribution is 2.326.

Step 4: Calculate the Margin of Error (ME)

The margin of error (ME) is calculated as ME = critical value × SE = 1.263.

Step 5: Construct the Confidence Interval

The 98% confidence interval for the population mean is (21.14, 23.66).

Final Answer: The 98% confidence interval for the population mean, rounded to 2 decimal places, is (21.14, 23.66).

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