Question

The data set (75, 85, 58, 72, 70, 75) is a random sample from the normal distribution No(µ, σ). Determine a 95% two-sided confidence interval for the mean µ .

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To calculate a 95% two-sided confidence interval for the mean µ, we need to use the formula:
Confidence interval = Mean ± (critical value * standard deviation / √n)
First, we need to calculate the critical value for a 95% confidence interval. Since it is a two-sided confidence interval, we need to find the z-value for (1 - (1 - 0.95)/2) = 0.975 in the standard normal distribution.
Using a standard normal distribution table or a calculator, the z-value for a 95% confidence interval is approximately 1.96.
Next, we need to calculate the mean and standard deviation of the given data set.
Mean (µ) = (75 + 85 + 58 + 72 + 70 + 75) / 6 = 72.5
Standard Deviation (σ) = 8.8
Now, we can plug the values into the confidence interval formula:
Confidence interval = 72.5± (1.96 * 8.8/ √6) = 72.5± 7.04
Therefore, the 95% two-sided confidence interval for the mean µ is approximately (65.46, 79.54).

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