Step 1 :Given the sample heights of [65.8, 66.9, 73, 69.7, 65.3, 62.7, 71.5, 73.3], we first calculate the mean and standard deviation of these heights.
Step 2 :The mean of the sample heights is calculated as \( \frac{65.8 + 66.9 + 73 + 69.7 + 65.3 + 62.7 + 71.5 + 73.3}{8} = 68.525 \)
Step 3 :The standard deviation of the sample heights is calculated as \( \sqrt{\frac{(65.8-68.525)^2 + (66.9-68.525)^2 + (73-68.525)^2 + (69.7-68.525)^2 + (65.3-68.525)^2 + (62.7-68.525)^2 + (71.5-68.525)^2 + (73.3-68.525)^2}{8-1}} = 3.918 \)
Step 4 :Next, we calculate the margin of error for a 90% confidence interval. The Z-score for a 90% confidence interval is approximately 1.645.
Step 5 :The margin of error is calculated as \( 1.645 \times \frac{3.918}{\sqrt{8}} = 2.278 \)
Step 6 :Finally, we calculate the 90% confidence interval for the average height. The lower bound is calculated as \( 68.525 - 2.278 = 66.247 \) and the upper bound is calculated as \( 68.525 + 2.278 = 70.803 \)
Step 7 :So, the 90% confidence interval for the average height is approximately \(\boxed{(66.25, 70.80)}\)