(a)
83.5
Sample mean formula: \( \bar{x} = \frac{\sum x}{n} \).
Step: \( \sum x = 2506 \), \( n = 30 \), so \( \bar{x} = \frac{2506}{30} \approx 83.5333 \approx 83.5 \).
Result: 83.5 [1]

(b)
13.9
Variance formula: \( S^2_{n-1} = \frac{\sum x^2 – \frac{(\sum x)^2}{n}}{n-1} \).
Step: \( \frac{2506^2}{30} \approx 209,267.8667 \), \( \sum x^2 – \frac{(\sum x)^2}{n} = 209,738 – 209,267.8667 \approx 470.1333 \).
Step: \( \frac{470.1333}{29} \approx 13.9126 \approx 13.9 \).
Result: 13.9 [2]

(c)
(82.1, 84.9)
Confidence interval formula: \( \bar{x} \pm t \cdot \frac{s}{\sqrt{n}} \).
Step: \( s = \sqrt{13.9126} \approx 3.729 \), \( \frac{s}{\sqrt{30}} \approx 0.681 \), t-value (95%, df=29) \( \approx 2.045 \).
Step: Margin \( 2.045 \cdot 0.681 \approx 1.3927 \), interval \( 83.5333 \pm 1.3927 \approx (82.1406, 84.9260) \approx (82.1, 84.9) \).
Result: (82.1, 84.9) [2]

(d)
Talha would suggest that the manufacturer’s claim is incorrect because 85 is outside the 95% confidence interval (82.1, 84.9)
95% CI (82.1, 84.9) does not include 85, suggesting the true mean is likely not 85.
Result: Talha would suggest that the manufacturer’s claim is incorrect because 85 is outside the 95% confidence interval (82.1, 84.9) [1]