(a)
83.5

Sample mean: \( \bar{x} = \frac{\sum x}{n} \)

\( \sum x = 2506 \), \( n = 30 \)

\( \bar{x} = \frac{2506}{30} \approx 83.5333 \)

Round to 1 decimal place: 83.5

Result: 83.5 [1]

(b)
13.9

Sample variance: \( s_{n-1}^2 = \frac{\sum x^2 – \frac{(\sum x)^2}{n}}{n-1} \)

\( \sum x^2 = 209738 \), \( \sum x = 2506 \), \( n = 30 \)

\( (\sum x)^2 = 2506^2 = 6280036 \)

\( \frac{(\sum x)^2}{n} = \frac{6280036}{30} \approx 209334.5333 \)

\( \sum x^2 – \frac{(\sum x)^2}{n} = 209738 – 209334.5333 \approx 403.4667 \)

\( s_{n-1}^2 = \frac{403.4667}{29} \approx 13.9126 \)

Round to 1 decimal place: 13.9

Result: 13.9 [2]

(c)
(82.1, 84.9)

95% confidence interval: \( \bar{x} \pm t \times \frac{s}{\sqrt{n}} \)

\( \bar{x} = 83.5333 \), \( s = \sqrt{13.9126} \approx 3.7298 \)

\( \frac{s}{\sqrt{n}} = \frac{3.7298}{\sqrt{30}} \approx 0.6810 \)

\( t \approx 2.045 \) (for 95% confidence, 29 df)

Margin of error: \( 2.045 \times 0.6810 \approx 1.3926 \)

Lower bound: \( 83.5333 – 1.3926 \approx 82.1407 \)

Upper bound: \( 83.5333 + 1.3926 \approx 84.9259 \)

Round to 1 decimal place: (82.1, 84.9)

Result: (82.1, 84.9) [2]

(d)
85 is outside the confidence interval; manufacturer’s claim is incorrect

Confidence interval: (82.1, 84.9)

Claim: \( \mu = 85 \)

85 is not within (82.1, 84.9)

Result: Manufacturer’s claim is incorrect [1]