Question

A manufacturer claims that their battery for a particular cell phone model ​lasts10
hours.
​
​However, consumers complained that the batteries lasted ​less than 10
hours.
​
​The manufacturer randomly sampled
24 batteries and ​found that the mean battery life of the sample was
9.4 hours with a
​standard deviation of
0.25 hours.
​
​Based on the data given, is there evidence ​that the manufacturer’s claim is false?
​Justify your answer.
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Answer 1

Step 1 — hypotheses
- H0: μ = 10 (manufacturer’s claim)
- Ha: μ < 10 (consumers’ complaint)

Step 2 — test statistic (one-sample t-test)
n = 24, x̄ = 9.4, s = 0.25, df = 23
t = (x̄ − μ0) / (s/√n) = (9.4 − 10) / (0.25/√24)
= −0.6 / (0.25/4.899) ≈ −11.76

Step 3 — p-value and conclusion
For df = 23, P(T ≤ −11.76) is essentially 0 (p < 0.0001). At α = 0.05 (or any conventional α), p ≪ α, so reject H0. There is very strong evidence that the mean battery life is less than 10 hours.

Optional check — 95% CI for μ:
x̄ ± t0.975,23 · s/√n = 9.4 ± 2.069·0.0510 ≈ (9.294, 9.506),
which does not include 10 and agrees with the conclusion.

Assumptions: the sample was random and the t-test is reasonably robust to nonnormality for n = 24. Under these conditions, the data provide strong evidence the manufacturer’s claim is false.