Question

A manufacturer claims that their battery for a particular cell phone model lasts 10 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?

Answer 1

Set up the test
- H0: μ = 10 hours (manufacturer's claim)
- Ha: μ < 10 hours (consumers' complaint)

Data: n = 24, x̄ = 9.4, s = 0.25. Use a one-sample t-test (sample is random and assume approximate normality).

Test statistic
t = (x̄ − μ0) / (s/√n) = (9.4 − 10) / (0.25/√24)
= −0.6 / 0.05102 ≈ −11.76
df = 23.

p-value
The p-value for t ≤ −11.76 with 23 df is essentially 0 (p < 0.0001).

Conclusion
At any usual significance level (α = 0.05, 0.01, etc.) we reject H0. There is very strong evidence that the mean battery life is less than 10 hours.

Additional: a 95% CI for μ is
9.4 ± t0.975,23*(0.25/√24) ≈ 9.4 ± 0.1055 → (9.295, 9.506),
which does not include 10, consistent with the test.