Question

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

Answer 1

I don’t have the numeric values from your problem (the claimed lifetime, the sample size n, the sample mean x̄, and the sample standard deviation s were left blank). Give those four numbers and I can run the test and give a conclusion.

In the meantime here is the exact test procedure you should use (consumers complain the batteries last less than the claimed value, so this is a one‑sided test):

1. Specify hypotheses
- Null: H0: μ = μ0 (or H0: μ ≥ μ0) where μ0 is the manufacturer’s claimed lifetime.
- Alternative: H1: μ < μ0 (consumers claim less).

2. Test statistic (use t because σ unknown)
t = (x̄ − μ0) / (s / √n)
degrees of freedom = n − 1.

3. Decision rule
- Find the p‑value = P(T_{n−1} ≤ t) (left‑tail probability).
- At significance level α (commonly 0.05): if p ≤ α, reject H0 (evidence the true mean is less than claimed); if p > α, do not reject H0.

Or use critical t: reject H0 if t ≤ t_{α, n−1} (the α quantile of the t distribution).

4. State conclusion in context
- If rejected: “There is statistically significant evidence at the α level that the mean battery life is less than μ0 hours.”
- If not rejected: “There is not enough evidence to conclude the mean battery life is less than μ0 hours.”

If you want, paste the four numbers (μ0, n, x̄, s) and your chosen α and I’ll compute the t value, p‑value, and give the conclusion.