Question

answer in the simplest way and like a teen would understand
A manufacturer claims that their battery for a particular cell phone model ​lasts 10 hours.
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​However, consumers complained that the batteries lasted ​less than 10 hours.
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​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.
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​Based on the data given, is there evidence ​that the manufacturer’s claim is false?
​Justify your answer.

also actually explain the math because you sometimes put random slashes and confuse me

Answer 1

Short answer: Yes. There is very strong evidence the batteries last less than 10 hours.

Step-by-step (simple math, no weird slashes):

1) Set up the test
- Claim says mean life mu = 10 hours.
- Consumers say it’s less, so we test:
- Null hypothesis H0: mu = 10
- Alternative hypothesis Ha: mu < 10 (one-sided)

2) Use a t-test (because the population standard deviation is unknown and sample size is 24)
- Sample size n = 24
- Sample mean x̄ = 9.4 hours
- Sample standard deviation s = 0.25 hours
- Degrees of freedom = n − 1 = 23

3) Compute the standard error (how much the sample mean typically varies)
- sqrt(24) ≈ 4.899
- standard error = s divided by sqrt(n) = 0.25 / 4.899 ≈ 0.05103

4) Compute the t-statistic
- t = (x̄ − 10) divided by standard error
- That is t = (9.4 − 10) / 0.05103 = −0.6 / 0.05103 ≈ −11.76

5) Decide using t and p-value
- With 23 degrees of freedom, a t of −11.76 is extremely far into the left tail.
- The p-value is basically 0 (much smaller than 0.01).
- Even using a common cutoff like 0.05, this t is way beyond the critical value (about −1.714), so we reject H0.

6) Conclusion in plain teen talk
- The data show the average battery life is significantly less than 10 hours. The manufacturer’s 10-hour claim is not supported by this sample — it’s very likely false.