Hypothesis testing: Test statistic, P-value and significance levels

Question

A manufacturer claims his light bulbs have a mean life $μ = 1800$ hours. A consumer group tested a random sample of $n = 250$ bulbs and found them to have a sample mean life $\bar{x} = 1790$ hours and a sample standard deviation $s = 50$ hours. Assess the manufacturer's claim.

1. what is $H_0$?
2. What is $H_a$?
3. What is the value of the test statistic?
4. In what range does the P-value reside?
5. Are the results statistically significant at the .05 level of significance?

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So I have the following information:

$μ = 1800 = H_O$

$n = 250$

$\bar{x} = 1790 = H_a$

$s = 50$

My test statistic would follow from the equation $\dfrac{\bar x - μ}{s/\sqrt{n}}$, giving $\dfrac{1790 - 1800}{50/\sqrt{250}} = -3.16$. This gives me a P-value of $0.0008$.

This means that the P-value resides in the range $P ≤ 0.01$. I think???

For the last question, I simply don't know what to do. Does it involve finding $Z_{0.05}$, then comparing it with the P-Value?

Answer 1

This depends on how you define range, but you have computed the test statistic and p-value correctly. The last question is much easier than you think. In order for the results in a hypothesis test to be statistically significant (in other words, to be able to reject the null hypothesis with sufficient statistical evidence), if we let $p$ be the p-value and $\alpha$ the significance level, we must have $p \leq \alpha$. Since you have calculated $p = 8 \times 10^{-4}$ here, it is clearly true for $\alpha = 0.05$ that $p \leq \alpha$. Thus, the answer to the last question is yes. Note that there are several ways to evaluate statistical significance, but this is one that is convenient in this context. To echo the comment of @AndréNicolas, I recommend changing the alternative hypothesis to $H_a : \mu \neq 1800$ (remember this is two-sided and the p-value is computed differently than in a standard one-sided test) or $H_a : \mu < 1800$, since this is the convention. The sample mean has nothing to do with the choice of hypotheses in practice.