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The lifetime of special light-bulbs is normally distributed. A sample of 81 bulbs has produced a mean of 738 hours and a standard deviation of 38.2 hours.

Test a hypothesis $𝐻_0 ∶ 𝜇 = 747.5$ versus $𝐻-1 ∶ 𝜇 ≠ 747.5,$ at the significance levels of $𝛼_1 = 0.05$ and $𝛼_2 = 0.01.$ Find the p-value of the test, and use the p-value to verify your answers.

This is how I approached it but not sure if this was right:

$z=\frac{738-747.5}{38.2/√81} = -2.23.$

p-value = $P(z=-2.23)$ --> (1-0.9871) = 0.01

Reject at 0.05 because it is higher than p-value. Accept at 0.01 because == p-value.

I am extremely unsure about this so any help/verification would be great. Thank you for reading!

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You calculated z correctly, however, I noticed when I calculated, I got z=-2.238, so I'd go round to z=-2.24 (rather than z=-2.23), though it doesn't change any results.

You also calculated the p-value correctly. I don't know how many decimals you're required to give, but it may be useful to give one more (as then you can see $p>\alpha_2$).

When it comes to testing hypotheses, if $p<\alpha$ then you reject the null hypothesis, $H_o$, and accept the alternative hypothesis, $H_1$.

So, your answers are all correct. Good job.

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    $\begingroup$ do we always reject the null hypotheses if p<α? $\endgroup$
    – jawedib
    Dec 1, 2017 at 2:54
  • $\begingroup$ For observed $z= -2.23,$ the P-value of a two-sided test is $P(|Z| > 2.23) = 2P(Z < -2.23).$ Reject $H_0$ at level $\alpha$ if the P-value is less than $\alpha.$ $\endgroup$
    – BruceET
    Dec 1, 2017 at 17:37
  • $\begingroup$ Can the person who downrated my answer explain why? $\endgroup$
    – M. Yates
    Dec 1, 2017 at 22:09

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