Question

according to a report done by s & j power, the mean lifetime of the light bulbs it manufactures is 55 months. a consumer group suspects that the actual mean lifetime of light - bulbs made by this manufacturer is less than 55 months. the group tests this by selecting 100 bulbs at random. for the bulbs in the sample, the mean lifetime is 53.4 months and the standard deviation of the lifetimes is 6 months. can we conclude, at the 0.01 level of significance, that the population mean lifetime μ of light - bulbs made by this manufacturer differs from 55 months? perform a two - tailed test. then complete the parts below. carry your intermediate computations to three or more decimal places, and round your responses as specified in the questions. (a) state the null hypothesis h0 and the alternative hypothesis h1. h0: μ = 55 h1: μ < 55 (b) determine the type of test statistic to use. z (c) find the value of the test statistic. (round to three or more decimal places.) (d) find the p - value. (round to three or more decimal places.) (e) can we conclude that the population mean lifetime of light bulbs made by this manufacturer differs from 55 months? yes no

Explanation:

Step1: Identify given values

$\mu_0 = 55$ (hypothesized population mean), $\bar{x}=6$ (sample mean), $n = 100$ (sample size), assume $\sigma$ is known (since we use z - test). Let's assume $\sigma = 6$ (not given in the problem, for illustration purposes. If population standard - deviation is unknown and $n\geq30$, we can approximate with sample standard - deviation).

Step2: Calculate the z - test statistic

The formula for the z - test statistic in a one - sample mean test is $z=\frac{\bar{x}-\mu_0}{\frac{\sigma}{\sqrt{n}}}$. Substituting the values: $z=\frac{6 - 55}{\frac{6}{\sqrt{100}}}=\frac{- 49}{\frac{6}{10}}=\frac{-49}{0.6}\approx - 81.667$.

Step3: Calculate the p - value

Since it is a two - tailed test, the p - value is $2P(Z<|z|)$. For $z\approx - 81.667$, $P(Z < - 81.667)\approx0$. So the p - value is approximately $2\times0 = 0$.

Step4: Make a decision

The significance level $\alpha=0.01$. Since the p - value ($0$) is less than $\alpha = 0.01$, we reject the null hypothesis.

Answer:

(a) The null hypothesis $H_0:\mu = 55$ and the alternative hypothesis for a two - tailed test should be $H_1:\mu
eq55$ (the given $H_1:\mu < 55$ is incorrect for a two - tailed test).
(b) The type of test statistic to use is $z$.
(c) The value of the test statistic is approximately $-81.667$.
(d) The p - value is approximately $0$.
(e) Yes