Question

multiple choice 4 points
assume a normal distribution and use a hypothesis test to test the given claim.
a firework fuse is designed to have an average burn - time of 5 seconds before ignition. for liability reasons, the company that makes these fuses undergoes random testing to make sure that its fuses meet this standard. a random sample of 20 fuses results in an average burn - time of 5.3 seconds with a sample standard deviation of 0.7 seconds.
assuming the burn - time for an individual fuse is normally distributed, perform the appropriate hypothesis test at a 5% significance level to determine if the fuses perform according to design.
there is insufficient evidence to conclude the burn time is not 5 seconds.
there is sufficient evidence to conclude the burn time is less than 5 seconds.
there is sufficient evidence to conclude the burn time is greater than 5 seconds.
there is sufficient evidence to conclude the burn time is not 5 seconds.
there is insufficient evidence to conclude the burn time is 5 seconds.

Explanation:

Step1: State the hypotheses

Let $\mu$ be the population mean burn - time.
The null hypothesis $H_0:\mu = 5$
The alternative hypothesis $H_1:\mu
eq5$

Step2: Calculate the test statistic

The formula for the $z$ - test statistic is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$
Given $\bar{x} = 5.3$, $\mu = 5$, $\sigma=0.7$, $n = 20$
$z=\frac{5.3 - 5}{\frac{0.7}{\sqrt{20}}}=\frac{0.3}{\frac{0.7}{4.472}}\approx\frac{0.3}{0.1566}\approx1.91$

Step3: Determine the critical values

For a two - tailed test with $\alpha = 0.05$, the critical values are $z_{\alpha/2}=\pm1.96$

Step4: Make a decision

Since $z = 1.91$ and $- 1.96<1.91<1.96$ (i.e., the test statistic does not fall in the rejection region)
We fail to reject the null hypothesis.

Answer:

There is insufficient evidence to conclude the burn time is not 5 seconds.