Question

23 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 greater than 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 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.
\(H_0:\mu = 5\) (null hypothesis, the fuses meet the design standard)
\(H_1:\mu
eq5\) (alternative hypothesis, a two - tailed test)

Step2: Calculate the test statistic

The formula for the \(t\) - test statistic is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\)
where \(\bar{x} = 5.3\), \(\mu = 5\), \(s = 0.7\), \(n = 20\)
\(t=\frac{5.3 - 5}{0.7/\sqrt{20}}\)
\(t=\frac{0.3}{0.7/4.472}\)
\(t=\frac{0.3}{0.1565}\approx1.917\)

Step3: Determine the critical value

For a two - tailed test with \(\alpha = 0.05\) and \(n-1=19\) degrees of freedom, the critical values are \(t_{\alpha/2,n - 1}=\pm 2.093\)

Step4: Make a decision

Since \(|t|=1.917<2.093\), we fail to reject the null hypothesis.

Answer:

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