To determine if the company should market the wax, we will conduct a hypothesis test.
First, we need to set up our null and alternative hypotheses. Given that the company needs the mean time to finish the course to be less than 55 seconds, we establish the following hypotheses:
- Null hypothesis, [tex]\(H_0\)[/tex]: The mean time to finish the course is equal to or greater than 55 seconds, i.e., [tex]\( \mu = 55 \)[/tex].
- Alternative hypothesis, [tex]\(H_A\)[/tex]: The mean time to finish the course is less than 55 seconds, i.e., [tex]\( \mu < 55 \)[/tex].
Thus, the correct choice of the hypotheses is:
D. [tex]\(H_0: \mu = 55\)[/tex]
[tex]\(H_A: \mu < 55\)[/tex]
Next, we will calculate the test statistic. For this, we need the sample mean, sample standard deviation, and sample size.
Given:
- The sample times are: [tex]\( [55.6, 65.1, 46.1, 51.2, 49.9, 46.8, 50.1, 44.6] \)[/tex]
- Sample size ([tex]\( n \)[/tex]) = 8
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 51.175 seconds (from the result)
- Sample standard deviation ([tex]\( s \)[/tex]) = 6.594 seconds (from the result)
We use the t-statistic formula for hypothesis testing with one sample:
[tex]\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \][/tex]
where:
- [tex]\( \bar{x} \)[/tex] is the sample mean
- [tex]\( \mu_0 \)[/tex] is the population mean (55 seconds)
- [tex]\( s \)[/tex] is the sample standard deviation
- [tex]\( n \)[/tex] is the sample size
Substitute the values into the formula:
[tex]\[ t = \frac{51.175 - 55}{6.594 / \sqrt{8}} \][/tex]
Compute the denominator:
[tex]\[ \frac{6.594}{\sqrt{8}} \approx 2.332 \][/tex]
Now calculate the test statistic:
[tex]\[ t = \frac{51.175 - 55}{2.332} \approx \frac{-3.825}{2.332} \approx -1.641 \][/tex]
So, the test statistic is:
[tex]\[ t \approx -1.641 \][/tex]
Thus, the test statistic is [tex]\(-1.641\)[/tex] rounded to three decimal places.
