Question

it currently takes users a mean of 44 minutes to install the most popular computer program made by rodetech, a software design company. after changes have been made to the program, the company executives want to know if the new mean is now different from 44 minutes so that they can change their advertising accordingly. a simple random sample of 51 new customers are asked to time how long it takes for them to install the software. the sample mean is 47.8 minutes with a standard deviation of 10.2 minutes. perform a hypothesis test at the 0.10 level of significance to see if the mean installation time has changed. step 3 of 3: draw a conclusion and interpret the decision. answer we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance that the new mean installation time is different from 44 minutes and that rodetech may need to change their advertising. we reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance that the new mean installation time is different from 44 minutes and that rodetech may need to change their advertising. we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.10 level of significance that the new mean installation time is different from 44 minutes and that rodetech may need to change their advertising. we reject the null hypothesis and conclude that there is insufficient evidence at a 0.10 level of significance that the new mean installation time is different from 44 minutes and that rodetech may need to change their advertising.

Explanation:

Step1: Calculate the test - statistic

The formula for the one - sample z - test statistic is $z=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}$, where $\bar{x} = 47.8$ (sample mean), $\mu = 44$ (population mean), $s = 10.2$ (sample standard deviation), and $n = 51$ (sample size).
$z=\frac{47.8 - 44}{\frac{10.2}{\sqrt{51}}}=\frac{3.8}{\frac{10.2}{7.1414}}=\frac{3.8}{1.4283}\approx2.66$

Step2: Determine the critical values

For a two - tailed test at $\alpha=0.10$ level of significance, the critical values are $z_{\alpha/2}=\pm 1.645$.

Step3: Make a decision

Since $|z| = 2.66>1.645$, we reject the null hypothesis. This means there is sufficient evidence at a 0.10 level of significance that the new mean installation time is different from 44 minutes and that RodeTech may need to change their advertising.

Answer:

We reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance that the new mean installation time is different from 44 minutes and that RodeTech may need to change their advertising.