A national organization has been working with utilities throughout the nation to find sites for large wind machines that generate electricity. Wind speeds must average more than 19 miles per hour (mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a particular site. Based on a sample of 45 wind speed recordings (taken at random intervals), the wind speed at the site averaged 19.9 mph, with a sample standard deviation of 4.5 mph. To determine whether the site meets the organization's requirements, consider the test Ha: μ > 19, where μ is the true mean wind speed at the site. For a significance level of α = 0.01, do we reject the null hypothesis? Which of the following is an appropriate conclusion?

Since the p value is higher than the significance level we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is not significantly higher than 19 mph

Step-by-step explanation:

Information given

[tex]\bar X=19.9[/tex] represent the sample mean

[tex]s=4.5[/tex] represent the sample standard deviation

[tex]n=45[/tex] sample size

[tex]\mu_o =19[/tex] represent the value to verify

[tex]\alpha=0.01[/tex] represent the significance level for the hypothesis test.

t would represent the statistic

[tex]p_v[/tex] represent the p value

Hypothesis to test

We want to verify if the true mean is higher than 19, the system of hypothesis would be:

Null hypothesis:[tex]\mu \leq 19[/tex]

Alternative hypothesis:[tex]\mu > 19[/tex]

The statistic would be given by:

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)

Replacing the info given we got:

[tex]t=\frac{19.9-19}{\frac{4.5}{\sqrt{45}}}=1.342[/tex]

The degrees of freedom are given by:

[tex] df=n-1= 45-1=44[/tex]

And the p value would be;

[tex]p_v =P(t_{44}>1.342)=0.0932[/tex]

Since the p value is higher than the significance level we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true mean is not significantly higher than 19 mph