A certain vehicle emission inspection station advertises that the wait time for customers is less than 8 minutes. A local resident wants to test this claim and collects a random sample of 64 wait times for customers at the testing station. She finds that the sample mean is 7.43 minutes, with a standard deviation of 3.6 minutes. Does the sample evidence support the inspection station's claim? Use the alphaequals0.005 level of significance to test the advertised claim that the wait time is less than 8 minutes.

For this case since the p value is higher than the significance level we have enough evidence to conclude that the true mean for the wait time is not significantly less than 8 minutes

Step-by-step explanation:

Information given

[tex]\bar X=7.43[/tex] represent the sampke mean in minutes

[tex]s=3.6[/tex] represent the sample standard deviation

[tex]n=64[/tex] sample size

[tex]\mu_o =8[/tex] represent the value to verify

[tex]\alpha=0.005[/tex] represent the significance level

t would represent the statistic (variable of interest)

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

System of hypothesis

For this case we are trying to proof if inspection station advertises that the wait time for customers is less than 8 minutes, the system of hypothesis would be:

Null hypothesis:[tex]\mu \geq 8[/tex]

Alternative hypothesis:[tex]\mu < 8[/tex]

Since the population deviation is not known the statistic can be calculated with:

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

The statistic for this case is given by:

[tex]t=\frac{7.43-8}{\frac{3.6}{\sqrt{64}}}=-1.27[/tex]

The degrees of freedom are given by:

[tex]df=n-1=64-1=63[/tex]

The p value can be calculated like this:

[tex]p_v =P(t_{(63)}<-1.27)=0.104[/tex]

For this case since the p value is higher than the significance level we have enough evidence to conclude that the true mean for the wait time is not significantly less than 8 minutes