Question

A safety officer wants to prove that μ = the average speed of cars driven by a school is less than 25 mph. Suppose that a random sample of 14 cars shows an average speed of 24.0 mph, with a sample standard deviation of 2.2 mph. Assume that the speeds of cars are normally distributed. For a significance level of α = 0.05, do we reject the null hypothesis? Which of the following is an appropriate conclusion?

Answer 1

Answer:

$t=\frac{24-25}{\frac{2.2}{\sqrt{14}}}=-1.70$    

$p_v =P(t_{(13)}<-1.70)=0.0565$  

If we compare the p value and the significance level given $\alpha=0.05$ and we can see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL reject the null hypothesis, so we can't conclude that the true average speed is lower than 25 mph at 5% of signficance.  

Step-by-step explanation:

Data given and notation  

$\bar X=24$ represent the sample mean

$s=2.2$ represent the sample standard deviation

$n=14$ sample size  

$\mu_o =25$ represent the value that we want to test

$\alpha=0.05$ represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

$p_v$ represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the true mean for the average speed is lower than 25 mph, the system of hypothesis would be:  

Null hypothesis:$\mu \geq 25$  

Alternative hypothesis:$\mu < 25$  

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$  (1)  

Calculate the statistic

We can replace in formula (1) the info given like this:  

$t=\frac{24-25}{\frac{2.2}{\sqrt{14}}}=-1.70$    

P-value

The first step is calculate the degrees of freedom, on this case:  

$df=n-1=14-1=13$  

Since is a one side test the p value would be:  

$p_v =P(t_{(13)}<-1.70)=0.0565$  

Conclusion  

If we compare the p value and the significance level given $\alpha=0.05$ and we can see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL reject the null hypothesis, so we can't conclude that the true average speed is lower than 25 mph at 5% of signficance.