An automobile manufacturer claims that their van has a 39.3 miles/gallon (MPG) rating. An independent testing firm has tested 17 vans and found a mean MPG of 39.7 with a variance of 6.25. Is there sufficient evidence at the 0.05 level that the vans outperform the manufacturer's MPG rating?

Step 1 of 5: State the null and alternative hypotheses.

Null Hypothesis ([tex]H_0[/tex]): The mean MPG of the vans is equal to 39.3.

Alternative Hypothesis ([tex]H_a[/tex]): The mean MPG of the vans is greater than 39.3.

In this scenario, we are dealing with a hypothesis test to determine if there is sufficient evidence to support that the vans tested have a better miles per gallon (MPG) performance than the manufacturer's claim.

Step 1: State the null and alternative hypotheses.

Null Hypothesis (H₀): The null hypothesis is a statement that indicates no effect or no difference. In this context, it implies that the vans do not outperform the manufacturer's MPG rating. The null hypothesis can be defined as:
[tex]H_0: \mu \leq 39.3[/tex]
This means that the true mean MPG of the vans is less than or equal to 39.3.
Alternative Hypothesis (H₁): The alternative hypothesis is what we want to provide evidence for. It suggests that the vans outperform the manufacturer's MPG rating. The alternative hypothesis can be defined as:
[tex]H_1: \mu > 39.3[/tex]
This means that the true mean MPG of the vans is greater than 39.3.

With these hypotheses set, further steps of the hypothesis test would involve choosing the significance level (already provided as 0.05), calculating the test statistic, and making a decision based on the comparison of the calculated statistic to the critical value from the relevant statistical distribution, typically a t-distribution in this case due to the sample size and unknown population standard deviation.