A nutritionist wants to compare the effects of two different diets on weight loss. After $8$ weeks, the following statistics are recorded:

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Use degrees of freedom equal to the smaller of $n_1-1$ and $n_2-1$. At the $0.05$ significance level, test the claim that there is no significant difference in mean weight loss between the two diets.

An ecologist records the average daily temperature in degrees Celsius and the number of bird species observed in $7$ different locations: $(12,5),(15,7),(18,9),(10,4),(20,11),(14,6),(16,8)$. At a significance level of $\alpha$ equals $0.05$, is there evidence of a linear correlation between temperature and the number of bird species? Use the critical value approach.

A tire manufacturer claims that the mean mileage of its competitor’s tires is less than $40,000$ miles. You are asked to conduct a hypothesis test to evaluate this claim. How would you state the null and alternative hypotheses if you were representing the competitor and wanted to reject the claim?