A nutritionist asserts that the average calorie count of breakfast sandwiches from two different chains, chain A and chain B, is the same. For chain A, the mean calorie count from $30$ sandwiches is $420$ with a population standard deviation of $50$. For chain B, the mean from $28$ sandwiches is $445$ with a population standard deviation of $60$. At $\alpha =0.10$, is there sufficient evidence to reject the nutritionist's assertion?

A nutritionist randomly assigned $65$ students to receive either a regular ($150\text{ mL}$) or extra-large ($300\text{ mL}$) smoothie before class. After class, the nutritionist measured how much smoothie each student drank. The $32$ students with the regular smoothie drank an average of $92.4\text{ mL}$ with a standard deviation of $18.2\text{ mL}$. The $33$ students with the extra-large smoothie drank an average of $121.7\text{ mL}$ with a standard deviation of $15.5\text{ mL}$. Construct a $95\%$ confidence interval for the difference in mean smoothie consumption between the two groups.

A psychologist measures the difference in reaction time (in seconds) for $20$ participants before and after consuming caffeine. The mean difference ($\text{after - before}$) is $0.82$ seconds, with a standard deviation of $1.20$ seconds. Assume the differences are normally distributed with no outliers. Test at the $0.05$ significance level whether the mean difference in reaction time is greater than $0$.

A researcher is studying the variability in blood glucose levels between two groups of diabetic patients. Group A ($n=51$) receives a new medication, while group B ($n=51$) receives a placebo. The standard deviation of blood glucose levels in group A is $22.5\text{ mg/dL}$, and in group B it is $15.3\text{ mg/dL}$. Assume blood glucose levels are normally distributed. At the $\alpha =0.01$ significance level, is the standard deviation in group A significantly higher than in group B?

A researcher wishes to use the $z$-test to compare the means of two groups. Which requirement is not necessary for the $z$-test?