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 company compares the mean monthly sales (in units) of two sales teams.

Team X: ${\bar{x}}_1=155$, ${\sigma }_1=15$, $n_1=30$.

Team Y: ${\bar{x}}_2=148$, ${\sigma }_2=12$, $n_2=35$.

Construct a $90\%$ confidence interval for the difference in mean sales (${\mu }_1-{\mu }_2$).

A company wants to test if the difference in average weekly hours worked by employees in its city A and city B offices is equal to $3$ hours. The following data are collected:

City A: ${\bar{x}}_1=42$, $n_1=25$, ${\sigma }_1=5$

City B: ${\bar{x}}_2=38$, $n_2=30$, ${\sigma }_2=6$

Test $H_0:{\mu }_1-{\mu }_2=3$ versus $H_a:{\mu }_1-{\mu }_2\ne 3$ at $\alpha =0.01$. What is your conclusion?