A researcher compares the average math scores of $55$ randomly selected sophomore students and $60$ randomly selected junior students. The mean score for sophomores is $88$, with a population standard deviation of $12$. The mean score for juniors is $91$, with a population standard deviation of $15$. The claim is that the mean score for sophomores is greater than the mean score for juniors. Should the hypothesis test be left-tailed, right-tailed, or two-tailed, and should a $z$-test or $t$-test be used?

A researcher wants to know if the average test score at school X is higher than at school Y. For $60$ students at school X, the mean score is $78$, with a population standard deviation of $10$. For $55$ students at school Y, the mean is $74$, with a population standard deviation of $12$. What is the standardized test statistic?

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 (${\mu }_1-{\mu }_2$).

Determine the critical value(s) for the test: $H_a:{\mu }_1\ne {\mu }_2$, $\alpha =0.02$, $n_1=12$, $n_2=14$. Assume random, independent samples, normal populations, and equal population variances.

A botanist is studying the average height of a particular plant species in two different forests. In forest X, a sample of $22$ plants has a mean height of $54.3$ $\text{cm}$ and a standard deviation of $8.7$ $\text{cm}$. In forest Y, a sample of $28$ plants has a mean height of $58.9$ $\text{cm}$ and a standard deviation of $10.2$ $\text{cm}$. At $\alpha =0.05$, can you conclude that there is a significant difference in mean plant height between the two forests? Assume equal population variances.

Suppose we are comparing the average monthly grocery expenditures (in dollars) for households in three different regions. Independent random samples of size $6$ were taken from each region. The computed test statistic is $F=1.07$ for testing whether the mean monthly grocery expenditure differs between the three regions. Assume the populations are normally distributed and the population variances are equal. Use a significance level of $\alpha =0.10$. What conclusion can be drawn about the mean monthly grocery expenditures across the three regions?

A researcher wants to compare the variances of two different teaching methods using a two-sample $F$-test. Which condition must be satisfied before using this test?

Researchers are studying blood lead levels in two groups: children living near a highway with $n=35$, $\bar{x}=8.2\mu \text{g/dL}$, and $s=3.9\mu \text{g/dL}$, and children living in rural areas with $n=35$, $\bar{x}=2.7\mu \text{g/dL}$, and $s=1.6\mu \text{g/dL}$. At the $0.05$ significance level, test the claim that the variation in blood lead levels is greater for children living near the highway than for those in rural areas.