Question

studies show that smaller class sizes have a direct impact on student success. researchers selected a random sample of 50 elementary school classrooms across the united states and found that the mean class size was 22.66 students with a standard deviation of 0.9 students. they also selected a random sample of 54 high school classrooms across the united states and found that the mean class size was 22.79 with a standard deviation of 2.1 students.

check each of conditions that have been met to construct a two-sample \\(t\\) confidence interval for the difference in mean class sizes between elementary and high schools.

• the sample of elementary classrooms is random
• the average classroom size is at least 10
• the sample size for elementary classrooms is at least 30
• the sample size for high school classrooms is at least 30
• the samples are independent
• the sample of high school classrooms is random

Explanation:

Verify the random sampling conditions

Using the Two-Sample t-Interval Conditions knowledge point

• The problem states: "Researchers selected a random sample of 50 elementary school classrooms..."
• The problem states: "They also selected a random sample of 54 high school classrooms..."
• Thus, both "The sample of elementary classrooms is random" and "The sample of high school classrooms is random" are met.

Verify the independence conditions

Using the Two-Sample t-Interval Conditions knowledge point

• The two samples (elementary school classrooms and high school classrooms) are selected independently of each other.
• Thus, "The samples are independent" is met.

Verify the normality/sample size conditions

Using the Two-Sample t-Interval Conditions knowledge point

• For a two-sample \(t\)-interval, the sampling distributions must be approximately normal. Since the population distributions are unknown, we check the Central Limit Theorem (CLT) condition where each sample size must be at least 30.
• For elementary classrooms: \(n_1 = 50 \ge 30\).
• For high school classrooms: \(n_2 = 54 \ge 30\).
• Thus, both "The sample size for elementary classrooms is at least 30" and "The sample size for high school classrooms is at least 30" are met.

Answer:

• ☑ The sample of elementary classrooms is random (Correct answer)
• ☐ The average classroom size is at least 10
• ☑ The sample size for elementary classrooms is at least 30 (Correct answer)
• ☑ The sample size for high school classrooms is at least 30 (Correct answer)
• ☑ The samples are independent (Correct answer)
• ☑ The sample of high school classrooms is random (Correct answer)