Question

to determine if his teaching method increases students learning, a professor administers a pretest to his class at the beginning of the semester and then a posttest at the end of the semester. the results from 10 randomly chosen students are given below. let population 1 be the pretest scores and population 2 be the posttest scores.

test scores
pretest | posttest
50 | 71
73 | 82
64 | 80
50 | 62
86 | 92
76 | 71
60 | 74
71 | 80
76 | 81
70 | 68

step 2 of 2: interpret the confidence interval obtained in step 1.

answer

• since the confidence interval does not contain zero, the data do not provide evidence of a significant increase in students scores using the professors teaching method at a 90% confidence level.
• since the confidence interval contains zero, the data do not provide evidence of a significant increase in students scores using the professors teaching method at a 90% confidence level.
• we are 90% confident that the mean decrease in students scores from the pretest to the posttest is between 3.9 and 13.1 points for the population of students in this professors class.
• we are 90% confident that the mean increase in students scores from the pretest to the posttest is between 3.9 and 13.1 points for the population of students in this professors class.

Explanation:

Identify the parameter of interest and the direction of change

Population 1 represents the Pretest scores and Population 2 represents the Posttest scores. The professor wants to determine if his teaching method increases students' learning, which corresponds to an increase from Pretest to Posttest (i.e., Posttest scores are higher than Pretest scores, meaning \(\mu_2 > \mu_1\), or a mean increase \(\mu_d = \mu_2 - \mu_1 > 0\)).

Analyze the confidence interval values

The confidence interval obtained in Step 1 is between \(3.9\) and \(13.1\) points. Since both bounds are positive, this interval represents a positive difference, indicating a mean increase in scores from pretest to posttest.

Evaluate the multiple-choice options

• Option 1 and Option 2 discuss whether the interval contains zero, which is used for hypothesis testing conclusions rather than standard direct interval interpretation.
• Option 3 states "We are 90% confident that the mean decrease...". This is incorrect because the positive interval indicates an increase.
• Option 4 states "We are 90% confident that the mean increase in students' scores from the pretest to the posttest is between 3.9 and 13.1 points for the population of students in this professor's class." This correctly interprets the confidence level, the direction of change (increase), the interval bounds, and the target population.

Answer:

• Since the confidence interval does not contain zero, the data do not provide evidence of a significant increase in students' scores using the professor's teaching method at a 90% confidence level.
• Since the confidence interval contains zero, the data do not provide evidence of a significant increase in students' scores using the professor's teaching method at a 90% confidence level.
• We are 90% confident that the mean decrease in students' scores from the pretest to the posttest is between 3.9 and 13.1 points for the population of students in this professor's class.
• We are 90% confident that the mean increase in students' scores from the pretest to the posttest is between 3.9 and 13.1 points for the population of students in this professor's class. (Correct answer)