Students at a local elementary school were randomly selected to participate in a reading fluency program. The program is designed to increase their reading fluency. A total of 17 students each took a pretest before the program and posttest after the program. The mean differences in the posttest and pretest is 11. The administration decided that all students in the school would participate in the program next school year. Let µD denote the mean difference in the student scores. The 90 percent confidence interval estimate of the mean difference for all students is (7, 15). What is an appropriate interpretation of the confidence interval?

The appropriate interpretation of the confidence interval is that we are 90% confident that the mean difference in the pre-test and post-test student scores for the entire the entire population of students in the school (µD) will be in the interval (7, 15).

It further means that the mean difference in the pre-test and post-test student scores for the entire school (µD) may or may not be in the interval, but we are 90% confident that, it exists and takes on a value in the interval (7, 15).

Step-by-step explanation:

Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence.

Mathematically, it is obtained from the relation,

Confidence Interval = (Sample mean) ± (Margin of error)

Margin of Error is the width of the confidence interval about the mean.

The margin of error is computed thus

Margin of Error = (Critical value) × (standard Error of the mean)

Critical value is obtained using the t-distribution. This is because there is no information provided for the population mean and standard deviation.

To find the critical value from the t-tables, we first find the degree of freedom and the significance level.

Degree of freedom = df = n - 1 = 17 - 1 = 16.

Significance level for 90% confidence interval

(100% - 90%)/2 = 5% = 0.05

t (0.05, 16)

And the standard error of the mean is computed from the samples of 17 students initially used.

At the end of the day, the confidence interval, (7,15) obtained, show that the sample mean was 11 and the margin of error was 4.

The confidence interval for the mean difference in pre-test and post-test student scores for the entire population of students in the school is then interpreted as defined above as the range of values that make up an interval where that mean difference in pre-test and post-test student scores for the entire population of students in the school, can be found with a confidence level of 90%.

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