Question

Suppose the math department chair at a large state university wants to estimate the average overall rating, out of five points, that students taking Introductory Statistics gave their lecturers on their end-of-term evaluations. She selects a random sample of 36 evaluations and records the summary statistics shown. Sample size Standard deviation of Confidence level Sample Sample standard Population standard mean deviation deviation x 3.8711 1.0755 1.0222 n S 0 с 02 0.1704 36 95% The department chair determines the standard deviation of the sampling distribution of x using o, the standard deviation calculated from all student evaluations submitted to her department. Assume the Statistics lecturers' ratings have the same standard deviation. Use this information to find the margin of error and the lower and upper limits of a 95% confidence interval for u, the mean overall rating students gave their Introductory Statistics lecturers. Round your answers to the nearest hundredth. Margin of error = Lower limit = Upper limit = Complete the following sentence to state the interpretation of the departinici Vran » confidence interval. Complete the following sentence to state the interpretation of the deparulCII Can O nvonfidence interval. The that the of the overall ratings given at the end of the term Answer Bank confidence interval the student evaluations in the sample probability is 95% all student evaluations of Statistics lecturers standard deviation is between the lower and upper limits mean standard deviation of x is 3.8711 department chair is 95% confident

Answer 1

Margin of error = 0.44, Lower limit = 3.85, Upper limit = 4.30. The 95% confidence interval suggests that the true mean overall rating of Introductory Statistics lecturers is likely between 3.85 and 4.30 based on the sample of 36 evaluations.

Step-by-step explanation:

The margin of error is calculated by multiplying the critical z-value for a 95% confidence interval by the standard deviation of the sampling distribution of the mean (calculated using the population standard deviation). In this case, the margin of error is 0.44.

To find the lower and upper limits of the confidence interval, we subtract and add the margin of error from the sample mean, respectively. The lower limit is 3.85, and the upper limit is 4.30.

In statistical terms, this means we are 95% confident that the true mean overall rating of Introductory Statistics lecturers lies between 3.85 and 4.30, based on the sample of 36 evaluations taken from the entire population. The margin of error represents the range within which we expect the true mean to fall, capturing the uncertainty associated with estimating population parameters from a sample.

Confidence intervals provide a range of values within which we can reasonably estimate the population parameter. They are essential in statistical inference, allowing researchers to make informed conclusions about a population based on a sample.

Answer 2

If we were to repeat this sampling and confidence interval construction process many times, 95% of the intervals we construct would contain the true population mean.

Margin of Error:

Formula: Margin of Error (ME) = Z-score * Standard Error

Z-score for 95% confidence level: 1.96 (found in standard normal distribution tables)

Standard Error: 0.1704 (provided)

ME = 1.96 * 0.1704 = 0.3336 (rounded to the nearest hundredth)

Lower Bound:

Formula: Lower Bound = Sample Mean - Margin of Error

Lower Bound = 3.8711 - 0.3336 = 3.5375 (rounded to the nearest hundredth)

Upper Bound:

Formula: Upper Bound = Sample Mean + Margin of Error

Upper Bound = 3.8711 + 0.3336 = 4.2047 (rounded to the nearest hundredth)

Interpretation of the Confidence Interval:

We are 95% confident that the mean overall rating given by students to their Introductory Statistics lecturers at the end of the term is between 3.54 and 4.20