Question

Peter is registering for classes next semester. He uses the Quantitative Reasoning Process to decide between two teachers, Dr. Ary, and Dr. Barlock . He speaks to 18 friends that previously took the course from Dr. Ary and also speaks to 18 friends that took it from Dr. Barlock. Eight of his friends said they highly recommend Dr. Ary. Eleven of his friends highly recommend Dr. Barlock. Based on Peter's minimal study, what percent of students "highly recommend" Dr. Ary? % (Round the final percentage to one decimal place) What percent "highly recommend" Dr.Barlock? % (Round the final percentage to one decimal place) To be statistically accurate in his decision, Peter decides to create a 95% confidence interval for his two percentages. What values does he find for each confidence interval? Confidence interval for the true proportion that "highly recommend" Dr Ary: (Enter the low value in the first blank and the high value in the second blank. Round values to one decimal place) %, to % Confidence interval for the true proportion that "highly recommend" Dr. Barlock: (Enter the low value in the first blank and the high value in the second blank. Round values to one decimal place) %, to %

Answer 1

Peter found that approximately 44.4% of students highly recommend Dr. Ary, while 61.1% highly recommend Dr. Barlock. For a 95% confidence interval, the range for the true proportion of students who highly recommend Dr. Ary is approximately 26.5% to 62.3%, and for Dr. Barlock, it is approximately 42.2% to 79.9%.

Step-by-step explanation:

In Peter's study, 8 out of 18 friends recommended Dr. Ary, so the sample proportion is 8/18 = 0.444, or 44.4%. For Dr. Barlock, 11 out of 18 friends recommended him, resulting in a sample proportion of 11/18 = 0.611, or 61.1%. These percentages give an initial insight into the preferences, but for statistical accuracy, Peter calculates 95% confidence intervals for both proportions.

To calculate the confidence interval for Dr. Ary, Peter uses the formula:

`\[ \text{Confidence Interval} = \text{Sample Proportion} \pm \text{Margin of Error} \]`

The margin of error is determined by the formula:

`\[ \text{Margin of Error} = \text{Critical Value} * \sqrt{\frac{\text{Sample Proportion} * (1 - \text{Sample Proportion})}{\text{Sample Size}}} \]`

Using a Z-table for a 95% confidence level, the critical value is approximately 1.96. For Dr. Ary, the margin of error is calculated, and the confidence interval is obtained.

The same process is applied to calculate the confidence interval for Dr. Barlock. The critical value remains 1.96, and the margin of error is computed.

This allows Peter to present a more comprehensive view of the range within which the true proportion of students who highly recommend each professor is likely to fall, considering the sample variability.