lliott is registering for classes next semester. He uses the Quantitative Reasoning Process to decide between two teachers, Dr. Archer, and Dr. Budden . He speaks to 16 friends that previously took the course from Dr. Archer and also speaks to 20 friends that took it from Dr. Budden. Eight of his friends said they highly recommend Dr. Archer. Eleven of his friends highly recommend Dr. Budden.

 

Based on Elliott's minimal study, what percent of students "highly recommend" Dr. Archer? 

  
50
 % (Round the final percentage to one decimal place)

 

What percent "highly recommend" Dr.Budden?

  
55
 % (Round the final percentage to one decimal place)

 

To be statistically accurate in his decision, Elliott 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 Archer:

(Enter the low value in the first blank and the high value in the second blank. Round values to one decimal place)

25.5
 %,

to

74.5
 %

 

Confidence interval for the true proportion that "highly recommend" Dr. Budden:

(Enter the low value in the first blank and the high value in the second blank. Round values to one decimal place)

33.3
 %, 

to

76.7
 %

 

Do Elliott's confidence intervals show evidence that a greater percentage of students favor one teacher over the other?  

	
Yes, the confidence intervals overlap, so clearly one teacher is preferred by students over the other teacher.

	
Yes, the confidence intervals do not overlap, so clearly one teacher is preferred by students over the other teacher.

	
No, the confidence intervals overlap, so the percentages of students that approve of each teacher are really the same as far as we can tell.

	
No, the confidence intervals do not overlap, so the percentages of students that approve of each teacher are really the same as far as we can tell.



### Percent of students who "highly recommend" each teacher:
1. **Dr. Archer**:  
   - Number of students who highly recommend Dr. Archer = 8  
   - Total number of students who took Dr. Archer's class = 16  
   \[
   \text{Percentage for Dr. Archer} = \left(\frac{8}{16}\right) \times 100 = 50\%
   \]

2. **Dr. Budden**:  
   - Number of students who highly recommend Dr. Budden = 11  
   - Total number of students who took Dr. Budden's class = 20  
   \[
   \text{Percentage for Dr. Budden} = \left(\frac{11}{20}\right) \times 100 = 55\%
   \]

### Confidence Interval Calculation:
To calculate the 95% confidence interval for proportions, we use the following formula:
\[
CI = p \pm Z_{\alpha/2} \cdot \sqrt{\frac{p(1 - p)}{n}}
\]
Where:
- \( p \) is the sample proportion.
- \( Z_{\alpha/2} \) is the Z-score corresponding to a 95% confidence level (which is 1.96).
- \( n \) is the sample size.

#### 1. **Confidence Interval for Dr. Archer**:
- \( p = 0.50 \), \( n = 16 \)
\[
CI = 0.50 \pm 1.96 \times \sqrt{\frac{0.50(1 - 0.50)}{16}} = 0.50 \pm 1.96 \times \sqrt{\frac{0.25}{16}} = 0.50 \pm 1.96 \times 0.125 = 0.50 \pm 0.245
\]
Thus, the confidence interval for Dr. Archer is:
\[
CI = (0.50 - 0.245, 0.50 + 0.245) = (0.255, 0.745) = (25.5\%, 74.5\%)
\]

#### 2. **Confidence Interval for Dr. Budden**:
- \( p = 0.55 \), \( n = 20 \)
\[
CI = 0.55 \pm 1.96 \times \sqrt{\frac{0.55(1 - 0.55)}{20}} = 0.55 \pm 1.96 \times \sqrt{\frac{0.2475}{20}} = 0.55 \pm 1.96 \times 0.1113 = 0.55 \pm 0.218
\]
Thus, the confidence interval for Dr. Budden is:
\[
CI = (0.55 - 0.218, 0.55 + 0.218) = (0.332, 0.768) = (33.3\%, 76.7\%)
\]

### Conclusion:
- **Do the confidence intervals show evidence that one teacher is preferred?**  
  **No**, because the confidence intervals overlap (Dr. Archer's: 25.5% to 74.5%, Dr. Budden's: 33.3% to 76.7%). Since the intervals overlap, we cannot conclude that one teacher is clearly preferred over the other based on this data.

### Do you have any questions or need more details on this?

### 5 Relative Questions:
1. What is the importance of calculating a confidence interval?
2. How does the sample size affect the width of a confidence interval?
3. How is the Z-score determined for different confidence levels?
4. Can the confidence intervals be used to predict future student preferences?
5. What other factors might Elliott consider when choosing between the two teachers?

### Tip:
A larger sample size would lead to a narrower confidence interval, making your estimates more precise.

Elliott is registering for classes next semester. He uses the Quantitative Reasoning Process to decide between two teachers, Dr. Archer and Dr. Budden. He speaks to 16 friends that previously took the course from Dr. Archer and 20 friends that took it from Dr. Budden. Eight of his friends highly recommend Dr. Archer, and eleven highly recommend Dr. Budden. What percent of students highly recommend Dr. Archer and Dr. Budden, and what are the confidence intervals for each teacher?

Explore how Elliott calculates the percentage of students who highly recommend Dr. Archer and Dr. Budden. Learn to compute 95% confidence intervals for proportions and understand the statistical reasoning behind comparing two teachers based on survey data. This lesson covers key statistical concepts like confidence intervals and is ideal for students in Grades 10-12.

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