https://file.aiquickdraw.com/mathbot/file/1730084522823ntoaujir.jpg

Let's break down and solve the problem step-by-step.

---

### Part (a): Constructing the 95% Confidence Interval

1. **Determine the sample proportion (p-hat):**  
   \( p\hat{} = \frac{166}{449 + 166} = \frac{166}{615} \approx 0.2707 \)

2. **Formula for the Confidence Interval for a Proportion:**  
   \[
   \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
   \]
   where:
   - \( \hat{p} \) is the sample proportion = 0.2707  
   - \( n \) is the total sample size = 615  
   - \( Z \) for 95% confidence = 1.96

3. **Calculate the standard error:**  
   \[
   SE = \sqrt{\frac{0.2707 \cdot (1 - 0.2707)}{615}} = \sqrt{\frac{0.2707 \cdot 0.7293}{615}} \approx \sqrt{0.0003207} \approx 0.0179
   \]

4. **Compute the margin of error:**  
   \[
   ME = 1.96 \cdot 0.0179 \approx 0.0351
   \]

5. **Confidence interval:**  
   \[
   0.2707 \pm 0.0351
   \]
   Lower bound: \( 0.2707 - 0.0351 \approx 0.236 \)  
   Upper bound: \( 0.2707 + 0.0351 \approx 0.306 \)

   So, the **95% confidence interval** is:  
   \[
   0.236 < p < 0.306
   \]

---

### Part (b): Evaluating the Expectation of 25%

The expectation is that **25%** of the offspring peas would be yellow, meaning the hypothesized proportion \( p_0 = 0.25 \). We need to check if 0.25 lies within the confidence interval \( 0.236 < p < 0.306 \).

- Since **0.25 is within the interval**, the results **do not contradict** the expectation that 25% of the peas would be yellow.

---

Would you like further clarification on any part of this solution?

---

### 5 Related Questions:
1. What is the Z-value for other confidence levels, such as 99%?
2. How would the confidence interval change with a smaller sample size?
3. What if the proportion of yellow peas was 30% instead—would that contradict the hypothesis?
4. How does increasing the confidence level affect the width of the confidence interval?
5. Can confidence intervals be used for other types of data besides proportions?

### Tip:
Always interpret the confidence interval in the context of the problem. It tells you that you are 95% confident that the true population proportion lies within the interval you calculated.

A genetic experiment with peas resulted in one sample of offspring that consisted of 449 green peas and 166 yellow peas. 

(a) Construct a 95% confidence interval to estimate the percentage of yellow peas. 
(b) Based on the confidence interval, do the results of the experiment appear to contradict the expectation that 25% of the offspring peas would be yellow?

Learn how to construct a 95% confidence interval for the proportion of yellow peas in a genetic experiment, where 166 out of 615 peas are yellow. This solution uses the formula for confidence intervals of proportions, allowing us to estimate the percentage of yellow peas and assess if the results contradict a 25% expected proportion.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation