Question

a genetic experiment with peas resulted in one sample of offspring that consisted of 435 green peas and 151 yellow peas.
a. construct a 95% confidence interval to estimate of 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?
a. construct a 95% confidence interval. express the percentages in decimal form.
0.224 < p < 0.306 (round to three decimal places as needed.)
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?
yes, the confidence interval does not include 0.25, so the true percentage could not equal 25%.
no, the confidence interval includes 0.25, so the true percentage could easily equal 25%.

Explanation:

Step1: Calculate sample proportion

The sample proportion \( \hat{p}=\frac{151}{151 + 356}=\frac{151}{507}\approx0.298\)

Step2: Find critical value

For a 95% confidence interval, the critical value \(z = 1.96\)

Step3: Calculate margin of error

The margin of error \(E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
\(n = 507\), \(\hat{p}=0.298\), \(1-\hat{p}=0.702\)
\(E = 1.96\sqrt{\frac{0.298\times0.702}{507}}\approx1.96\times\sqrt{\frac{0.209}{507}}\approx1.96\times\sqrt{0.000412}\approx1.96\times0.0203\approx0.040\)

Step4: Construct confidence interval

The confidence interval is \(\hat{p}-E\(0.298-0.040 < p<0.298 + 0.040\)
\(0.258

Step5: Answer part b

Since \(0.25\) is not in the interval \(0.258 < p<0.338\), the confidence interval does not include \(0.25\), so the true percentage could not equal \(25\%\)

Answer:

a. \(0.26 < p<0.34\) (rounded to two decimal places)
b. Yes, the confidence interval does not include \(0.25\), so the true percentage could not equal \(25\%\)