Question

a genetic experiment with peas resulted in one sample of offspring that consisted of 447 green peas and 165 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.
\\( \square < p < \square \\) (round to three decimal places as needed.)

Explanation:

Step1: Calculate the sample proportion $\hat{p}$

The total number of peas \(n=447 + 165=612\). The sample proportion of yellow peas \(\hat{p}=\frac{165}{612}\approx0.27\).

Step2: Calculate the critical value \(z_{\alpha/2}\)

For a \(95\%\) confidence interval, \(\alpha = 1- 0.95=0.05\), and \(\alpha/2=0.025\). From the standard normal distribution table, \(z_{\alpha/2}=z_{0.025} = 1.96\).

Step3: Calculate the margin of error \(E\)

The formula for the margin of error for a proportion is \(E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\).
Substitute \(\hat{p}=0.27\), \(n = 612\), and \(z_{\alpha/2}=1.96\) into the formula:

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Step4: Calculate the confidence interval

The confidence interval for the population proportion \(p\) is \(\hat{p}-ESubstitute \(\hat{p}=0.27\) and \(E = 0.035\) into the formula: \(0.27- 0.035

Answer:

\(0.235 < p < 0.305\)