Questions: A genetic experiment with peas resulted in one sample of offspring that consisted of 442 green peas and 167 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. p < (Round to three decimal places as needed.)

Solution Steps

The sample consists of \(442\) green peas and \(167\) yellow peas. The total number of peas is:

\[ n = 442 + 167 = 609 \]

The sample proportion of yellow peas is calculated as:

\[ \hat{p} = \frac{167}{609} \approx 0.274 \]

To construct a \(95\%\) confidence interval for the proportion of yellow peas, we use the formula:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

where \(z\) is the critical value for \(95\%\) confidence, which is approximately \(1.96\).

Calculating the standard error:

\[ \text{SE} = \sqrt{\frac{0.274(1 - 0.274)}{609}} \approx 0.025 \]

Now, we can calculate the confidence interval:

\[ 0.274 \pm 1.96 \cdot 0.025 \]

Calculating the margin of error:

\[ 1.96 \cdot 0.025 \approx 0.049 \]

Thus, the confidence interval is:

\[ (0.274 - 0.049, 0.274 + 0.049) = (0.239, 0.31) \]

The expected proportion of yellow peas is \(0.25\). We check if this value lies within the confidence interval:

\[ 0.239 < 0.25 < 0.31 \]

Since \(0.25\) is within the interval \((0.239, 0.31)\), the results do not contradict the expectation that \(25\%\) of the offspring peas would be yellow.

Final Answer