A genetic experiment with peas resulted in one sample of offspring that consisted of 442 green peas and 152 yellow peas.
a. Construct a 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 of the offspring peas would be
a. Construct a confidence interval. Express the percentages in decimal form.
(Round to three decimal places as needed.)

Question

A genetic experiment with peas resulted in one sample of offspring that consisted of 442 green peas and 152 yellow peas.
a. Construct a 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 of the offspring peas would be
a. Construct a confidence interval. Express the percentages in decimal form.
(Round to three decimal places as needed.)

Ask by Estrada Ingram. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

We start by computing the sample proportion of yellow peas. Let $$ n=442+152=594 $$ and let $$ x=152 $$ be the number of yellow peas. Then the sample proportion is

$$
\hat{p}=\frac{152}{594}\approx0.256.
$$

The standard error (SE) for the proportion is computed as

$$
\text{SE}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.256\,(1-0.256)}{594}}.
$$

Calculating the product in the numerator:

$$
0.256 \times 0.744 \approx 0.190,
$$

so

$$
\text{SE}\approx\sqrt{\frac{0.190}{594}}\approx\sqrt{0.000320}\approx0.0179.
$$

For a $$95\%$$ confidence interval, we use the critical value $$ z=1.96 $$. Thus, the margin of error (ME) is

$$
\text{ME}=1.96\times0.0179\approx0.0351.
$$

The confidence interval for $$ p $$ is given by

$$
\hat{p}\pm\text{ME}\qquad \Longrightarrow \qquad 0.256\pm0.0351.
$$

This results in the interval

$$
0.2209 \text{ to } 0.2911.
$$

Rounded to three decimal places, the interval is

$$
0.221 < p < 0.291.
$$

For part (b), the expected percentage of yellow peas is $$25\%$$ or $$0.25$$ in decimal form, which lies within the interval $$0.221 < p < 0.291$$. Therefore, the experimental results do not contradict the expectation that $$25\%$$ of the offspring peas would be yellow.
a. The $$95\%$$ confidence interval for the percentage of yellow peas is $$0.221 < p < 0.291$$. b. The expected $$25\%$$ of yellow peas falls within the confidence interval, so the results do not contradict the expectation.

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