Questions: In a genetics experiment on peas, one sample of offspring contained 440 green peas and 598 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green is the result reasonably close to the value of 3/4 that was expected? The probability of getting a green pea is approximately (Type an integer or decimal rounded to three decimal places as needed) Is this probability reasonably close to 3/4 ? Choose the correct answer below A. No, it is not reasonably close B. Yes, it is reasonably close.

Solution Steps

In the genetics experiment, the number of green peas is \(440\) and the number of yellow peas is \(598\). The total number of peas is:

\[ n = 440 + 598 = 1038 \]

The sample proportion of green peas is calculated as:

\[ \hat{p} = \frac{440}{1038} \approx 0.424 \]

We are testing the null hypothesis \(H_0: p = \frac{3}{4}\) against the alternative hypothesis \(H_a: p \neq \frac{3}{4}\). The hypothesized population proportion is:

\[ p_0 = \frac{3}{4} = 0.75 \]

The test statistic \(Z\) is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values:

\[ Z = \frac{0.424 - 0.75}{\sqrt{\frac{0.75(1 - 0.75)}{1038}}} \approx -24.2638 \]

The P-value associated with the test statistic \(Z = -24.2638\) is:

\[ \text{P-value} = 0.0 \]

For a significance level of \(\alpha = 0.05\), the critical region for a two-tailed test is:

\[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

Since \(Z = -24.2638\) falls within the critical region, we reject the null hypothesis.

Final Answer