Question

a genetic experiment with peas resulted in one sample of offspring that consisted of 417 green peas and 156 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.
328 < p < 421 (round to three decimal places as needed.)

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}$ of yellow peas is $\hat{p}=\frac{156}{417 + 156}=\frac{156}{573}\approx0.272$.

Step2: Calculate the standard error

The standard error $SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $n = 573$.
$SE=\sqrt{\frac{0.272\times(1 - 0.272)}{573}}=\sqrt{\frac{0.272\times0.728}{573}}\approx\sqrt{\frac{0.198}{573}}\approx0.0186$.

Step3: Find the critical value

For a 95% confidence interval, the critical value $z$ (from the standard normal distribution) is $z = 1.96$.

Step4: Calculate the margin of error

The margin of error $ME=z\times SE=1.96\times0.0186\approx0.0365$.

Step5: Calculate the confidence interval

The lower limit is $\hat{p}-ME=0.272 - 0.0365 = 0.2355\approx0.236$.
The upper limit is $\hat{p}+ME=0.272+0.0365 = 0.3085\approx0.309$.

Answer:

a. $0.236 < p < 0.309$ (Note: The values you provided in the boxed area seem incorrect. The correct 95% confidence interval based on the formula is as calculated above).
b. Since the expected value of $0.25$ (25% in decimal form) lies within the confidence interval $(0.236,0.309)$, the results of the experiment do not appear to contradict the expectation that 25% of the offspring peas would be yellow.