One of Mendel’s famous genetics experiments yielded 580 peas, with 428 of them green and 152 yellow. a. Find a 99% confidence interval estimate of the percentage of green peas. b. Based on his theory of genetics, Mendel expected that 75% of the offspring peas would be green. Given that the percentage of offspring green peas is not 75%, do the results contradict Mendel’s theory? Why or why not?
The Correct Answer and Explanation is :
Answers:
(a) 99% Confidence Interval for the Percentage of Green Peas:
The confidence interval for the percentage of green peas is (69.09%, 78.50%).
(b) Do the results contradict Mendel’s theory?
No, the results do not contradict Mendel’s theory because the expected proportion of green peas (75%) falls within the 99% confidence interval (69.09% to 78.50%).
Explanation:
Gregor Mendel, the father of genetics, predicted that 75% of the offspring peas should be green based on his laws of inheritance. In this experiment, 428 out of 580 peas were green, which gives an observed proportion of:
[
\hat{p} = \frac{428}{580} \approx 0.7379 \text{ or } 73.79\%
]
To determine if this observed proportion significantly deviates from Mendel’s expected proportion, we calculate a 99% confidence interval for the true proportion of green peas in the population.
Using the normal approximation method, the standard error (SE) is calculated as:
[
SE = \sqrt{\frac{\hat{p} (1 – \hat{p})}{n}} = \sqrt{\frac{(0.7379)(1 – 0.7379)}{580}} \approx 0.0188
]
The Z-score for a 99% confidence level is 2.576.
The margin of error (ME) is:
[
ME = Z \times SE = 2.576 \times 0.0188 \approx 0.0487
]
Thus, the 99% confidence interval is:
[
\hat{p} \pm ME = 0.7379 \pm 0.0487 = (0.6909, 0.7849)
]
Converting to percentages:
[
(69.09\%, 78.50\%)
]
Since Mendel’s expected percentage of green peas (75%) is within this confidence interval, we cannot conclude that the experimental results contradict Mendel’s theory. The variation observed is likely due to natural random sampling variability rather than an error in Mendel’s predictions.