Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a

confidence interval estimate of p, then address the given question. Mendelian GeneticsOne 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 percentageof 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?

Short answer

a. The 99% confidence interval in terms of percentage of peas is between69.1% to 78.5%.

b. The result contradicts Mendel’s theory because 75% is included in the confidence interval.

Step-by-step solution

Given information

The number of peas from genetics experiments is recorded.

The number of sample values $n=580$.

The confidence interval is$99\%$

Check the requirement

The requirements for the test are:

1. The samples are selected randomly and normally distributed.
2. There are two categories of the outcome, either green or yellow.
3. The counts of successes and failures are 428 and 152respectively which are greater than 5

All the conditions are satisfied. Hence we can construct a confidence interval for the population proportion.

Calculate the sample proportion

a.

Thesample proportion of the green peas is:

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{428}{580} \\ =0.74 \end{gathered}$$

Therefore, the sample proportion is 0.74.

Then,

$$\begin{gathered} \hat{q}=1-\hat{p} \\ =1-0.74 \\ =0.26 \end{gathered}$$

Compute the critical value

At confidence interval, $\alpha =0.01$.

Using the standard normal table,

$$\begin{gathered} z_{crit}=z_{\frac{\alpha }{2}} \\ =z_{0.005} \\ =2.575 \end{gathered}$$

Compute margin of error

The margin of error is given by,

$$\begin{gathered} E=z_{crit}\times \sqrt{\frac{\hat{p}\hat{q}}{n}} \\ =2.575\times \sqrt{\frac{0.74\times 0.26}{580}} \\ =0.0472 \end{gathered}$$

The margin of error is 0.0472.

 Step 6: Compute the confidence interval

The formula for the 99% confidence interval is given by,

$$\begin{gathered} CI=\hat{p}-E<p<\hat{p}+E \\ =\left(0.74-0.0472<p<0.74+0.0472\right) \\ =(0.6907<p<0.7851) \end{gathered}$$

In percentage,99% confidence interval is between ..

Test the claim using the confidence interval

b.

It is claimed that the 99% confidence interval of the percentage of green peas is between 69.07% and 78.51%.

The interval includes 75% claimed value.

Therefore, we can expect that 75% of the offspring peas would be green.

Given the percentage of green peas is not 75%, the result contradicts Mendel’s theory.