Questions: Question 27, 6.3.18 Part 3 of 3 HW Score: 50.76%, 23.86 of 47 points Chapter 6 Review Homework Kennedy Pierre 11/04/24 8:04 PM A genetics experiment involves a population of fruit flies consisting of 3 males named Al, Billy, and Carl and 1 female named Dana. Assume that two fruit flies are randomly selected with replacement. a. After listing the possible samples and finding the proportion of males in each sample, use a table to describe the sampling distribution of the proportion of males. Proportion of males Probability 0 0.0625 0.5 0.375 1 0.5625 (Type integers or fractions.) b. Find the mean of the sampling distribution. μ=0.75 (Round to two decimal places as needed.) c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of males? If so, does the mean of the sampling distribution of proportions always equal the population proportion? A. No, the sample mean is equal to the population proportion of males. These values are not always equal, because proportion is a biased estimator. B. Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is a biased estimator. C. No, the sample mean is equal to the population proportion of males. These values are not always equal, because proportion is an unbiased estimator. D. Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is an unbiased estimator.

Question 27, 6.3.18 Part 3 of 3 HW Score: 50.76%, 23.86 of 47 points Chapter 6 Review Homework Kennedy Pierre 11/04/24 8:04 PM A genetics experiment involves a population of fruit flies consisting of 3 males named Al, Billy, and Carl and 1 female named Dana. Assume that two fruit flies are randomly selected with replacement. a. After listing the possible samples and finding the proportion of males in each sample, use a table to describe the sampling distribution of the proportion of males. Proportion of males Probability 0 0.0625 0.5 0.375 1 0.5625 (Type integers or fractions.) b. Find the mean of the sampling distribution. μ=0.75 (Round to two decimal places as needed.) c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of males? If so, does the mean of the sampling distribution of proportions always equal the population proportion? A. No, the sample mean is equal to the population proportion of males. These values are not always equal, because proportion is a biased estimator. B. Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is a biased estimator. C. No, the sample mean is equal to the population proportion of males. These values are not always equal, because proportion is an unbiased estimator. D. Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is an unbiased estimator.
Transcript text: Question 27, 6.3.18 Part 3 of 3 HW Score: $50.76 \%, 23.86$ of 47 points apter 6 Review Homework lannedy pierre 11/04/24 8:04 PM A genetics experiment involves a population of fruit flies consisting of 3 males named Al , Billy, and Carl and 1 female named Dana. Assume that two frit flies are randomly selected with replacement. a. After listing the possible samples and finding the proportion of males in each sample, use a table to describe the sampling distribution of the proportion of males. \begin{tabular}{cc} Proportion of males & Probability \\ \hline 0 & 0.0625 \\ 0.5 & 0.375 \\ 1 & 0.5625 \\ \hline \end{tabular} (Type integers or fractions.) b. Find the mean of the sampling distribution. $\mu=0.75$ (Round to two decimal places as needed.) c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of males? If so, does the mean of the sampling distribution of proportions always equal the population proportion? A. No, the sample mean is equal to the population proportion of males. These values are not always equal, because proportion is a biased estimator. B. Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is a biased estimator. C. No, the sample mean is equal to the population proportion of males. These values are not always equal, because proportion is an unbiased estimator. D. Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is an unbiased estimator.
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Solution

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Solution Steps

Step 1: Calculate the Mean of the Sampling Distribution

The mean of the sampling distribution is calculated using the formula:

\[ \text{Mean} = 0 \times 0.0625 + 0.5 \times 0.375 + 1 \times 0.5625 = 0.75 \]

Step 2: Calculate the Variance of the Sampling Distribution

The variance \( \sigma^2 \) is calculated as follows:

\[ \sigma^2 = (0 - 0.75)^2 \times 0.0625 + (0.5 - 0.75)^2 \times 0.375 + (1 - 0.75)^2 \times 0.5625 = 0.09 \]

Step 3: Calculate the Standard Deviation of the Sampling Distribution

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{0.09} = 0.3 \]

Step 4: Compare the Mean with the Population Proportion

The population proportion of males is given by:

\[ \text{Population proportion of males} = \frac{3}{4} = 0.75 \]

Since the mean of the sampling distribution \( 0.75 \) is equal to the population proportion of males \( 0.75 \), we conclude that:

\[ \text{Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is an unbiased estimator.} \]

Final Answer

The mean of the sampling distribution is \( 0.75 \), the population proportion of males is \( 0.75 \), and the conclusion is that the sample mean is equal to the population proportion of males.

Thus, the answer is:

\(\boxed{\text{Yes, the sample mean is equal to the population proportion of males. These values are always equal, because proportion is an unbiased estimator.}}\)

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