Questions: Help aline Live Cameron Terry 10/17/24 6:47 PM Question 28, 6.3.15 HW Score: 21.15%, 5.5 of 26 Part 1 of 3 Points: 0 of 1 Save When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg. Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion? For the entire population, assume the probability of having a boy is 1/2, the probability of having a girl is 1/2, and this is not affected by how many boys or girls have previously been born. Determine the probabilities of each sample proportion. Sample proportion of girls Probability V □ ▽ □ V □ (Type integers or simplified fractions.)

Solution Steps

To solve this problem, we need to determine the sample proportions of girls in two births and their corresponding probabilities. The sample space consists of four equally likely outcomes: bb, bg, gb, and gg. We will calculate the proportion of girls for each outcome and then determine the probability of each sample proportion. Finally, we will check if the mean of these sample proportions equals the population proportion of girls.

The sample space for two births regarding gender is given by the outcomes: \[ \text{Sample Space} = \{ \text{bb}, \text{bg}, \text{gb}, \text{gg} \} \]

For each outcome, we calculate the sample proportion of girls:

• For \(\text{bb}\): \( \frac{0}{2} = 0.0 \)
• For \(\text{bg}\): \( \frac{1}{2} = 0.5 \)
• For \(\text{gb}\): \( \frac{1}{2} = 0.5 \)
• For \(\text{gg}\): \( \frac{2}{2} = 1.0 \)

Thus, the sample proportions are: \[ \text{Proportions} = \{ 0.0, 0.5, 0.5, 1.0 \} \]

Next, we calculate the probabilities of each sample proportion:

• \( P(0.0) = \frac{1}{4} = 0.25 \)
• \( P(0.5) = \frac{2}{4} = 0.5 \)
• \( P(1.0) = \frac{1}{4} = 0.25 \)

Thus, the probabilities are: \[ \text{Probabilities} = \{ 0.0: 0.25, 0.5: 0.5, 1.0: 0.25 \} \]

The mean of the sample proportions is calculated as follows: \[ \text{Mean} = (0.0 \times 0.25) + (0.5 \times 0.5) + (1.0 \times 0.25) = 0.0 + 0.25 + 0.25 = 0.5 \]

The population proportion of girls is: \[ \text{Population Proportion} = \frac{1}{2} = 0.5 \]

Final Answer