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Chapter 6: Problem 15

When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg (where \(\mathrm{b}=\) boy and \(\mathrm{g}=\) girl). 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?

Short Answer

Expert verified
The mean of the sample proportions equals the population proportion of girls (0.5). This suggests that the sample proportion is an unbiased estimator of the population proportion.

Step by step solution

01

Define the Sample Space

The sample space for the genders of two births is: bb: both children are boys bg: first child is a boy, second child is a girl gb: first child is a girl, second child is a boy gg: both children are girls.
02

Identify the Sample Proportions

Calculate the proportion of girls for each outcome in the sample space:- For the outcome bb, the proportion of girls is \(\frac{0}{2} = 0\)- For the outcome bg, the proportion of girls is \(\frac{1}{2} = 0.5\)- For the outcome gb, the proportion of girls is \(\frac{1}{2} = 0.5\)- For the outcome gg, the proportion of girls is \(\frac{2}{2} = 1\)
03

Construct the Sampling Distribution Table

Construct a table for the proportion of girls:\[\begin{array}{c|c}\text{Outcome} & \text{Proportion of Girls} \hlinebb & 0 \bg & 0.5 \gb & 0.5 \gg & 1 \end{array}\]
04

Calculate the Mean of the Sample Proportions

Find the mean of the sample proportions:\[\bar{p} = \frac{0 + 0.5 + 0.5 + 1}{4} = \frac{2}{4} = 0.5\]
05

Compare with the Population Proportion of Girls

Since there are 2 girls out of a total of 4 (2 boys and 2 girls), the population proportion of girls is:\[p = \frac{2}{4} = 0.5\]Compare this to the mean of the sample proportions (0.5). Since they are equal, this suggests that the sample proportion is an unbiased estimator of the population proportion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Space
The concept of sample space is essential in probability and statistics. It represents all the possible outcomes of a given random experiment. In the exercise, we deal with the genders of two births. The sample space for this scenario is the set of all possible gender combinations:
  • bb: both children are boys
  • bg: first child is a boy, second child is a girl
  • gb: first child is a girl, second child is a boy
  • gg: both children are girls
Each combination has an equal likelihood. Understanding the sample space helps us determine probabilities and calculate other statistical measures efficiently.
Sample Proportion
The sample proportion is an essential concept in inferential statistics. It describes the proportion of a specific outcome in a random sample. In this exercise, we calculate the proportion of girls in each possible outcome from our sample space:
  • For bb, the proportion of girls is \(\frac{0}{2} = 0\)
  • For bg, the proportion of girls is \(\frac{1}{2} = 0.5\)
  • For gb, the proportion of girls is \(\frac{1}{2} = 0.5\)
  • For gg, the proportion of girls is \(\frac{2}{2} = 1\)
Each proportion gives us a snapshot of the specific sample's characteristic relative to the total sample size.
Unbiased Estimator
An unbiased estimator is a statistical measure that is expected to equal the parameter it estimates. In this exercise, we explore whether the sample proportion can serve as an unbiased estimator for the population proportion of girls. By comparing the mean of the sample proportions (0.5) with the population proportion (also 0.5), we observe that they are equal. This equality suggests that the sample proportion accurately estimates the population proportion, indicating it is an unbiased estimator.
Population Proportion
The population proportion represents the fraction of the total population that possesses a particular characteristic. In our scenario, the population proportion of girls among the two births is calculated as follows:
Since there are two girls and two boys in the sample space, the population proportion of girls is \(\frac{2}{4} = 0.5\). Comparing this population proportion with the sample proportions helps us determine how well a sample can estimate the broader population characteristic.
Mean of Sample Proportions
The mean of sample proportions provides a central value for the proportions obtained from different samples. To calculate this mean in our example, we use the proportions obtained for each outcome: 0, 0.5, 0.5, and 1. The mean is calculated as follows: \(\bar{p} = \frac{0 + 0.5 + 0.5 + 1}{4} = 0.5\). By comparing this mean with the population proportion, we can evaluate whether our sample proportion is unbiased. In this case, the mean of sample proportions (0.5) matches the population proportion (0.5), confirming the sample proportion is an unbiased estimator.

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Most popular questions from this chapter

Computers are commonly used to randomly generate digits of telephone numbers to be called when conducting a survey. Can the methods of this section be used to find the probability that when one digit is randomly generated, it is less than 3 ? Why or why not? What is the probability of getting a digit less than 3 ?

In the year that this exercise was written, there were 879 challenges made to referee calls in professional tennis singles play. Among those challenges, 231 challenges were upheld with the call overturned. Assume that in general, \(25 \%\) of the challenges are successfully upheld with the call overturned. a. If the \(25 \%\) rate is correct, find the probability that among the 879 challenges, the number of overturned calls is exactly 231 . b. If the \(25 \%\) rate is correct, find the probability that among the 879 challenges, the number of overturned calls is 231 or more. If the \(25 \%\) rate is correct, is 231 overturned calls among 879 challenges a result that is significantly high?

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2. About \(\quad \%\) of the area is between \(z=-3.5\) and \(z=3.5\) (or within \(3.5\) standard deviations of the mean).

The U.S. Air Force once used ACES-II ejection seats designed for men weighing between \(140 \mathrm{lb}\) and \(211 \mathrm{lb}\). Given that women's weights are normally distributed with a mean of \(171.1 \mathrm{lb}\) and a standard deviation of \(46.1 \mathrm{lb}\) (based on data from the National Health Survey), what percentage of women have weights that are within those limits? Were many women excluded with those past specifications?

The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. a. In a letter to "Dear Abby," a wife claimed to have given birth 308 days after a brief visit from her husband, who was working in another country. Find the probability of a pregnancy lasting 308 days or longer. What does the result suggest? b. If we stipulate that a baby is premature if the duration of pregnancy is in the lowest \(3 \%\), find the duration that separates premature babies from those who are not premature. Premature babies often require special care, and this result could be helpful to hospital administrators in planning for that care.
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