Question

The article "Boy or Girl: Which Gender Baby Would You Pick?" (LiveScience, March 23. 2005 , www.livescience.com) summarized the findings of a study that was published in Fertility and Sterility. The LiveScience article makes the following statements: "When given the opportunity to choose the sex of their baby, women are just as likely to choose pink socks as blue, a new study shows" and "Of the 561 women who participated in the study, 229 said they would like to choose the sex of a future child. Among these 229 , there was no greater demand for boys or girls." These statements are equivalent to the claim that for women who would like to choose the baby's sex, the proportion who would choose a girl is 0.50 or \(50 \%\). a. The journal article on which the LiveScience summary was based ("Preimplantation Sex-Selection Demand and Preferences in an Infertility Population," Fertility and Sterility [2005]: \(649-658\) ) states that of the 229 women who wanted to select the baby's sex, 89 wanted a boy and 140 wanted a girl. Does this provide convincing evidence against the statement of no preference in the LiveScience summary? Test the relevant hypotheses using \(\alpha=\) .05. Be sure to state any assumptions you must make about the way the sample was selected in order for your test to be appropriate. b. The journal article also provided the following information about the study: \- A survey with 19 questions was mailed to 1385 women who had visited the Center for Reproductive Medicine at Brigham and Women's Hospital. \- 561 women returned the survey. Do you think it is reasonable to generalize the results from this survey to a larger population? Do you have any concerns about the way the sample was selected or about potential sources of bias? Explain.

Short answer

To provide the short answer, we need to perform the calculations for Step 2 and Step 3. Depending on those calculations and critical reasoning for Step 4, the short answer can be established.

Step-by-step solution

Formulating the Hypotheses

The exercise gives a claim that states there's no preference amongst the women who'd like to choose their baby's sex, implying equal preference for a boy or a girl, or in other terms the proportion of preference is 0.50. We formulate the hypotheses as follows: Null Hypothesis (\(H_0\)): The proportion of women who would choose a girl is 0.50, Alternative Hypothesis (\(H_a\)): The proportion of women who would choose a girl is not 0.50.

Perform Hypothesis Testing

Let's perform a hypothesis testing using the given alpha level \(\alpha\)= 0.05. We will use a two-tailed z-test for one sample proportion: 1. Compute the sample proportion (\(p̂\)), which is the number of women who wanted a girl divided by the total number of women. (It's 140/229 = 0.61 approximately) 2. Calculate the test statistic (z), using the formula: z = \((p̂ - p0) / sqrt((p0*(1-p0)) / n)\) Where \(p̂\) is the sample proportion, \(p0\) is the expected proportion under null hypothesis, n is the sample size. (Our \(p̂\) is 0.61, \(p0\) is 0.50, n is 229) 3. Determine the critical z-value for a two-tailed z-test at \(\alpha\)= 0.05. (It is approximately ±1.96) 4. Compare the calculated test statistic with the critical value to decide about rejecting or not rejecting \(H_0\).

Assess the Result

Depending on the calculated z value and its comparison with critical value ±1.96, we can decide to reject or not reject the null hypothesis.

Generalize the Results

The next part deals with whether the results from this survey can be generalized to a larger population and any potential concerns about the way the sample was selected or about potential bias. To answer these questions, certain factors such as representativeness of the sample, responsiveness bias, non-response rate, etc., need to be considered.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental aspect of statistics, used to determine whether a statement about a population parameter is supported by sample data. In the given exercise, we are considering whether a preference exists between choosing a baby girl or a baby boy among women who have a choice.
When conducting hypothesis testing, we start by establishing two competing hypotheses:

• Null Hypothesis ( \( H_0 \)): This is a statement of no effect or no difference. In our case, it claims that exactly half (50%) of the women want to choose a girl.
• Alternative Hypothesis ( \( H_a \)): This suggests a different claim: the proportion is not equal to 50%.

The process involves calculating a test statistic, here a z-value, to compare the observed data to the hypothesis. If the calculated statistic is more extreme than a critical value, usually determined by a significance level (like \( \alpha = 0.05 \)), we reject the null hypothesis.
Thus, hypothesis testing allows us to make data-driven decisions regarding the preferences of the population from which the sample was drawn.

Survey Sampling

Survey sampling is a method used to collect data from a subset of the population to make inferences about the entire population. In this exercise, a survey was mailed to women who visited a medical center, and a portion of them responded.
Key considerations in survey sampling include:

• Random Sampling: Ensures that every individual has an equal chance of being selected, reducing bias.
• Sample Size: Larger samples tend to yield more accurate estimates of population parameters.

In the exercise, 561 out of 1385 women responded to the survey. While this is a good response rate, we should question whether the responding women are representative of the entire population of interest.
Survey sampling allows researchers to make generalizations from a manageable number of participants, but it's crucial to ensure the sample is as representative as possible to enhance the validity of the conclusions drawn.

Proportion Testing

Proportion testing is an aspect of hypothesis testing focused on comparing an observed sample proportion to a claimed population proportion. It's used when we want to see if the sample proportion is significantly different from a predefined value.

• Sample Proportion ( \( \hat{p} \)): Calculated by dividing the number of positive responses (like choosing a girl) by the total sample size. In this case, 140 out of 229 women, giving us \( \hat{p} \approx 0.61 \).
• Formula for the Test Statistic (z): \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \), helps in checking if the observed proportion significantly deviates from the hypothesized proportion (0.50 here).

A two-tailed test is often used in proportion testing to determine if the sample proportion is significantly higher or lower than the hypothesized value. The outcome of this test helps researchers understand if there's evidence suggesting a real preference in the population.

Sample Bias

Sample bias occurs when the collected survey responses do not accurately reflect the broader population due to the way the sample was chosen or who chose to participate. This can lead to skewed results and misleading conclusions.
In the scenario described in the exercise, there are several potential sources of sample bias:

• Selection Bias: Only women visiting a specific medical center were surveyed, which may not be a representative slice of all women.
• Response Bias: The possibility that women with strong opinions on sex selection are more likely to participate, thus distorting the results.
• Non-Response Bias: With only 561 out of 1385 responding, some views may be underrepresented, particularly those of non-responders.

Understanding the possibility of sample bias is crucial because it highlights any limitations in the generalizability of the study's findings. Addressing potential biases can improve both the reliability and the applicability of survey results.