91Ó°ÊÓ

Based on historical birthing records, the proportion of males born worldwide is \(0.51 .\) In other words, the commonly held belief that boys are just as likely as girls is false. Systematic lupus erythematosus (SLE), or lupus for short, is a disease in which one's immune system attacks healthy cells and tissue by mistake. It is well known that lupus tends to exist more in females than in males Researchers wondered, however, if families with a child who had lupus had a lower ratio of males to females than the general population. If this were true, it would suggest that something happens during conception that causes males to be conceived at a lower rate when the SLE gene is present. To determine if this hypothesis is true, the researchers obtained records of families with a child who had SLE A total of 23 males and 79 females were found to have SLE. The 23 males with SL \(E\) had \(a\) total of 23 male siblings and 22 female siblings The 79 females with SLE had a total of 69 male siblings and 80 female siblings Source L.N. Moorthy, M.G.E. Peterson, K.B. Onel, and T.J.A. Lehman. "Do Children with Lupus Have Fewer Male Siblings" Laprus 2008 \(17: 128-131,2008\) (a) Explain why this is an observational study. (b) Is the study retrospective or prospective? Why? (c) There are a total of \(23+69=92\) male siblings in the study How many female siblings are in the study? (d) Draw a relative frequency bar graph of gender of the siblings. (e) Find a point estimate for the proportion of male siblings in families where one of the children has SLE (f) Does the sample evidence suggest that the proportion of male siblings in families where one of the children has SLE is less than 0.51 , the accepted proportion of males born in the general population? Use the \(\alpha=0.05\) level of significance. (g) Construct a \(95 \%\) confidence interval for the proportion of male siblings in a family where one of the children has SLE.

Step by step solution

01

Why this is an observational study

An observational study is where researchers observe the effect of a variable without trying to change who is or isn't exposed to it. In this case, researchers are observing families with children who have SLE and the gender distribution of their siblings without any intervention.

02

Identify if the study is retrospective or prospective

The study is retrospective because it looks at historical birthing records and existing data on children with SLE and their siblings. The researchers are not setting up a study to follow these children into the future but are, instead, analyzing past records.

03

Calculate the number of female siblings

Given the total number of male siblings is 92, we calculate the total number of female siblings as follows. From the data given: 22 female siblings (males with SLE) + 80 female siblings (females with SLE) sums up to 102 female siblings.

04

Draw a relative frequency bar graph of gender of the siblings

A relative frequency bar graph represents the proportion of each category relative to the total. Calculate the relative frequency: \[ \text{Relative Frequency for Males} = \frac{92}{194} \approx 0.474 \] \[ \text{Relative Frequency for Females} = \frac{102}{194} \approx 0.526 \] The bar graph would have two bars, one for males (0.474) and one for females (0.526), on the y-axis with labels 'Males' and 'Females' on the x-axis.

05

Find a point estimate for the proportion of male siblings

The point estimate for the proportion of male siblings is calculated by dividing the number of male siblings by the total number of siblings. \[ \hat{p} = \frac{92}{194} \approx 0.474 \]

06

Determine if the sample evidence suggests a proportion less than 0.51

To determine if the proportion of male siblings is less than 0.51, conduct a hypothesis test. Set up the null hypothesis and alternative hypothesis: \( H_0: p = 0.51 \) \( H_a: p < 0.51 \). Calculate the test statistic using: \[ z = \frac{\hat{p} - 0.51}{\sqrt{\frac{0.51 (1-0.51)}{n}}} \] Where \( \hat{p} \) is the sample proportion = 0.474, \( p \) is the population proportion = 0.51, and \( n \) is the sample size = 194. Compute\( z \) and compare it with the critical value for a 0.05 significance level. If \( z \) is less than the critical value, reject the null hypothesis.

07

Construct a 95% confidence interval

To construct a 95% confidence interval for the proportion of male siblings, use: \[ \hat{p} \pm Z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] Where \( \hat{p} = 0.474 \), \( Z_{\frac{\alpha}{2}} = 1.96 \) for 95% confidence, and \( n = 194 \). Compute the margin of error and then the confidence interval.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Observational study

An observational study is a type of research where the investigator observes subjects and measures variables of interest without assigning treatments to the subjects. The essential feature of an observational study is that it investigates the relationship between variables without manipulation. In this specific case, researchers observed the families with children who have lupus (SLE) to study the gender distribution of their siblings. The researchers did not interfere with the families or their circumstances; they simply gathered existing data. This is a powerful method for understanding natural phenomena but it cannot establish causation since other underlying factors might affect the outcomes.

Retrospective study

A retrospective study looks backward in time, usually using medical records and interviews with patients who already have a known condition. This study on families with children who have SLE is retrospective because it analyzes pre-existing records of births. The researchers looked at data that had already been collected, rather than setting up an experiment and following subjects into the future. Conducting a retrospective study is efficient and cost-effective, but it can be limited by the accuracy and completeness of historical data.

Hypothesis testing

Hypothesis testing is a fundamental method in statistics used to determine the likelihood that a hypothesis about a population parameter is true, based on sample data. Here's how it's done in this study on SLE:

• First, researchers set up a null hypothesis (\( H_0 \)) that the proportion of male siblings is equal to the general population proportion of 0.51.
• The alternative hypothesis (\( H_a \)) proposes that the proportion of male siblings is less than 0.51.
• They then use a test statistic, in this case a z-score, to measure how far the sample proportion deviates from the null hypothesis proportion.
• The critical value approach is used to make a decision: if the computed z-value is less than the critical value at a specified significance level, they reject the null hypothesis in favor of the alternative hypothesis. This steps helps determine whether the sample evidence suggests a lower proportion of male siblings in families where one child has SLE.

Point estimate

A point estimate provides a single value as an estimate of a population parameter. In this study, the point estimate for the proportion of male siblings is calculated as the number of male siblings divided by the total number of siblings. The formula for the point estimate \( \text{pj =} = \frac{92}{194} \text{approx 0.474} \). This point estimate suggests that out of the total number of siblings (194), approximately 47.4% are male. While a point estimate gives a quick snapshot, it does not indicate the uncertainty or variability associated with the estimate.

Confidence interval

A confidence interval provides a range of values that is likely to contain the population parameter, giving more information than a point estimate about the uncertainty. In this study, we construct a 95% confidence interval for the proportion of male siblings using the formula:

• First, calculate the standard error of the point estimate
• Use the Z-score for a 95% confidence level (\( Z_{\frac{\text{alpha}}{2}} = 1.96 \))
< br >
• Combine these to form the interval:
\( 0.474 \text{ pm} 1.96 \text{ sqrt} \frac{0.474 (1 - 0.474)}{194}\)

• Compute the margin of error and determine the interval. The resulting confidence interval gives a range of values that, with 95% confidence, includes the true proportion of male siblings for families with a child with SLE, providing a more comprehensive understanding than a single point estimate.