Researcher Interested In The Relationship Between Body Mass

1 A Researcher Interested In The Relationship Between Body Mass Index

A researcher interested in the relationship between body mass index (BMI) and total serum cholesterol wished to fit a simple linear regression model in which total serum cholesterol is predicted from BMI using the following data. Use the data to compute the ten residuals for the subjects. Do the residual data support use of this model? Why or why not? How can you check to see if you calculated the residuals correctly? Hint for those doing this: Use SPSS to determine model. Calculate for each observation the residual. Look at SPSS options to plot residuals.

Subject #, Total cholesterol, BMI

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Additionally, the scenario involves testing a proportion concerning expatriates living overseas because of political or social attitudes:

Traditionally, 2 percent of the citizens of the United States live in a foreign country because they are disenchanted with U.S. politics or social attitudes. In order to test if this proportion has increased since the September 11, 2001, terror attacks, U.S. consulates contacted a random sample of 400 of these expatriates. The sample yields 12 people who report they are living overseas because of political or social attitudes. Can you conclude this data shows the proportion of politically motivated expatriates has increased?

a) State the null and alternative hypothesis.

b) Calculate the test statistic. Show your calculation.

c) Determine the p-value.

d) What is your decision regarding the null statement if α = 0.05?

e) Write a conclusion statement.

Paper For Above instruction

The investigation into the relationship between body mass index (BMI) and total serum cholesterol involves regression analysis to determine how well BMI predicts cholesterol levels among subjects. Additionally, an inquiry into the change in the proportion of U.S. expatriates living abroad due to political or social discontent considers hypothesis testing to assess if the proportion has increased since the 2001 terrorist attacks. This paper discusses methods for residual analysis in regression and conducting a hypothesis test for a population proportion, providing a comprehensive understanding of both statistical techniques.

Analysis of Residuals in BMI and Cholesterol Regression Model

Regression analysis aims to model the relationship between an independent variable, BMI, and a dependent variable, total serum cholesterol. After fitting a simple linear regression, residuals—the differences between observed and predicted values—are calculated to assess the suitability of the model. Residual analysis is crucial for diagnosing problems such as non-linearity, heteroscedasticity, or outliers, which can compromise the validity of inferences.

To compute residuals, the regression equation is required: Ŷ = b₀ + b₁X, where Ŷ is the predicted serum cholesterol, X is BMI, and b₀, b₁ are regression coefficients estimated from the data. For each subject, the residual is calculated as Residual = Observed Y - Predicted Y. These residuals are then examined for patterns. In SPSS, the residuals can be obtained by running the regression analysis and saving residuals through the "Save Residuals" option. Plotting these residuals against predicted values or each predictor variable helps diagnose any violations of regression assumptions.

Support for the appropriateness of the model hinges on residual behavior: residuals should display randomness, exhibit constant variance, and have no clear patterns when plotted. If residuals are randomly scattered around zero with no systematic structure, the linear model is likely suitable. Conversely, patterns like curvature or funnel shapes suggest model inadequacies.

Furthermore, calculating residuals manually or via statistical software also involves verifying correctness. This can be done by checking if the sum of residuals near zero or by cross-validating residuals with the observed values. Plotting residuals in SPSS and examining outliers or influential points enables validation of residual calculations.

Testing the Change in Proportion of Politically Motivated Expatriates

The second part of the analysis involves testing whether the proportion of US citizens residing abroad due to political or social attitudes has increased since 2001. The null hypothesis (H₀) posits no increase, maintaining that the proportion remains at 2%, while the alternative hypothesis (H₁) suggests an increase.

Formally, the hypotheses are:

  • H₀: p = 0.02
  • H₁: p > 0.02

Given the sample size n = 400, with x = 12 expatriates reporting living overseas due to political or social attitudes, the sample proportion is p̂ = x/n = 12/400 = 0.03.

The test utilizes a one-proportion z-test, calculated by:

z = (p̂ - p₀) / √(p₀(1 - p₀) / n)

where p₀ = 0.02. Substituting the values:

z = (0.03 - 0.02) / √(0.02 * 0.98 / 400) ≈ 0.01 / √(0.0196 / 400) ≈ 0.01 / √(0.000049) ≈ 0.01 / 0.007 ≈ 1.43

The p-value corresponds to the probability of observing a z-value of 1.43 or higher in the standard normal distribution. Using z-tables or statistical software, the p-value is approximately 0.076, which exceeds the significance level α = 0.05.

Based on the p-value, we fail to reject the null hypothesis at the 5% level. This indicates insufficient evidence to conclude that the proportion of expatriates living abroad due to political/social attitudes has increased since 2001.

In conclusion, although there is a slight increase in the observed proportion from 2% to 3%, the change is not statistically significant at the 0.05 level. Therefore, the data do not provide sufficient evidence to support the claim that the proportion has risen post-2001, emphasizing the importance of statistical testing in policy analysis.

References

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