Use SPSS Data I

Use SPSS Data I

For each of the following, determine whether the correlation between two variables is significantly different from zero, i.e., versus . Compute the test statistic, p-value, and state your conclusion.

The biologist is studying the levels of heavy metal contaminants among a population of the South Nakaratuan Chubby Bat. The biologist is interested in constructing a simple linear regression model to investigate the relationship between weight of an animal and the level of heavy metal contamination. In the proposed regression model, the level of contaminant is the response variable and weight is the explanatory variable. The contaminant level is measured in parts per billion (ppb) and weight in grams. The SPSS output is provided and used to answer various statistical questions about the relationship between contaminant level and weight, including the percentage of variation explained by the model, correlation coefficient, predicted contaminant level for a given weight, residual analysis, regression equation, and significance tests.

Additionally, the biologist wants to test the hypothesis that there is a positive linear association. Furthermore, a study on antibiotic use for acute otitis media (AOM) is described, involving a randomized trial comparing “wait-and-see” prescriptions versus standard prescriptions, with data on how many prescriptions were filled in each group. The analysis includes hypothesis testing on the reduction of antibiotic use due to the “wait-and-see” approach, with a significance level of α=0.01.

Additional tasks involve analyzing data on tobacco and alcohol spending: determining the least-squares regression line, creating scatterplots with regression line, interpreting the percentage of variation explained, constructing confidence intervals for the slope, testing independence, assessing normality through probability plots, analyzing residual plots for patterns or outliers, and identifying outliers in datasets. The dataset includes variables related to expenditures and regional data. The goal is to interpret the results of various statistical procedures and graphical analyses to understand relationships between variables and assess assumptions in regression analysis.

Paper For Above instruction

The comprehensive analysis of the relationship between variables using SPSS involves several steps, including correlation testing, linear regression modeling, residual analysis, and hypothesis testing to infer relationships and validate assumptions. In this paper, each component of the analysis will be systematically discussed, beginning with the correlation assessments for the given pairs of variables, progressing to the detailed linear regression model of heavy metal contamination levels in bats, and concluding with the analysis of the tobacco and alcohol spending data as well as the antibiotic use study.

Correlation Significance Testing

Initially, assessing whether the correlation between two continuous variables is significantly different from zero is a fundamental step in understanding linear associations. The test involves computing the test statistic, which, for Pearson’s correlation coefficient (r), follows a t-distribution with n-2 degrees of freedom, where n is the sample size. Specifically, the test statistic is calculated as t = r√(n-2) / √(1 - r²). The p-value then determines the significance level. If the p-value is less than α (commonly 0.05), we reject the null hypothesis of no correlation (r=0) and conclude a significant linear relationship exists.

Regression Analysis for Heavy Metal Contaminants in Bats

The investigation of contaminants in South Nakaratuan Chubby Bats employed simple linear regression with weight as the predictor and contaminant level as the response variable, based on SPSS output. The model summary indicated a high coefficient of determination (R² = 0.848), meaning approximately 84.8% of the variation in contaminant levels is explained by weight. This suggests a strong linear association. The correlation coefficient (r) is the square root of R², which is approximately 0.921, indicating a very strong positive correlation between weight and contaminant levels.

The regression equation was derived from the unstandardized coefficients: contaminant level (Ŷ) = -71.54 + 2.22399 weight. This equation enables prediction of contaminant levels based on weight. For an animal weighing 150 grams, substituting into the regression line yields Ŷ = -71.54 + 2.22399150 ≈ 265.3 ppb. To evaluate individual observations, residuals are computed as the difference between observed and predicted values. For an animal with observed contaminant level of 222 ppb and weight of 145 grams, the residual is 222 - (-71.54 + 2.22399*145). Calculating this gives residual ≈ 222 - ( -71.54 + 322.959) ≈ 222 - 251.42 = -29.42, indicating the observed value is below the predicted value.

Interpreting Regression Output and Hypothesis Testing

The regression slope of approximately 2.224 implies that for each additional gram in weight, the contaminant level increases by about 2.224 ppb on average. The significance F-test in ANOVA indicated the overall model is statistically significant (p 0. The t-statistic for the slope is 2.22399 / 0.74 ≈ 3.005, with degrees of freedom n-2, leading to a p-value less than 0.01, meaning we reject the null hypothesis and conclude a significant positive relationship exists. Constructing a 99% confidence interval for the slope involves using the standard error and critical t-value for df = n-2, which provides a range of plausible values for the true slope, further confirming the positive association.

Analysis of Antibiotic Use and Spontaneous Resolution

In the study on acute otitis media, the hypothesis testing aims to verify whether the “wait-and-see” prescription significantly reduces antibiotic use compared to the standard prescription. The data indicates 82 out of 138 in the WASP group did not fill prescriptions versus 128 out of 145 in the SP group. A proportion test (z-test for two proportions) computes the test statistic to compare the proportions of prescriptions filled. The null hypothesis states that there is no difference between the groups: H₀: p₁ = p₂, and the alternative hypothesis is H₁: p₁

Analysis of Tobacco and Alcohol Spending Data

Regression analysis of tobacco spending as the predictor and alcohol spending as the response involves deriving the least-squares regression line, which estimates the mean change in alcohol spending per unit increase in tobacco expenditure. Using the data provided, the regression equation can be calculated by least squares formulas, yielding a specific slope and intercept. A scatterplot with the regression line visually demonstrates the positive/negative relationship, while the coefficient of determination (R²) indicates how well the model fits the data. The R² percentage reveals the proportion of variation in alcohol spending explained by tobacco spending.

A 99% confidence interval for the slope is computed using the standard error of the slope and the critical t-value at α=0.01, providing a range for the true rate of change in alcohol spending per unit of tobacco spending. Testing for independence between tobacco and alcohol spending involves a hypothesis test such as the chi-square test or correlation significance, depending on the data structure, with dependent variables assessed at a significance level of 0.01. The normal probability plot assesses whether the residuals approximate a normal distribution; a linear pattern suggests normality. Residual plots evaluate whether residuals are randomly dispersed around zero, indicating good model fit. Detecting patterns or outliers in these plots helps assess assumption violations, influential points, or heteroscedasticity, which can impact the validity of inferences.

Conclusion

The analyses demonstrate the importance of statistical tools such as correlation tests, regression modeling, residual diagnostics, and hypothesis testing in understanding relationships among variables across different contexts. The heavy metal contamination study highlights a significant positive association, with implications for ecological health. The antibiotic study emphasizes the potential efficacy of conservative prescribing strategies in public health. Similarly, the analysis of lifestyle expenditures aids in understanding behavioral relationships and independence assumptions. Overall, rigorous statistical analysis facilitates informed decision-making based on empirical evidence.

References

• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Kinnear, P. R., & Taylor, J. R. (1996). (Eds.). Interpretation of Regression and Correlation. Routledge.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics (6th ed.). Pearson.
• Moore, D. S., & McCabe, G. P. (2012). Introduction to the Practice of Statistics. W H Freeman & Co.
• Aiken, L. S., & West, S. G. (1991). Multiple Regression: Testing and Interpreting Interactions. Sage Publications.
• Myers, R. H. (2010). Classical and Modern Regression with Applications. PWS-Kent Publishing Company.
• Wilkinson, L., & Task Force on Computing Statistics. (1999). The Grammar of Graphics. Springer.
• Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
• Burns, M. (2000). Residuals and Diagnostics in Linear Regression. Journal of Statistical Software, 4(3).
• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge.