Math 104 Fall 14 Lab As

Sheet1bobopeer140121121859122322380224math 104 Fall 14lab Assignment

Analyze multiple statistical problems involving hypothesis testing, confidence intervals, correlation, and regression analysis based on real-world data scenarios. Use significance levels, test statistics, and interpret results to determine relationships and effects among variables, including medical, financial, demographic, and behavioral data. Specific tasks include performing Z-tests and F-tests for differences in proportions, means, and variances; constructing confidence intervals; calculating correlation coefficients; assessing statistical significance; and interpreting the practical implications of findings.

Paper For Above instruction

In this comprehensive analysis, various statistical methods are employed to interpret data from multiple real-world scenarios. The overarching goal is to understand relationships among variables, assess differences across groups, and determine the significance and practical implications of findings using appropriate statistical tests and confidence intervals.

Statistical Testing of Lipitor's Infection Rates

The first case involves evaluating whether Lipitor, a cholesterol-lowering drug, affects infection rates. A two-proportion Z-test was conducted comparing infection rates between treated (7 out of 94) and placebo (27 out of 270) groups. The null hypothesis posited no difference in proportions, with an alternative hypothesis suggesting a difference exists. The calculated proportions were approximately 0.074 and 0.1, respectively. The Z test statistic was -0.7326, with a p-value of approximately 0.4638, exceeding the significance level of 0.05. Therefore, we fail to reject the null hypothesis, concluding that there is no statistically significant difference in infection rates between the groups. This suggests Lipitor does not significantly alter infection risks compared to placebo.

However, statistical significance does not imply clinical insignificance. The effect size and confidence intervals around the difference in proportions should be considered for practical implications. This analysis supports the conclusion that Lipitor’s effect on infection rates is negligible within the sample and at the 0.05 significance level.

Comparison of FICO Scores and Mortgage Interest Rates

Next, data on high-interest and low-interest mortgage borrowers was examined using a two-sample Z-test for means, given large sample sizes. The high-interest group (n=40) had a mean FICO score of approximately 594.8 (SD=12.2), whereas the low-interest group had a mean of 785.2 (SD=16.3). The null hypothesis tested whether the two populations had equal means, with the alternative hypothesizing the high-interest group’s mean score was lower. The calculated Z statistic was extremely high in magnitude (-59.1451), with a p-value effectively zero, leading to rejection of the null hypothesis. This indicates a statistically significant difference in FICO scores, with lower scores associated with higher mortgage interest rates.

This relationship implies that FICO scores significantly influence mortgage interest rates, with lower scores correlating to higher rates. Such a finding aligns with credit scoring models where poorer credit is linked to higher borrowing costs. Consequently, individuals with higher FICO scores typically benefit from more favorable mortgage terms, highlighting the critical role of credit rating in financial decision-making.

Impact of Cocaine Use on Birth Weight

Further, the analysis compared birth weights of babies born to mothers who did and did not use cocaine. The sample of non-cocaine-using mothers had 186 babies with a mean birth weight of approximately 3103 grams (SD=696), whereas the cocaine-using group had 190 babies with a mean of 2700 grams (SD=645). An F-test for equality of variances indicated no significant difference, allowing the use of pooled variance t-tests. The t-test yielded a statistic of approximately 5.825 with a p-value virtually zero (

This evidence strongly suggests that prenatal cocaine exposure contributes to lower birth weights. Clinically, these outcomes have implications for neonatal health, emphasizing the importance of substance abuse prevention during pregnancy.

Comparison of Delinquent Account Amounts in Ohio and Illinois

Investigating differences in credit delinquency among states used variance and mean tests. An F-test revealed unequal variances (p=0.0283), justifying the use of separate variance t-tests. The mean delinquent amount in Ohio (n=15) was approximately $524 (SD=38), while in Illinois (n=20), it was about $473 (SD=22). The t-test produced a statistic around 4.647 with a p-value of 0.0002, indicating a statistically significant difference in means, with Ohio’s delinquent amounts slightly higher. The 95% confidence interval for the mean difference ranged from about $28 to $136, indicating a modest but significant difference in delinquent amounts.

Practically, although the difference is statistically significant, the magnitude suggests limited practical importance, as it spans a relatively narrow dollar range. Nonetheless, the statistical evidence supports differing credit behaviors or economic conditions influencing delinquency levels in these states.

Correlation and Regression Fundamentals

In other studies, the correlation coefficient measures linear association strength between variables, such as hours studied and GPA. A high positive correlation indicates that increased studying tends to associate with higher GPA, whereas a negative correlation indicates an inverse relationship. Calculating and interpreting these coefficients requires understanding the coefficient of determination, which quantifies shared variance, and the coefficient of alienation, representing unexplained variance.

For example, a reported correlation of 0.567 between speed and strength (sample size=20) was tested for significance at the 0.01 level, revealing it to be significant. Conversely, a correlation of -0.45 between test scores and completion time was examined with 80 children, also found significant at the 0.05 level. This illustrates how sample size and correlation magnitude influence significance testing.

Linear regression extends these analyses by enabling prediction of one variable from another, with the regression equation incorporating slope (rate of change) and intercept (baseline). In contrast, analysis of variance (ANOVA) compares means across multiple groups. The correlation coefficient indicates the strength of relationships, whereas regression models quantify and predict the nature of those relationships.

Predicting Health Outcomes and Educational Success

Betsy’s scenario on predicting Alzheimer’s disease involves selecting meaningful predictors such as cognitive assessments and physical health measures. Criteria for predictor selection include theoretical relevance, statistical significance, and unique contribution beyond existing variables. Additional predictors might include genetic markers or lifestyle factors.

Regression models incorporating multiple predictors can estimate future disease onset, aiding early intervention. The model would be represented as:

Alzheimer’s risk = β₀ + β₁(education level) + β₂(physical health) + β₃(genetic markers) + β₄(lifestyle factors) + ε

Predicting Super Bowl Outcomes

Joe’s inquiry into whether the average number of wins predicts Super Bowl victories involves categorical outcome analysis, such as logistic regression. The predictor variable (average wins) is continuous, while the outcome (win or lose) is categorical. The usefulness can be assessed through measures like likelihood ratios or classification accuracy. The advantage of categorical dependent variables, modeled via logistic regression, includes interpretability and suitability for binary outcomes.

Additional predictors may include team roster quality, coaching experience, or injury rates, which could improve prediction accuracy. These variables capture factors influencing team performance beyond mere wins, enabling a nuanced understanding of championship prospects.

Analysis of Children's Aggressive Behavior

In the final practical example, linear regression predicts playground aggression based on hitting a Bobo doll. Regression output provides the slope (predictor’s effect), intercept (baseline aggression), mean predicted value, correlation coefficient, and standard error. This analysis quantifies how increases in hitting the doll relate to subsequent aggressive behavior, informing behavioral interventions and understanding aggression contagion.

Understanding Correlation and Regression

Scatterplots visually depict relationships—positive, negative, strong, weak—and contextual examples enhance comprehension. The coefficient of determination (R²) indicates variance shared between variables, while the coefficient of alienation measures the unexplained variance. Quantifying shared variance is critical for interpreting the practical significance of correlations.

In predicting student success, variables such as prior GPA, standardized test scores, socioeconomic status, and extracurricular engagement may be examined using multiple regression to identify key predictors. The p-value associated with the correlation tests the null hypothesis of no association, guiding inference about the significance of observed relationships.

Conclusion

This analysis demonstrates the importance of selecting appropriate statistical tests based on data type, sample size, and research questions. Proper interpretation of p-values, confidence intervals, and effect sizes is essential for meaningful conclusions. These statistical tools collectively enable researchers to assess relationships, differences, and effects in varied applied contexts, informing policy and practice across fields such as medicine, finance, education, and behavioral science.

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