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Question 1: Explain the assumptions of linear regression and why they are important in SPSS analysis.
Solution:
Linear regression is widely used in SPSS to explore the relationship between variables. However, to ensure valid results, certain assumptions must be met. These include linearity, independence of errors, homoscedasticity (equal variances), normality of residuals, and absence of multicollinearity. Linearity ensures a straight-line relationship, while independence prevents errors from being correlated. Homoscedasticity implies that the spread of residuals remains constant across predicted values. Normal residuals support reliable significance testing, and multicollinearity checks help avoid inflated standard errors. Understanding and checking these assumptions in SPSS is crucial for accurate interpretation.
Question 2: What is the purpose of conducting a one-way ANOVA, and how do you interpret the results in SPSS?
Solution:
A one-way ANOVA is used when comparing the means of three or more groups to determine whether at least one group mean differs significantly. In SPSS, the F-statistic and corresponding p-value guide the interpretation. A significant result (p < 0.05) indicates that not all group means are equal. However, the ANOVA alone doesn’t tell us which groups differ, so post-hoc tests like Tukey's HSD are used for further comparison. The analysis helps in identifying group-level differences with statistical confidence.
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Question 1: Application of Regression Analysis in Academic Research
Scenario: A university is conducting a study on the impact of study hours and class attendance on students' final grades. The dataset consists of students’ study hours per week, the number of classes attended, and their final exam scores. The goal is to determine whether these two independent variables significantly influence academic performance.
Solution:
To analyze this scenario, multiple linear regression is applied. The general regression model is:
Where:
is the final exam score (dependent variable),
represents study hours per week,
represents the number of classes attended,
is the intercept,
and are the coefficients for the independent variables,
is the error term.
Step 1: Checking Assumptions
Before running a regression analysis, key assumptions such as linearity, homoscedasticity, and normality of residuals are verified. Residual plots and histograms help in diagnosing any violations.
Step 2: Estimating the Model
Using statistical software like R or SPSS, the regression coefficients are estimated. If the p-values for and are below 0.05, they are considered statistically significant predictors of final exam scores.
Step 3: Interpreting the Results
A positive coefficient for (study hours) suggests that increasing study hours improves scores.
A positive coefficient for (attendance) implies that attending more classes is beneficial.
The coefficient of determination () indicates how well the model explains the variance in scores.
The findings highlight the importance of study habits and classroom engagement, reinforcing the role of structured learning in academic success.
Question 2: Hypothesis Testing in Medical Research
Scenario: A medical researcher wants to test whether a new drug effectively reduces blood pressure. A sample of patients was given the drug, and their blood pressure was recorded before and after the treatment. The researcher seeks to determine if there is a significant reduction in blood pressure using a paired sample t-test.
Solution:
Step 1: Defining the Hypotheses
Null Hypothesis (): The drug has no effect on blood pressure (mean difference = .
Alternative Hypothesis (): The drug reduces blood pressure (mean difference < .
Step 2: Conducting the Paired t-Test
A paired sample t-test is appropriate since measurements are taken from the same subjects before and after treatment.
Test statistic formula:
Where:
is the mean difference,
is the hypothesized difference (,
is the standard deviation of differences,
is the sample size.
Step 3: Decision Making
If the computed t-value falls in the critical region (determined by the significance level, ), the null hypothesis is rejected, concluding that the drug significantly lowers blood pressure.
Step 4: Interpretation
A significant result would indicate that the drug is effective, providing statistical evidence to support its use in medical treatment.
Conclusion
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