Points And Facts About Correlation: Answer The Following Que

5 Pointsfacts About Correlationanswer The Following Questions Ab

Answer the following questions about correlation (r). a. What is the strongest the correlation can ever be? b. If there is no relationship, r is equal to c. The correlation coefficient ranges from __________to __________ d. If the points fall in an almost perfect, negative linear pattern, r is close to: __________ e. If the points fall in an almost perfect, positive linear pattern, r is close to: __________

Data has been collected on 219 STAT 200 students, measuring weight in pounds and height in inches. A linear regression was performed on height and weight, with the output provided. Tasks include writing the regression equation, identifying response and predictor variables, interpreting the slope, testing the significance of the slope, predicting weight for a student 65 inches tall, and explaining fitted values and residuals with specific data points.

Additionally, an analysis involves examining the relationship between eighth-grade IQ and ninth-grade math scores for 20 students. Tasks include creating scatter plots, conducting linear regression, interpreting R-squared and correlation, evaluating model assumptions via residual and probability plots, addressing outliers, re-fitting the model after removing an outlier, and comparing model fits before and after outlier removal.

Paper For Above instruction

Correlation is a statistical measure that expresses the extent to which two variables are linearly related. The correlation coefficient, denoted as r, ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, -1 a perfect negative linear relationship, and 0 indicates no linear relationship.

The strongest correlation possible is an absolute value of 1, representing a perfect linear relationship (either positive or negative). When the data points closely align along a straight line, the correlation coefficient approaches either 1 or -1, depending on the direction of the relationship.

In the context of this analysis, the correlation coefficient ranges from -1 (perfect negative relationship) to 1 (perfect positive relationship). If the points fall in an almost perfect, negative linear pattern, the correlation coefficient r is close to -1. Conversely, if they form a near-perfect, positive linear pattern, r is close to 1. When there is no apparent linear relationship, r is approximately 0.

In the regression analysis involving height and weight for 219 students, the descriptive statistics indicate relationships that can be modeled linearly. The regression equation typically takes the form:

Weight = Intercept + (Slope × Height)

From the regression output, one can determine the specific intercept and slope. The response variable, or dependent variable, is weight, as it depends on height, the predictor or independent variable.

The slope indicates the expected change in weight associated with a one-inch increase in height. For example, if the slope is 2.5, then for every additional inch in height, the weight increases by approximately 2.5 pounds. This interpretation emphasizes how the variables are linearly related.

The significance of the slope can be tested through hypothesis testing: the null hypothesis (H0) states that the slope is zero (no relationship), and the alternative hypothesis (Ha) states that the slope is not zero (there is a relationship). The test statistic (t-value), p-value, and decision rule determine whether the slope is statistically significant. A small p-value (less than the significance level, e.g., 0.05) leads to rejecting H0, indicating a significant relationship between height and weight.

Using the regression equation, the weight of a student with a height of 65 inches can be predicted by plugging the value into the equation. Since the regression line estimates average weight at a given height, this prediction is valid if the model assumptions hold.

The fitted values are the predicted weights obtained from the regression equation for each student based on their height. Residuals are the differences between observed weights and the corresponding fitted values—they measure the deviations of actual data points from the regression line. For instance, if a student with height 54 inches weighs 110 pounds, the fitted value is the predicted weight for 54 inches based on the model. The residual is then calculated as: residual = observed weight - fitted value.

In the case of examining the relationship between eighth-grade IQ and ninth-grade math scores for 20 students, scatter plots visually reveal the nature of the relationship, whether positive, negative, or nonexistent. Conducting linear regression provides an equation that models how IQ predicts math scores, with the R-squared value indicating the proportion of variance explained by the model. For example, an R-squared of 0.75 suggests that 75% of the variability in math scores is accounted for by IQ.

The correlation coefficient can be calculated from R-squared by taking its square root, considering the direction of the relationship (positive or negative).

Assessing model assumptions involves checking residual plots for constant variance (homoscedasticity) and using Q-Q plots or probability plots to evaluate normality. Residuals should scatter randomly around zero without patterns, supporting the validity of the assumptions. Outliers can influence these assumptions significantly, especially in small samples.

The high IQ student (student 17) appears as an outlier, which can be visually identified through residual and Q-Q plots. If the data point exerts undue influence, it may be justified to remove it. After removing the outlier, re-fitting the regression model often results in improved fit and more reliable estimates. The regression equation may change, and the R-squared may increase, reflecting better model fit.

Comparing the original and revised models involves examining the slopes, R-squared values, and residual diagnostics. The goal is to determine whether excluding the outlier yields a more accurate and valid model that better meets regression assumptions and provides reliable predictions.

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