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Determining Sample Correlation Coefficients

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1. The following statistics were calculated from pairs of observations where X represents the independent variable and Y represents the dependent variable.

Σx = 511 Σy = 314 Σxy = 19,064
Σx^2 = 34,234.5 Σy^2 = 13,036 n = 8

Determine the least squares line.
Determine the sample correlation coefficient between X and Y.
Determine if there is a linear relationship between X and Y at the .10 significance level.
Find a 90% confidence interval for the slope of the regression line.
Find a 90% confidence interval for the mean of Y if X = 60.
Find a 90% prediction interval for Y if X = 60.

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Solution Preview

1. The following statistics were calculated from pairs of observations where X represents the independent variable and Y represents the dependent variable.

Σx = 511 Σy = 314 Σxy = 19,064
Σx^2 = 34,234.5 Σy^2 = 13,036 n = 8

a. Determine the least squares line.

Assume the equation of line is y=mx+b

m=(8*19064-511*314)/(8*34234.5-511^2)= -0.62266

b=(314+0.62266*511)/8=79.022
Therefore, the line of equation is: y=-0.62266x+79.022

b. Determine the sample correlation coefficient between X and Y.

R=(8*19064-511*314)/sqrt((8*34234.5-511^2)*(8*13036-314^2))= -0.932

c. Determine if ...

Solution Summary

The expert determines the sample correlation coefficients. The linear relationship between X and Y at the 01 significance level is provided.

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Meryl's Apparel

Meryl's Apparel is an upscale chain of women's clothing stores, located primarily in the southwest United States. Due to recent success, Meryl's top management is planning to expand by locating new stores in other regions of the country. The director of planning has been asked to study the relationship between yearly sales and the store size. As part of the study, the director selects a sample of 25 stores and determines the size of the store in square feet and the sales for last year. The sample data follow. The use of statistical software is suggested.

a. Draw a scatter diagram. Use store size as the independent variable. Does there appear to be a relationship between the two variables. Is it positive or negative?
b. Determine the correlation coefficient and the coefficient of determination. Is the relationship strong or weak? Why?
c. At the 0.5 significance level, can we conclude there is a significant positive correlation?

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