Question

The following table shows student’s test scores on the first two tests in an introductory biology class.

First test, x	Second test, y
55	58
40	43
71	68
82	86
90	87
50	51
83	87
75	70
65	67
52	55
77	77
93	90

(a) Draw a scatter plot using one the following website(s): http://www.alcula.com/calculators/statistics/scatter-plot/ or https://www.meta-chart.com/scatter
(b) Estimate the correlation in words (positive, negative, or no correlation)
(c) Calculate the correlation coefficient, r.
(d) Determine whether r is statistically significant at the 0.01 level of significance.
(e) Determine whether r is statistically significant at the 0.05 level of significance.
(f) Calculate the coefficient of determination, r2.
(g) Interpret the meaning of r2 for the given set of data.
(h) Find the equation of the least-squares regression line, if appropriate.

Answer 1

Scatter plot:

b) Correlation:

There is a perfect a positive straight line linear correlation between x and y variables

c) r = 0.98

(d) Determine whether r is statistically significant at the 0.01 level of significance.

Run a regression analysis in excel followed by below procedures:

Go to data tab --> choose data analysis --> choose regression

P-value: 0.00000000913

If the p-valuep-value is less than the significance level (α=0.01):

• Decision: Reject the null hypothesis.
• Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero."

(e) Determine whether r is statistically significant at the 0.05 level of significance.

If the p-valuep-value is less than the significance level (α=0.05):

• Decision: Reject the null hypothesis.
• Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between xx and yy because the correlation coefficient is significantly different from zero."

(f) Calculate the coefficient of determination, r2.

r-squared = 0.97

g) meaning of r-squared:

• The R-squared value, denoted by R2, is the square of the correlation. It measures the proportion of variation in the dependent variable that can be attributed to the independent variable.

  The R-squared value R2 is always between 0 and 1 inclusive.

The points are exactly on the trend line. Correlation r = 1; R-squared = 1.00 (Perfect positive linear correlation)

(h) Find the equation of the least-squares regression line, if appropriate.

y = 0.9145x + 6.4355