# Regression, Correlation, and Hypothesis Testing

True / False

_______ 1. The usual objective of regression analysis is to predict estimate the value of one variable when the value of another variable is known.

_______ 2. Correlation analysis is concerned with measuring the strength of the relationship between two variables.

_______ 3. The term ei in the simple linear regression model indicates the amount of change in Y for a unit change in X.

_______ 4. In the sample regression equation y = a + bx, b is the slope of the regression line.

_______ 5. The coefficient of determination can assume any value between -1 and +1.

_______ 6. In the least squares model, the explained sum of squares is always smaller than the regression sum of squares.

_______7. The sample correlation coefficient and the sample slope will always have the same sign.

_______ 8. Given the sample regression equation y = -3 + 5x, we know that in the sample X and Y are inversely related.

_______ 9. Given the sample regression equation y = 5 - 6x, we know that when X = 2, Y = 17.

_______ 10. An important relationship in regression analysis is = .

_______ 11. Regression analysis is concerned with the form of the relationship among variables, whereas correlation analysis is concerned with the strength of the relationship.

_______ 12. The correlation coefficient indicates the amount of change in Y when X change by one unit.

_______ 13. In simple linear regression analysis, when the slope is equal to zero, the independent variable does not explain any of the variability in the dependent variable.

_______14. One of the purposes of regression analysis is to estimate a mean of the independent variable for given values of the dependent variable.

_______ 15. The variable that can be manipulated by the investigator is called the independent variable.

_______ 16. When b = 0, X and Y are not related.

_______ 17. If zero is contained in the 95% confidence interval for b, we may reject Ho: b = 0 at the 0.05 level of significance.

_______ 18. If in a regression analysis the explained sum of squares is 75 and the unexplained sum of square is 25, r2 = 0.33.

_______ 19. In general, the smaller the dispersion of observed points about a fitted regression line, the larger the value of the coefficient of determination.

_______ 20. When small values of Y tend to be paired with small values of X, the relationship between X and Y is said to be inverse.

_______ 21. An alternative hypothesis (Ha) is a theory that contradicts the null hypothesis. The alternative hypothesis will be accepted when there is strong evidence leading us to reject the null hypothesis.

_______ 22. The p-value of a test depends on the observed data, but the critical values of a test do not.

_______ 23. Other things being equal, decreasing a (alpha) increases beta .

_______ 24. The larger the p-value associated with a test of hypothesis, the stronger the support for the null hypothesis.

_______ 25. The probability that the test statistic will fall in the critical region, given that H0 is true, represents the probability of making a type II error.

_______ 26. When the null hypothesis is true, the probability that the test statistic will fall in the critical region is call the level of significance of the test.

_______ 27. When we reject a true null hypothesis, we commit a Type I error.

_______ 28. The alternative hypothesis is the hypothesis that is tested.

_______ 29. The larger the p-value, the stronger the evidence against the null hypothesis.

_______ 30. A small p-value provides evidence supporting the alternative hypothesis.

_______ 31. The p-value of a test is the probability of getting a test statistic as extreme as or more extreme than the observed one. The probability is calculated based on the assumption that the null hypothesis is false.

_______ 32. If the p-value for a test is greater than or equal to the level of significance, we may reject H0.

_______ 33. A type I error can occur only when the statistician decides to reject the null hypothesis.

_______ 34. The alternative hypothesis always contains a statement of equality.

_______ 35. If we fail to reject the null hypothesis, we conclude that the null hypothesis may be true.

_______ 36. In general, in most practical hypothesis-testing situations, the level of significance and the p-value will be the same.

_______ 37. Mathematically, b0 represents the Y-intercept of the line relating X and Y. But practically speaking, it often times is not interpretable. Only if the value of 0 is within the relevant range on the X-values we can interpret b0 as the average value of Y when X=0. Relevant range refers to the region between the minimum and maximum X-values observed in the sample.

_______38. The quantity sigma ^2 is the same as the mean square error (MSE) or . It is the variability of the data points around the line.

_______39. A correlation coefficient of 0.65 indicates a stronger linear relationship between two variables than does a correlation coefficient of -0.65.

______ 40. In general, the larger the dispersion of observed points about a fitted regression line, the larger the value of the coefficient of determination of 0.42.

______ 41. Given the statistically significant sample regression equation = 4-3x, we know that in the sample x and y and inversely related.

______ 42. The correlation coefficient is the proportion of total variation in Y that is explained by X.

______ 43. Given Ho: beta 1 = 0 and Ha: beta 1 not equal to 0, intuitively, we would be unable to reject the null hypothesis if the sample slope was close to "0". The farther it was from "0" the more plausible the alternative hypothesis would be.

______ 44. Rejection of H0 in regression analysis implies that there is not a statistically significant relationship between X and Y.

______ 45. Type II error is the probability or risk assumed by accepting the null hypothesis when it is actually false.

______ 46. A z or t statistic tells us the number of standard deviation our sample slope is from Zero under the null. The larger the z, the more plausible Ha:

______ 47. Given the statistically significant sample regression equation y = -3 + 5x, we know that in the sample X and Y are inversely related.

______ 48. A type I error is the rejection of a true null hypothesis.

______ 49. In general, the smaller the dispersion of observed points about fitted regression line, the larger the value of the coefficient of determination.

_______ 50. At the 5% level, b1 is said to deviate form beta 1 by more than 1.96 standard deviations, at most 5% of the time (hence, favors Ha)

_______ 51. Given Ho: µ = 8.8, Ha: µ not equal to 8.8 and Z = = -4 indicates that the value 7.2 is 4 standard error below 8.8.

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#### Solution Preview

True / False

True 1. The usual objective of regression analysis is to predict estimate the value of one variable when the value of another variable is known.

True 2. Correlation analysis is concerned with measuring the strength of the relationship between two variables.

False 3. The term ei in the simple linear regression model indicates the amount of change in Y for a unit change in X.

The term ei is the error term.

True 4. In the sample regression equation y = a + bx, b is the slope of the regression line.

False 5. The coefficient of determination can assume any value between -1 and +1.

The coefficient of determination can assume any value between 0 and +1.

False 6. In the least squares model, the explained sum of squares is always smaller than the regression sum of squares.

The explained sum of squares is the same as the regression sum of squares.

True 7. The sample correlation coefficient and the sample slope will always have the same sign.

For positive correlation there is positive slope and for negative correlation the slope is negative.

False 8. Given the sample regression equation y = -3 + 5x, we know that in the sample X and Y are inversely related.

Since the slope term (5) is positive the sample X and Y are directly related.

False 9. Given the sample regression equation y = 5 - 6x, we know that when X = 2, Y = 17.

Y = 5-6*2 =5-12 = -7

True 10. An important relationship in regression analysis is = .

= : Total Variation = Explained variation + Unexplained Variation

True 11. Regression analysis is concerned with the form of the relationship among variables, whereas correlation analysis is concerned with the strength of the relationship.

False 12. The correlation coefficient indicates the amount of change in Y when X change by one unit.

The slope of regression equation indicates the amount of change in Y when X change by one unit.

True 13. In simple linear regression analysis, when the slope is equal to zero, the independent variable does not explain any of the variability in the dependent variable.

Y = a or constant in this case.

False 14. One of the purposes of regression analysis is to estimate a mean of the independent variable for given values of the dependent variable.

Regression estimates the dependent variable for given value of independent variable.

True 15. The variable that can be manipulated by the investigator is called the independent variable.

True 16. When b = 0, X and Y are not related.

False 17. ...

#### Solution Summary

Answers to 51 True or False Questions on regression analysis, Correlation analysis, least squares model, hypothesis testing, p-value, Type I error, level of significance, Type II error, standard error etc.

Scatterplot, Regression Analysis and Correlation Hypothesis Test

Construct a Scatterplot of the raw data. Then, run the Correlation/Regression program.

Costs of Television: Listed below are prices (in dollars) and quality rating scores of rear-projection televisions. All of the televisions have screen sizes of 55 or 56 inches. Is there sufficient evidence to conclude that there is a linear correlation between the price and the quality rating score of rear-projection televisions? Does it appear that as the price increases, the quality score also increases? Do the results suggest that as you pay more, you get better quality?

Price 2300 1800 2500 2700 2000 1700 1500 2700

Quality 74 73 70 66 63 62 52 68

Identify the correlation coefficient: r = __________

Identify the CV at alpha = 0.05: __________

Is there significant linear correlation?: ________

How do you know? ____________________________