Exam 17: Correlation

In a multiple regression problem, the regression sum of squares is $\sum(\hat{y}-\bar{y})^{2}=12,000$ and the total sum of squares is $\sum(y-\bar{y})^{2}=20,000$ . Find the value of the multiple correlation coefficient.

Table 17.1 The mathematics S.A.T. scores and college grade point averages of eight students are given below. -Use Table 17.1 to solve the following: a. Calculate the correlation coefficient $r$ . b. What is the proportion of total variation in $y$ that is accounted for by $x$ ? c. Test the null hypothesis of no relationship between the two variables at $\alpha=0.05$ . d. Calculate a $95 \%$ confidence interval for $\mathrm{Q}$ .

The proportion of total variation which is unexplained can be denoted in symbols by __________.

A statistic that measures the relationship between variables while eliminating the effects of other variables is called the

If the multiple correlation coefficient is 0.80 , then the proportion of total variation in $y$ that can be attributed to the $x$ 's is __________.

A financial economist wants to evaluate how interest rates affect the inflation rate. Here are some results on yearly prime interest rates $(x)$ and inflation rates $(y)$ for the 13 years 1965 through 1977. $\sum x y=500, \sum y=78$ , $\sum x^{2}=450, \sum y^{2}=600, \sum x=65$ . For the situation above: a. Calculate the sample coefficient of determination and correlation coefficient. b. Give the proportion of variation explained by the regression line. c. Test the null hypothesis of no correlation against a two-tailed alternative at the 0.01 significance level.

For two sets of data $x$ and $y, r=0.75$ . Suppose each value of $x$ is measured in feet. If each $x$ value is converted to inches by multiplying by 12, what is the new value of $r$ ? Suppose, instead, a constant is added to each value of $x$ . What is the new $r$ value?

Table 17.3 The table below gives the number of housing starts and the unemployment rates for a sequence of quarters: -Use Table 17.3 to solve the following: a. Calculate the correlation coefficient. b. What is the proportion of the total variation in unemployment rates that is explained by the number of housing starts? c. Test the null hypothesis that $\mathrm{Q}=-0.1$ at $\alpha=0.05$ .

The correlation coefficient is equal to

Table 17.2 Below is a list of countries with their 1979 "excess monetary growth*" ( along with their mid -1980 inflation rate). *Excess monetary growth is the extent to which money supply growth exceeds to the growth of constant dollar GNP; specifically, the ratio of 1979 M2 to 1978 M2 is divided by the ratio of 1979 real GNP to 1978 real GNP. -Use Table 17.2 to solve the following: a. Calculate the correlation coefficient. b. What is the proportion of the total variation in inflation rate that is explained by excess monetary growth? c. Test the null hypothesis that $\mathrm{Q}=0.3$ at $\alpha=0.01$ . d. Calculate a $99 \%$ confidence interval for $\mathrm{Q}$ .

If all data points lie on the regression line, then the standard error of estimate is __________.

The value of $z$ is a natural logarithm.

In regression analysis, the quantity that gives the amount by which $y$ changes for a unit change in $x$ is called the

The sum $\sum(\hat{y}-\bar{y})^{2}$ is called the __________ sum of squares.

The sample coefficient $r$ is an estimate of __________.

If $r$ has a value of 0.60 , then $60 \%$ of the variation in $y$ can be explained by the variation in $x$ .

The sum $\sum(y-\bar{y})^{2}$ is called the __________ sum of squares.

If all data points lie on a regression line having a nonzero slope, then $r^{2}$ equals __________.

If the correlation between $x$ and $y$ is not zero, then the variability about the estimated regression line will be less than the total variability in $y$ .

The least squares equation $y=3+4 x_{1}+5 x_{2}$ is considered to be a better predictor of $y$ than the least squares equation $y=7+5 x_{3}$ , if the multiple correlation coefficient is __________ than the absolute value of the correlation coefficient involving $x_{1}$ and $x_{3}$ .