Correlation Analysis

The post is about MCQs correlation and Regression Analysis with Answers. There are 20 multiple-choice questions covering topics related to correlation and regression analysis, coefficient of determination, testing of correlation and regression coefficient, Interpretation of regression coefficients, and the method of least squares, etc. Let us start with MCQS Correlation and Regression Analysis with answers.

Online Multiple-Choice Questions about Correlation and Regression Analysis with Answers

1. Which one of the following statements is true?

 The estimated and the true regression lines are always the same

 The $Y$’s must be normally distributed about true regression line before the sample coefficient of determination can be calculated

 The correlation coefficient can b calculated only after the estimated regression line has been found

 The units in which $X$ and $Y$ are measured will not affect the value of $r$

2. The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by

 Rs. 10.00

 Rs. 1.00

 Rs. 0.50

 Rs. 0.10

3. If the coefficient of determination is 0.49, the correlation coefficient may be

 0.51

 0.24

 0.70

 0.49

4. The strength of the linear relationship between two numerical variables may be measured by the

 Coefficient of Correlation

 $Y$ intercept

 Slope

 Scatter diagram

5. What do we mean when a simple linear regression model is “statistically” useful?

 All the statistics computed from the sample make sense

 The model is practically useful for predicting $Y$

 The model is better predictor of $Y$ than the sample mean $\overline{Y}$

 The model is an excellent predictor of $Y$

6. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is

 $1-\frac{0.256}{2}$

 $1-0.256$

 $\frac{0.256}{2}$

 $(0.256)^2$

7. Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$

 There is no correlation

 The variance of $X$ is negative

 Variable $X$ is larger than variable $Y$

 The slope $b_1$ is negative

8. The true correlation coefficient $\rho$ will be zero only if

 $b=0$

 The slope of the true regression line is equal to zero

 $r=0$

 The Y intercept of the true regression line is equal to zero

9. Testing for the existence of correlation is equivalent to

 Testing for the existence of the $Y$ intercept ($\beta_0$)

 The confidence interval estimated for predicting $Y$

 None of these

 Testing for the existence of the slope ($\beta_1$)

10. Which of the following does the least squares method minimize?

 SSR

 SST

 SSE

 All of these

11. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is

 $1-0.256$

 $\frac{0.256}{2]$

 $(0.256)^2$

 $1-\frac{0.256}{2}$

12. If the correlation coefficient ($r=1.00$) then

 All the data points must fall exactly on a straight line with a negative slope

 All the data points must fall exactly on a straight line with a slope that equals 1.00

 All the data points must fall exactly on a horizontal straight line with a zero slope

 All the data points must fall exactly on a straight line with a positive slope

13. The $Y$ intercept ($b_0$) represents the

 Change in estimated average $Y$ per unit change in $X$

 Variation around the sample regression line

 Predicted value of $Y$

 Predicted value of $Y$ when $X=0$

14. If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a

 A t-test for independence

 A Z-test for the difference in two proportions

 $\chi^2$ test for independence

 $\chi^2$ test the difference in two proportions

15. Which one of the following situations is inconsistent?

 $Y=10+2X$ and $r=0.50$

 $Y=-200 + 0.9X$ and $r=-0.86$

 $Y=500+0.01X$ and $r=0.75$

 $Y=-8-3X$ and $r=-0.95$

16. The correlation coefficient

 May be smaller than $-1$ only when $X$ and $Y$ are inversely related

 Equals the standard error of the estimate divided by the square root of the sample size

 Equals the positive or negative square root of the coefficient of determination

 Can be calculated only after regression analysis is performed

17. In a simple linear regression problem, $r$ and $\beta_1$

 May have opposite signs

 Must have the same signs

 are equal

 Must have opposite signs

18. The slope ($b_1$) represents

 Predicted value of $Y$ when $X=0$

 The estimated average change in $Y$ per unit change in $X$

 The predicted value of $Y$

 Variation around the line of regression

19. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is

 $\frac{0.256}{2}$

 $1-0.256$

 $(0.256)\times 2$

 $1-\frac{0.256}{2}$

20. If the correlation coefficient $r=1.00$ then

 The explained variation equals the unexplained variation

 The Y-intercept ($b_0$) must equal to 0

 There is no explained variation

 There is no unexplained variation

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MCQs Correlation and Regression Analysis

• The $Y$ intercept ($b_0$) represents the
• The slope ($b_1$) represents
• Which of the following does the least squares method minimize?
• What do we mean when a simple linear regression model is “statistically” useful?
• If the correlation coefficient $r=1.00$ then
• If the correlation coefficient ($r=1.00$) then
• Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$
• Testing for the existence of correlation is equivalent to
• The strength of the linear relationship between two numerical variables may be measured by the
• In a simple linear regression problem, $r$ and $\beta_1$
• The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is
• The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is
• If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a
• The correlation coefficient
• If the coefficient of determination is 0.49, the correlation coefficient may be
• The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by
• Which one of the following situations is inconsistent?
• Which one of the following statements is true?
• The true correlation coefficient $\rho$ will be zero only if

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The coefficient of correlation is a statistic used to measure the strength and direction of the linear relationship between two Quantitative variables.

Properties of Correlation Coefficient

Understanding these properties helps us to interpret the correlation coefficient accurately and avoid misinterpretations. The following are some important Properties of Correlation Coefficient.

• The correlation coefficient ($r$) between $X$ and $Y$ is the same as the correlation between $Y$ and $X$. that is the correlation is symmetric with respect to $X$ and $Y$, i.e., $r_{XY} = r_{YX}$.
• The $r$ ranges from $-1$ to $+1$, i.e., $-1\le r \le +1$.
• There is no unit of $r$. The correlation coefficient $r$ is independent of the unit of measurement.
• It is not affected by the change of origin and scale, i.e., $r_{XY}=r_{YX}$. If a constant is added to each value of a variable, it is called a change of origin and if each value of a variable is multiplied by a constant, it is called a change of scale.
• The $r$ is the geometric mean of two regression coefficients, i.e., $\sqrt{b_{YX}\times b_{XY}}$.
  In other words, if the two regression lines of $Y$ on $X$ and $X$ on $Y$ are written as $Y=a+bX$ and $X=c+dy$ respectively then $bd=r^2$.
• The sign of $r_{XY}, b_{YX}$, and $b_{XY}$ is dependent on covariance which is common in the three as given below:
• $r=\frac{Cov(X, Y)}{\sqrt{Var(X) Var(Y)}},\,\, b_{YX} = \frac{Cov(Y, X)}{Var(X)}, \,\, b_{XY}=\frac{Cov(Y, X)}{Var(Y)}$

Hence, $r_{YX}, b_{YX}$, and $b_{XY}$ have the same sign.

• If $r=-1$ the correlation is perfectly negative, meaning as one variable increases the other increases proportionally.
• If $r=+1$ the correlation is perfectly positive, meaning as one variable increases the other decreases proportionally.
• If $r=0$ there is no correlation, i.e., there is no linear relationship between the variables. However, a non-linear relationship may exist but it does not necessarily mean that the variables are independent.

Theorem: Correlation: Independent of Origin and Scale. Show that the correlation coefficient is independent of origin and scale, i.e., $r_{XY}=r_{uv}$.

Proof: The formula for correlation coefficient is,

$$r_{XY}=\frac{\varSigma(X-\overline{X})((Y-\overline{Y})) }{\sqrt{[\varSigma(X-\overline{X})^2][\varSigma(Y-\overline{Y})^2]}}$$

\begin{align*}
\text{Let}\quad u&=\frac{X-a}{h}\\
\Rightarrow X&=a+hu \Rightarrow \overline{X}=a+h\overline{u} \\
\text{and}\quad v&=\frac{Y-b}{K}\\
\Rightarrow Y&=b+Kv \Rightarrow \overline{Y}=b+K\overline{v}\\
\text{Therefore}\\
r_{uv}&=\frac{\varSigma(u-\overline{u})((v-\overline{v})) }{\sqrt{[\varSigma(u-\overline{u})^2][\varSigma(v-\overline{v})^2]}}\\
&=\frac{\varSigma (a+hu-a-h\overline{u}) (b+Kv-b-K\overline{v})} {\sqrt{\varSigma(a+hu-a-h\overline{u})^2\varSigma(b+Kv-b-K\overline{v})^2}}\\
&=\frac{\varSigma(hu-h\overline{u})(Kv-K\overline{v})}{\sqrt{[\varSigma(hu-h\overline{u})^2][\varSigma(Kv-K\overline{v})^2]}}\\
&=\frac{hK\varSigma(u-\overline{u})(v-\overline{v})}{\sqrt{[h^2 K^2 \varSigma(u-\overline{u})^2] [\varSigma(v-\overline{v})^2]}}\\
&=\frac{hK\varSigma(u-\overline{u})(v-\overline{v})}{hK\,\sqrt{[\varSigma(u-\overline{u})^2] [\varSigma(v-\overline{v})^2]}}\\
&=\frac{\varSigma(u-\overline{u})(v-\overline{v}) }{\sqrt{[\varSigma(u-\overline{u})^2][\varSigma(v-\overline{v})^2]}}=
r_{uv}
\end{align*}

Note that

1. Non-causality: Correlation does not imply causation. If two variables are strongly correlated, it does not necessarily mean that changes in one variable cause changes in the other. This is because the correlation only measures the strength and direction of the linear relationship between two quantitative variables, not the underlying cause-and-effect relationship.
2. Sensitive to Outliers: The correlation coefficient can be sensitive to outliers, as outliers can disproportionately influence the correlation calculation.
3. Assumption of Linearity: The correlation coefficient measures the linear relationship between variables. It may not accurately capture non-linear relationships between variables.
4. Scale Invariance: The correlation coefficient is independent of the scale of the data. That is, multiplying or dividing all the values of one or both variables by a constant will not affect the strength and direction of correlation coefficient. This makes it useful for comparing relationships between variables measured in different units.
5. Strength vs. Causation: A high correlation does not necessarily imply causation. It is because two variables are strongly correlated does not mean one causes the other. There might be a third unknown factor influencing both variables. Correlation analysis is a good starting point for exploring relationships, but further investigation is needed to establish causality.

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