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Chapter 4: Problem 56

Suppose that two variables, \(X\) and \(Y\), are negatively associated. Does this mean that above-average values of \(X\) will always be associated with below- average values of \(Y ?\) Explain.

Short Answer

Expert verified
No, it indicates a tendency, not a strict rule.

Step by step solution

01

Understand Negative Association

When two variables, such as \(X\) and \(Y\), are said to be negatively associated, it means that as one variable increases, the other variable tends to decrease. This is generally indicated by a negative correlation coefficient.
02

Above-average Values

An above-average value of \(X\) is any value of \(X\) that is greater than the mean (or average) value of the distribution of \(X\).
03

Relationship to Below-average Values of \(Y\)

A below-average value of \(Y\) is any value of \(Y\) that is less than the mean (or average) value of the distribution of \(Y\).
04

Apply Negative Association

Given the negative association, above-average values of \(X\) are generally associated with below-average values of \(Y\), but this does not guarantee that it will always be the case. It indicates a tendency rather than a strict rule.
05

Conclusion

Therefore, while above-average values of \(X\) are likely to be associated with below-average values of \(Y\) due to the negative association, exceptions are possible, and 'always' is too strong a term.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Correlation
When we talk about negative correlation, we're referring to a relationship between two variables where one variable increases as the other decreases. This is commonly represented by a negative correlation coefficient. For example, if X and Y are our variables, and they are negatively correlated, you will often find that when X goes up, Y comes down.

Imagine you are comparing the time you spend watching TV (X) and your grades in school (Y). A negative correlation would suggest that as you spend more time watching TV, your grades decrease. However, remember it's not a perfect rule; it's more about the overall trend.
Mean Value
The mean value, or average, is a central concept in statistics. It represents the central point of a set of numbers. To find the mean, add up all the numbers and divide by the number of values you added. For example, if your test scores are 70, 80, and 90, the mean is \[ \frac{70 + 80 + 90}{3} = 80 \].

This concept is crucial when discussing above-average and below-average values. If X has a mean of 50 and you have a value of X that is 60, this value is above-average. Similarly, if Y has a mean of 50 and you have a value of Y that is 40, this value is below-average.
Distribution
Distribution describes how values are spread over a range. Think of it as a way to visualize or think about data. Common types of distributions include normal (bell-curve) and skewed distributions.

A normal distribution is symmetric, with most values clustering around the mean and fewer as you move away. On the other hand, a skewed distribution has a longer tail on one side or the other. Understanding distribution helps in identifying patterns, such as whether more students scored around the mean value or if there were lots of very high and very low scores.

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Most popular questions from this chapter

The General Social Survey asks questions about one's happiness in marriage. Is there an association between gender and happiness in marriage? Use the data in the table to determine if gender is associated with happiness in marriage. Treat gender as the explanatory variable. $$\begin{array}{lrrr} & \text { Male } & \text { Female } & \text { Total } \\\\\hline \text { Very happy } & 7,609 & 7,942 & 15,551 \\\\\hline \text { Pretty happy } & 3,738 & 4,447 & 8,185 \\\\\hline \text { Not too happy } & 259 & 460 & 719 \\\\\hline \text { Total } & 11.606 & 12.849 & 24.455\end{array}$$

Use the linear correlation coefficient given to determine the coefficient of determination, \(R^{2} .\) Interpret each \(R^{2}\) (a) \(r=-0.32\) (b) \(r=0.13\) (c) \(r=0.40\) (d) \(r=0.93\)

Professor Katula feels that there is a relation between the number of hours a statistics student studies each week and the student's age. She conducts a survey in which 26 statistics students are asked their age and the number of hours they study statistics each week. She obtains the following results: $$ \begin{array}{ll|ll|ll} \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } \\ \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} \\ \hline 18 & 4.2 & 19 & 5.1 & 22 & 2.1 \\ \hline 18 & 1.1 & 19 & 2.3 & 22 & 3.6 \\ \hline 18 & 4.6 & 20 & 1.7 & 24 & 5.4 \\ \hline 18 & 3.1 & 20 & 6.1 & 25 & 4.8 \\ \hline 18 & 5.3 & 20 & 3.2 & 25 & 3.9 \\ \hline 18 & 3.2 & 20 & 5.3 & 26 & 5.2 \\ \hline 19 & 2.8 & 21 & 2.5 & 26 & 4.2 \\ \hline 19 & 2.3 & 21 & 6.4 & 35 & 8.1 \\ \hline 19 & 3.2 & 21 & 4.2 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data. Comment on any potential influential observations. (b) Find the least-squares regression line using all the data points. (c) Find the least-squares regression line with the data point (35,8.1) removed. (d) Draw each least-squares regression line on the scatter diagram obtained in part (a). (e) Comment on the influence that the point (35,8.1) has on the regression line.

(a) By hand, draw a scatter diagram treating \(x\) as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the leastsquares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part ( \(d\) ). $$ \begin{array}{rrrrrr} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 7 & 6 & 3 & 2 & 0 \\ \hline \end{array} $$

Consider the following data set: $$ \begin{array}{lllllllll} \hline x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\\ \hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\ \hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\ \hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5 \\ \hline \end{array} $$ (a) Draw a scatter diagram with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20 . (b) Compute the linear correlation coefficient. (c) Now multiply both \(x\) and \(y\) by 2 . (d) Draw a scatter diagram of the new data with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20. Compare the scatter diagrams. (e) Compute the linear correlation coefficient. (f) Conclude that multiplying each value in the data set by a nonzero constant does not affect the correlation between the variables.
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