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Chapter 4: Problem 22

Name the Relation, Part II For each of the following statements, explain whether you think the variables will have positive correlation, negative correlation, or no correlation. Support your opinion. (a) Number of cigarettes smoked by a pregnant woman each week and birth weight of her baby (b) Years of education and annual salary (c) Number of doctors on staff at a hospital and number of administrators on staff (d) Head circumference and IQ (e) Number of movie goers and movie ticket price

Short Answer

Expert verified
(a) Negative correlation; (b) Positive correlation; (c) Positive correlation; (d) No correlation; (e) Negative correlation.

Step by step solution

01

Identify the Relation for Each Pair

To determine the type of correlation between each pair of variables, consider whether an increase in one variable tends to cause an increase, decrease, or does not affect the other variable.
02

Analyze Statement (a)

(a) Number of cigarettes smoked by a pregnant woman each week and birth weight of her baby. In general, the more cigarettes a pregnant woman smokes, the lower the birth weight of her baby tends to be. Thus, this is a negative correlation.
03

Analyze Statement (b)

(b) Years of education and annual salary. Typically, more years of education are associated with higher annual salaries. This would indicate a positive correlation.
04

Analyze Statement (c)

(c) Number of doctors on staff at a hospital and number of administrators on staff. Hospitals with more doctors usually also have more administrators to manage them. This suggests a positive correlation.
05

Analyze Statement (d)

(d) Head circumference and IQ. No consistent relationship has been established between head circumference and IQ. Hence, there is likely no correlation.
06

Analyze Statement (e)

(e) Number of movie goers and movie ticket price. Typically, as movie ticket prices increase, fewer people may attend the movies, indicating a negative correlation.

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Key Concepts

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

positive correlation
Positive correlation occurs when an increase in one variable is associated with an increase in another variable.
In this scenario, both variables move in the same direction. The more one variable increases, the more the other variable is likely to increase as well.
For example:
  • Years of education and annual salary: Typically, more years of education are linked with higher annual salaries. This means that as education increases, salary also tends to increase, demonstrating a positive correlation.
  • Number of doctors on staff at a hospital and number of administrators on staff: Having more doctors often requires more administrators to manage them. Hence, as the number of doctors increases, the number of administrators also goes up.
negative correlation
Negative correlation happens when an increase in one variable is associated with a decrease in another variable.
In this type of relationship, the variables move in opposite directions. As one variable goes up, the other tends to go down.
For example:
  • Number of cigarettes smoked by a pregnant woman each week and birth weight of her baby: Generally, the more cigarettes a pregnant woman smokes, the lower the birth weight of her baby. This negative correlation illustrates how an increase in smoking leads to a decrease in birth weight.
  • Number of moviegoers and movie ticket price: As the price of movie tickets increases, fewer people are likely to attend. This negative correlation shows that higher prices typically result in lower attendance.
no correlation
No correlation means there is no consistent relationship between two variables.
In this situation, changes in one variable do not predictably lead to changes in the other variable.
For example:
  • Head circumference and IQ: There is no established relationship between these two variables. This means that changes in head circumference do not predictably affect a person's IQ, demonstrating no correlation.
It is important to note that no correlation doesn't mean that the variables can't have an unusual pattern; it simply means there is no clear, consistent relationship.
variable relationship
Understanding variable relationships is key to correlation analysis.
There are three primary types of relationships: positive correlation, negative correlation, and no correlation.
  • Positive correlation: Both variables increase together.
  • Negative correlation: One variable increases as the other decreases.
  • No correlation: The variables have no predictable, consistent relationship.
By examining these relationships, we can predict trends and make educated decisions. For instance, knowing a positive correlation exists between education and salary could encourage further education for higher earnings.

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Most popular questions from this chapter

(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 ( \(d\) ). $$ \begin{array}{lrrrrr} \hline x & 20 & 30 & 40 & 50 & 60 \\ \hline y & 100 & 95 & 91 & 83 & 70 \\ \hline \end{array} $$

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.

(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 ( \(d\) ). $$ \begin{array}{llllll} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 2 & 4 & 7 & 11 & 18 \\ \hline \end{array} $$

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.

31\. Putting It Together: A Tornado Model Is the width of a tornado related to the amount of distance for which the tornado is on the ground? Go to www.pearsonhighered.com/sullivanstats to obtain the data file \(4_{-} 3_{-} 31\) using the file format of your choice for the version of the text you are using. The data represent the width (yards) and length (miles) of tornadoes in the state of Oklahoma in \(2013 .\) (a) What is the explanatory variable? (b) Explain why this data should be analyzed as bivariate quantitative data. (c) Draw a scatter diagram of the data. What type of relation appears to exist between the width and length of a tornado? (d) Determine the correlation coefficient between width and length. (e) Is there a linear relation between a tornado's width and its length on the ground? (f) Find the least-squares regression line. (g) Predict the length of a tornado whose width is 500 yards. (h) Was the tornado whose width was 180 yards and length was 1.9 miles on the ground longer than would be expected? (i) Interpret the slope. (j) Explain why it does not make sense to interpret the intercept. (k) What proportion of the variability in tornado length is explained by the width of the tornado? (I) Plot residuals against the width. Does the residual plot suggest the two variables are linearly related? (m) Draw a boxplot of the residuals. Are there any outliers? (n) A major tornado was 4576 yards wide that had a length of 16.2 miles. Is this an influential tornado? Explain.
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