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Chapter 4: Problem 60

Suppose a doctor telephones those patients who are in the highest \(10 \%\) with regard to their recently recorded blood pressure and asks them to return for a clinical review. When she retakes their blood pressures, will those new blood pressures, as a group (that is, on average), tend to be higher than, lower than, or the same as the earlier blood pressures, and why?

Short Answer

Expert verified
On the second check, the blood pressures of this group will tend to be lower than the earlier, not due to improvement in health conditions, but because of the statistical phenomenon known as 'regression toward the mean'.

Step by step solution

01

Understanding the General Concept

The scenario involves a phenomenon known as 'regression to the mean'. This statistical concept describes how, if a variable is extreme on its first measurement, it will tend to be closer to the average on its second measurement. This is not due to any specific cause, but simply a statistic probability. It occurs because of the random variations that naturally happen.
02

Applying the Concept to the Exercise

Since the doctor is only calling back the highest 10% of patients with regard to their blood pressure, their initial readings were relatively extreme. There is no specific medical intervention suggested between the two readings that might lower the patients' blood pressure. Therefore, it can be expected that when these patients return for a clinical review, their new blood pressure readings will, on average, be closer to the mean, or lower, than the previous ones.
03

Explaining the Outcome

This doesn't mean that the patients' health conditions have improved. The statistical phenomenon of regression toward the mean suggests that if you select a group based on extreme measurement, on follow-up measurement they are likely to appear less extreme due to natural variation. It's important to note that regression to the mean does not occur if the readings are not initially extreme or if there is significant consistency between measurements.

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Most popular questions from this chapter

The following table shows the heights and weights of some people. The scatterplot shows that the association is linear enough to proceed. $$ \begin{array}{|c|c|} \hline \text { Height (inches) } & \text { Weight (pounds) } \\ \hline 60 & 105 \\ \hline 66 & 140 \\ \hline 72 & 185 \\ \hline 70 & 145 \\ \hline 63 & 120 \\ \hline \end{array} $$ a. Calculate the correlation, and find and report the equation of the regression line, using height as the predictor and weight as the response. b. Change the height to centimeters by multiplying each height in inches by \(2.54\). Find the weight in kilograms by dividing the weight in pounds by \(2.205 .\) Retain at least six digits in each number so there will be no errors due to rounding. c. Report the correlation between height in centimeters and weight in kilograms, and compare it with the correlation between the height in inches and weight in pounds. d. Find the equation of the regression line for predicting weight from height, using height in centimeters and weight in kilograms. Is the equation for weight in pounds and height in inches the same as or different from the equation for weight in kilograms and height in centimeters?

The following table gives the number of miles per gallon in the city and on the highway for some of the most fuel efficient cars according to Consumer Reports. Make a scatterplot of the data using city mileage as the predictor variable. Find the regression equation and use it to predict the highway mileage for a fuel-efficient car that gets 40 miles per gallon in city driving. Would it be appropriate to use the regression equation to predict the highway mileage for a fuel-efficient car that got 60 miles per gallon in city driving? If so, make the prediction. If not, explain why it would be inappropriate to do so. $$ \begin{array}{|lcc|} \hline \text { Model } & \text { City Mileage } & \text { Highway Mileage } \\\ \hline \text { Toyota Prius 3 } & 43 & 59 \\ \hline \text { Hyundai Ioniq } & 42 & 60 \\ \hline \text { Toyota Prius Prime } & 38 & 62 \\ \hline \text { Kia Niro } & 33 & 52 \\ \hline \text { Toyota Prius C } & 37 & 48 \\ \hline \text { Chevrolet Malibu } & 33 & 49 \\ \hline \end{array} $$$$ \begin{array}{|lcc|} \hline \text { Model } & \text { City Mileage } & \text { Highway Mileage } \\\ \hline \text { Ford Fusion } & 35 & 41 \\ \hline \text { Hyundai Sonata } & 31 & 46 \\ \hline \text { Toyota Camry } & 32 & 43 \\ \hline \text { Ford C-Max } & 35 & 38 \\ \hline \end{array} $$

Data on the number of home runs, strikeouts, and batting averages for a sample of 50 Major League Baseball players were obtained. Regression analyses were conducted on the relationships between home runs and strikeouts and between home runs and batting averages. The StatCrunch results are shown below. (Source: mlb.com) Simple linear regression results: Dependent Variable: Home Runs Independent Variable: Strikeouts Home Runs \(=0.092770565+0.22866236\) Strikeouts Sample size: 50 \(\mathrm{R}\) (correlation coefficient) \(=0.63591835\) \(\mathrm{R}-\mathrm{sq}=0.40439215\) Estimate of error standard deviation: \(8.7661607\) Simple linear regression results: Dependent Variable: Home Runs Independent Variable: Batting Average Home Runs \(=45.463921-71.232795\) Batting Average Sample size: 50 \(\mathrm{R}\) (correlation coefficient) \(=-0.093683651\) \(\mathrm{R}-\mathrm{sq}=0.0087766264\) Estimate of error standard deviation: \(11.30876\) Based on this sample, is there a stronger association between home runs and strikeouts or home runs and batting average? Provide a reason for your choice based on the StatCrunch results provided.

In Exercise \(4.1\) there is a graph of the relationship between SAT score and college GPA. SAT score was the predictor and college GPA was the response variable. If you reverse the variables so that college GPA was the predictor and SAT score was the response variable, what effect would this have on the numerical value of the correlation coefficient?

Move studios try to predict how much money their movies will make. One possible predictor is the amount of money spent on the production of the movie. The table shows the budget and amount of money made for a sample of movies made in 2017 . The budget (amount spent making the movie) and gross (amount earned by ticket sales) are shown in the table. Make a scatterplot of the data and comment on what you see. If appropriate, do a complete linear regression analysis. If it is not appropriate to do so, explain why not. (Source: IMDB) $$ \begin{array}{|lcc|} \hline \text { Film } & \begin{array}{c} \text { Gross(in \$ } \\ \text { millions) } \end{array} & \begin{array}{c} \text { Budget (in \$ } \\ \text { millions) } \end{array} \\ \hline \text { Wonder Woman } & 412.6 & 149 \\ \hline \text { Beauty and the Beast } & 504 & 160 \\ \hline \text { Guardians of the Galaxy Vol. } 2 & 389.8 & 200 \\ \hline \text { Spider-Man: Homecoming } & 334.2 & 175 \\ \hline \text { It } & 327.5 & 35 \\ \hline \text { Despicable Me 3 } & 264.6 & 80 \\ \hline \text { Logan } & 226.3 & 97 \\ \hline \text { The Fate of the Furious } & 225.8 & 250 \\ \hline \text { Dunkirk } & 188 & 100 \\ \hline \text { The LEGO Batman Movie } & 175.8 & 80 \\ \hline \text { Thor Ragnarok } & 310.7 & 180 \\ \hline \text { Get Out } & 175.5 & 5 \\ \hline \text { Dead Men Tell No Tales } & 172.6 & 230 \\ \hline \text { Cars } 3 & 152 & 175 \\ \hline \end{array} $$
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