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Chapter 3: Problem 55

Does it pay to study for an exam? The number of hours studied, \(x,\) is compared to the exam grade received, \(y:\).$$\begin{array}{l|ccccc}\hline \boldsymbol{x} & 2 & 5 & 1 & 4 & 2 \\ \boldsymbol{y} & 80 & 80 & 70 & 90 & 60 \\\\\hline\end{array}$$.a. Find the equation for the line of best fit. b. Draw the line of best fit on the scatter diagram of the data drawn in Exercise 3.15 (p. 133 ). c. \(\quad\) Based on what you see in your answers to parts a and b, does it pay to study for an exam? Explain.

Short Answer

Expert verified
a. The equation for the line of best fit is \(y = 2.5x + 69\). \nc. Yes, it pays to study for an exam as the positive slope indicates that more study hours generally leads to higher scores.

Step by step solution

01

Calculate Mean of x and y

First, calculate the mean of x (study hours) and y (marks). The mean is calculated as the sum of all items divided by the number of items.\n Mean of x, \( \bar{x} = \frac{2+5+1+4+2}{5} = 2.8 \)\n Mean of y, \( \bar{y} = \frac{80+80+70+90+60}{5} = 76 \)
02

Calculate Slope

Next, calculate the slope of the line using the formula \( b = \frac{ \Sigma xy - n\bar{x}\bar{y} }{ \Sigma x虏 - n \bar{x}虏 } \).\n Here, \( \Sigma xy = 2*80+5*80+1*70+4*90+2*60 = 960 \), \( \Sigma x虏 = 2虏+5虏+1虏+4虏+2虏 = 30 \).\n So, \( b = \frac{ 960 - 5*2.8*76 }{ 30 - 5 * 2.8虏 } = 2.5 \) Future steps will estimate y-intercept.
03

Calculate Y Intercept

Finally, calculate the y-intercept (a) which yields the initial value of dependent variable when x is zero. It is defined by the formula \( a = \bar{y} - b\bar{x} \). \n So, \( a = 76 - 2.5 * 2.8 = 69 \)
04

Formulate the Equation

Our line of best fit, or regression line, is therefore \( y = 2.5x + 69 \)
05

Interpretation

After analyzing results from steps 1 through 4, to answer part c, one could say that it generally pays to study for an exam. The positive slope of 2.5 in the regression equation indicates that for each additional study hour, the expected increase in the exam score is around 2.5 points. Although studying longer does not guarantee a perfect score, it does demonstrate a positive trend.

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Key Concepts

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

Mean Calculation
To understand linear regression, we first need to calculate the mean values of the data sets we have. In our example, the mean will show us the average number of study hours and the average exam score.

  • For the mean of study hours, \( \bar{x} \), you add up all the study hours and divide by the total number of data points. Here, that's \( \frac{2+5+1+4+2}{5} = 2.8 \).
  • Similarly, for the exam grades, \( \bar{y} \), add up the scores and divide by five, which results in \( \frac{80+80+70+90+60}{5} = 76 \).
Calculating the mean gives a central value that represents the average tendency of the data, which is crucial for further calculations in regression analysis.
Slope of Line
To find how the number of study hours relates to the exam score, we need to calculate the slope of the line. The slope tells us how much the exam score is expected to change with each additional hour of study. It's calculated as:

\[ b = \frac{ \Sigma xy - n\bar{x}\bar{y} }{ \Sigma x^2 - n \bar{x}^2 } \]
This formula uses the sum of products of corresponding x and y values (\( \Sigma xy \)), the total number of points (\( n \)), and their squared sums (\( \Sigma x^2 \)).

In our case:
  • \( \Sigma xy = 960 \) and \( \Sigma x^2 = 30 \).
  • With \( n = 5 \), \( \bar{x} = 2.8 \), and \( \bar{y} = 76 \), the slope is \( b = 2.5 \).
The slope of 2.5 indicates that each additional hour of studying is associated with a 2.5-point increase in the exam score.
Y-Intercept
The y-intercept gives us the baseline performance, representing the expected exam score if no hours are spent studying. This is calculated using the formula:

\[ a = \bar{y} - b\bar{x} \]
Here, \( \bar{y} \) is the mean exam score, \( b \) is the slope, and \( \bar{x} \) is the mean study hours.

For our data, this becomes:
  • \( a = 76 - 2.5 \times 2.8 \)
  • The result gives \( a = 69 \).
This means that if a student does not study at all, they are expected to score 69, assuming this linear relationship holds.
Line of Best Fit
The line of best fit provides a visual representation of the data trend. It essentially shows the average relationship between study hours and exam scores.
The equation for the line of best fit, derived from our calculations, is:
\[ y = 2.5x + 69 \]
  • Here, \( y \) represents the exam score.
  • \( x \) refers to study hours.
  • \( 2.5 \) is the slope, indicating the rate of score change per study hour.
  • \( 69 \) is the y-intercept, the score without study hours.
This line helps predict future outcomes or evaluate potential changes in one variable as another changes.
Regression Analysis
Regression analysis is a statistical way of discovering the relationship between variables. By analyzing the data of hours studied versus exam scores, it helps establish a predictability pattern.

With a positive slope from our regression line \( y = 2.5x + 69 \), the analysis suggests that studying more hours generally results in a higher exam score. This kind of analysis is crucial, as it can:
  • Reveal trends and correlations.
  • Help predict outcomes based on linear relationships.
  • Guide decision-making in educational planning and advice.
However, it's important to remember that correlation does not imply causation. While the data shows a trend, it doesn't guarantee outcomes for every individual case.

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Most popular questions from this chapter

Does studying for an exam pay off? a. Draw a scatter diagram of the number of hours studied, \(x,\) compared with the exam grade received, \(y\).$$\begin{array}{l|rrrrr}\hline \boldsymbol{x} & 2 & 5 & 1 & 4 & 2 \\ \boldsymbol{y} & 80 & 80 & 70 & 90 & 60 \\\\\hline\end{array}$$.b. Explain what you can conclude based on the pattern of data shown on the scatter diagram drawn in part a. (Retain these solutions to use in Exercise \(3.55,\) p. 157 )

An experimental psychologist asserts that the older a child is, the fewer irrelevant answers he or she will give during a controlled experiment. To investigate this claim, the following data were collected. Draw a scatter diagram. (Retain this solution to use in Exercise \(3.38, p .143 .)\). $$\begin{array}{l|rrrrrrrrrr}\hline \text { Age, } x & 2 & 4 & 5 & 6 & 6 & 7 & 9 & 9 & 10 & 12 \\ \text { Irr Answers, } y & 12 & 13 & 9 & 7 & 12 & 8 & 6 & 9 & 7 & 5 \\\\\hline\end{array}$$.

Draw a scatter diagram for these data.$$\begin{array}{l|llllllllllllllllll}\hline x & 2 & 12 & 4 & 6 & 9 & 4 & 11 & 3 & 10 & 11 & 3 & 1 & 13 & 12 & 14 & 7 & 2 & 8 \\\y & 4 & 8 & 10 & 9 & 10 & 8 & 8 & 5 & 10 & 9 & 8 & 3 & 9 & 8 & 8 & 11 & 6 & 9 \\\\\hline\end{array}$$.Would you be justified in using the techniques of linear regression on these data to find the line of best fit? Explain.

A sample of 15 upper-class students whe commute to classes was selected at registration. They were asked to estimate the distance \((x)\) and the time \((y)=\) required to commute each day to class (see the following table).$$\begin{array}{cc|cc}\begin{array}{c}\text { Distance, } x \\\\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\\\text { (nearest }5 \text { minutes }) \end{array} & \begin{array}{c}\text { Distance, } x \\\\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\\\text { (nearest } 5 \text { minutes } \end{array} \\\\\hline 18 & 20 & 2 & 5 \\\8 & 15 & 15 & 25 \\\20 & 25 & 16 & 30 \\\5 & 20 & 9 & 20 \\\5 & 15 & 21 & 30 \\\11 & 25 & 5 & 10 \\\9 & 20 & 15 & 20 \\\10 & 25 & & \\\\\hline \end{array}$$.a. Do you expect to find a linear relationship between the two variables commute distance and commute time? If so, explain what relationship you expect. b. Construct a scatter diagram depicting these data. c. Does the scatter diagram in part b reinforce what you expected in part a?

Consider the saying, "Build it and they will come." This notable saying from a movie may very well apply to shopping malls. Just be sure when you build that there is room for not only the mall but also for those that will come-and thus include enough space for parking. Consider the random sample of major malls in Irvine, California.$$\begin{array}{ccc} \text { Square Feet } & \text { Parking Spaces } & \text { Number of Stores } \\\ \hline 270,987 & 3128 & 65 \\ 258,761 & 1500 & 43 \\ 1,600,350 & 8572 & 120 \\ 210,743 & 793 & 59 \\ 880,000 & 7100 & 95 \\ 2,700,000 & 15,000 & 300 \\ \hline\end{array}$$ a. Draw a scatter diagram with "parking spaces" as the dependent variable, \(y,\) and "square feet" as the independent variable, \(x .\) (Suggestion: Use 1000 s of square feet.) b. Does the scatter diagram in part a suggest that a linear regression will be useful? Explain. c. Calculate the equation for the line of best fit. d. Draw the line of best fit on the scatter diagram you obtained in part a. Explain the role of a positive slope for this pair of variables. e. Do you see a potential lurking variable? Explain its possible role. f. Draw a scatter diagram with "parking spaces" as the dependent variable, \(y,\) and "number of stores" as the independent variable, \(x\) g. Does the scatter diagram in part e suggest that a linear regression will be useful? Explain. h. Calculate the equation for the line of best fit. i. Draw the line of best fit on the scatter diagram you obtained in part e. j. Do you see a potential lurking variable? Explain its possible role. k. Draw a scatter diagram with "number of stores" as the dependent variable, \(y,\) and "square feet" as the predictor variable, \(x\) 1\. Does the scatter diagram in part \(\mathrm{k}\) suggest that a linear regression will be useful? Explain. m. Calculate the equation for the line of best fit. n. Draw the line of best fit on the scatter diagram you obtained in part \(\mathrm{k}\)
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