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Chapter 12: Problem 1

Find the regression equation for predicting final score from midterm score, based on the following information: $$\begin{array}{cl}{\text { average midterm score }=70,} & {\text { SD }=10} \\\ {\text { average final score }=55,} & {\text { SD }=20, \quad r=0.60}\end{array}$$

Short Answer

Expert verified
The regression equation is \( y = -29 + 1.2x \).

Step by step solution

01

Recognize the formula for regression equation

The regression equation for predicting one variable from another is given by the formula: \( y = a + bx \), where \(a\) is the y-intercept and \(b\) is the slope of the line. To find \(b\), the formula is \( b = r \times \frac{SD_y}{SD_x} \), where \(r\) is the correlation coefficient, \(SD_y\) is the standard deviation of the y variable, and \(SD_x\) is the standard deviation of the x variable.
02

Calculate the slope (b)

Substitute the given values into the formula for the slope: \( b = 0.60 \times \frac{20}{10} \). This calculation yields \( b = 0.60 \times 2 = 1.2 \). Hence, the slope \(b\) is 1.2.
03

Find the y-intercept (a)

The y-intercept can be calculated using the formula \( a = \bar{y} - b\bar{x} \), where \(\bar{x}\) and \(\bar{y}\) are the averages of the midterm and final scores, respectively. Substitute the values into the equation: \( a = 55 - 1.2 \times 70 \). Compute \( a = 55 - 84 = -29 \). Therefore, the y-intercept \(a\) is -29.
04

Write the regression equation

Now that we have both the slope and the y-intercept, we can write the regression equation to predict final scores from midterm scores: \( y = -29 + 1.2x \). Here, \(y\) represents the predicted final score and \(x\) is the midterm score.

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Key Concepts

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

Correlation Coefficient
The correlation coefficient, often represented by the symbol \( r \), is a key concept in statistics. It measures the strength and direction of a linear relationship between two variables. In our example, the midterm and final scores. The value of \( r \) is always between -1 and 1.
  • A correlation coefficient of 1 implies a perfect positive correlation, indicating that as one variable increases, so does the other.
  • A coefficient of -1 denotes a perfect negative correlation, meaning that as one variable increases, the other decreases.
  • An \( r \) of 0 suggests no linear correlation.
In the problem above, \( r = 0.60 \), indicating a moderate positive correlation between the midterm and final scores. This suggests that, generally, higher midterm scores are associated with higher final scores, but not in a perfect one-to-one manner.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, or average, value, while a high standard deviation indicates a wide range of values.
  • In our context, the midterm scores have a standard deviation \( SD_x = 10 \).
  • The final scores have a \( SD_y = 20 \).
These values offer insight into how much the scores vary. The larger standard deviation of the final scores shows that they are more spread out around the mean than the midterm scores. Understanding this variation helps when using the scores in calculations, like finding the slope in our regression equation.
Slope Calculation
The slope in a regression equation indicates how much the final score is expected to change for each one-point increase in the midterm score. It is calculated using the formula:\[b = r \times \frac{SD_y}{SD_x}.\]Here, \( r \) is the correlation coefficient, \( SD_y \) is the standard deviation of the final scores, and \( SD_x \) is the standard deviation of the midterm scores.
In this exercise:
  • \( b = 0.60 \times \frac{20}{10} = 0.60 \times 2 = 1.2 \).
This means for every additional point on the midterm, the final score is expected to increase by 1.2 points, highlighting the positive relationship between these two variables.
Y-Intercept Calculation
The y-intercept is the predicted value of the final score when the midterm score is zero. It helps set the starting point in the regression equation, represented by:\[a = \bar{y} - b\bar{x}.\]In our problem:
  • \( \bar{x} \) is the average midterm score, which is 70.
  • \( \bar{y} \) is the average final score, which is 55.
  • We already found \( b = 1.2 \), the slope.
With these, we calculate the y-intercept:\[a = 55 - 1.2 \times 70 = 55 - 84 = -29.\]This implies that theoretically, a midterm score of zero would result in a predicted final score of -29, which, although not practical, helps position the regression line on a graph.

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Most popular questions from this chapter

In a large study (hypothetical) of the relationship between parental income and the IQs of their children, the following results were obtained: $$\begin{aligned} \text { average income } & \approx \$ 90,000, & \mathrm{SD} \approx \$ 45,000 \\ \text { average } \mathrm{IQ} & \approx 100, & \mathrm{SD} \approx 15, \quad r \approx 0.50 \end{aligned}$$ For each income group \((\$ 0-\$ 9999, \$ 10,000-\$ 19,999, \$ 20,000-\$ 29,999,\) etc. \(.\) the average IQ of children with parental income in that group was calculated and then plotted above the midpoint of the group \((55,000, \$ 15,000, \$ 25,000,\) etc.). It was found that the points on this graph followed a straight line very closely. The slope of this line (in IQ points per dollar) would be about: \(6,000 \quad 3,000 \quad 1,500 \quad 1,500 \quad 1 / 500 \quad 1 / 1,500 \quad 1 / 3,000 \quad 1 / 6,000\) can't say from the information given

A study is made of working couples. The regression equation for predicting wife's income from husband's income is wife's income \(=0.1667 \times\) husband's income \(+\$ 24,000.\) Another investigator solves this equation for husband's income, and gets husband's income \(=6 \times\) wife's income \(-\$ 144,000.\) True or false, and explain: the second investigator has found the regression equation for predicting husband's income from wife's income. If you want to compute anything, $$\begin{aligned} \text { husband's average income } &=\$ 54,000, \quad S D=\$ 39,000 \\ \text { wife's average income } &=\$ 33,000, \quad S D=\$ 26,000, \quad r=0.25 \end{aligned}$$

For men age \(18-24\) in the HANES5 sample, the regression equation for predicting height from weight ispredicted height \(=(0.0267\) inches per pound) \(\times\) (weight) \(+65.2\) inches (Height is measured in inches and weight in pounds.) If someone puts on 20 pounds, will he get taller by20 pounds \(\times 0.0267\) inches per pound \(\approx 0.5\) inches? If not, what does the slope mean?

Many epidemiologists think that a high level of salt in the diet causes high blood pressure. INTERSALT is often cited to support this view. INTERSALT was a large study done at 52 centers in 32 countries. \(^{15}\) Each center recruited 200 subjects in 8 age- and sex-groups. Salt intake was measured, as well as blood pressure and several possible confounding variables. After adjusting for age, sex, and the other confounding variables, the authors found a signif- icant association between high salt intake and high blood pressure. However, a more detailed analysis showed that in 25 of the centers, there was a positive association between blood pressure and salt; in the other \(27,\) the association was negative. Do the data support the theory that high levels of salt cause high blood pressure? Answer yes or no, and explain briefly.

For women age \(25-34\) in the HANES5 sample, the relationship between height and income can be summarized as follows: \(^{2}\) $$\begin{aligned} \text { average height } \approx 64 \text { inches, } & \text { SD } \approx 2.5 \text { inches } \\ \text { average income } \approx \$ 21,000, & \text { SD } \approx \$ 20,000, \quad r \approx 0.2 \end{aligned}$$ What is the regression equation for predicting income from height? What does the equation tell you?
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