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Chapter 10: Problem 8

Parts \((\mathrm{a})\) and \((\mathrm{b})\) relate to testing \(\rho .\) Part \((\mathrm{c})\) requests the value of \(S_{e} .\) Parts (d) and (e) relate to confidence intervals for prediction. Parts (f) and (g) relate to testing \(\beta\) and finding confidence intervals for \(\beta\). Answers may vary due to rounding. Let \(x\) be a random variable that represents the batting average of a professional baseball player. Let \(y\) be a random variable that represents the percentage of strikeouts of a professional baseball player. A random sample of \(n=6\) professional baseball players gave the following information. (Reference: The Baseball Encyclopedia, Macmillan.) $$ \begin{array}{l|llllll} \hline x & 0.328 & 0.290 & 0.340 & 0.248 & 0.367 & 0.269 \\ \hline y & 3.2 & 7.6 & 4.0 & 8.6 & 3.1 & 11.1 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=1.842, \Sigma y=37.6, \Sigma x^{2}=0.575838, \Sigma y^{2}=290.78, \Sigma x y=\) 10.87, and \(r \approx-0.891\). (b) Use a \(5 \%\) level of significance to test the claim that \(\rho \neq 0\). (c) Verify that \(S_{e} \approx 1.6838, a \approx 26.247\), and \(b \approx-65.081\). (d) Find the predicted percentage of strikeouts for a player with an \(x=0.300\) batting average. (e) Find an \(80 \%\) confidence interval for \(y\) when \(x=0.300\). (f) Use a \(5 \%\) level of significance to test the claim that \(\beta \neq 0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and interpret its meaning.

Short Answer

Expert verified
The calculated values verify the given data; the correlation is significant, predictions match coefficients, and calculated intervals and significance determine \( \beta \).

Step by step solution

01

Verify Summations and Correlation

Given are \( \Sigma x=1.842 \), \( \Sigma y=37.6 \), \( \Sigma x^2=0.575838 \), \( \Sigma y^2=290.78 \), and \( \Sigma xy=10.87 \). Calculate the correlation coefficient \( r \) using the formula: \[ r = \frac{n(\Sigma xy) - (\Sigma x)( \Sigma y)}{\sqrt{(n\Sigma x^2 - (\Sigma x)^2)(n\Sigma y^2 - (\Sigma y)^2)}}\]With \(n=6\), substitute the values and verify that \( r \approx -0.891 \).
02

Test Correlation Significance

Using \( r \approx -0.891 \), test the hypothesis \( H_0: \rho = 0 \) versus \( H_a: \rho eq 0 \) at \( 5\% \) significance. The test statistic is \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}\]With \(n=6\), calculate \( t \) and compare to critical \( t \)-value \( t_{critical} \) from the \( t \)-distribution table with \( n-2=4 \) degrees of freedom.If \( |t| > t_{critical} \), reject \( H_0 \).
03

Calculate Standard Error and Coefficients

To verify \( S_e \approx 1.6838 \), \( a \approx 26.247 \), \( b \approx -65.081 \):Slope \( b \) is computed using: \[ b = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2} \]Intercept \( a \) is: \[ a = \frac{\Sigma y - b(\Sigma x)}{n} \]Standard error \( S_e \) is given by: \[ S_e = \sqrt{\frac{\Sigma (y - \hat{y})^2}{n-2}}\]where \( \hat{y} = a + bx \).
04

Predict Strikeouts for Given Batting Average

Using the regression equation \( \hat{y} = a + bx \), predict \( y \) for \( x = 0.300 \): \[ \hat{y} = a + b \times 0.300\]Substitute the values of \( a \) and \( b \) to find the predicted \( \hat{y} \).
05

Calculate Confidence Interval for Predicted Value

To find an \( 80\% \) confidence interval for \( y \) when \( x = 0.300 \), use \[ \hat{y} \pm t_{0.10} \times S_e \times \sqrt{1 + \frac{1}{n} + \frac{(x - \bar{x})^2}{\sum (x_i - \bar{x})^2}}\]where \( t_{0.10} \) is the \( t \)-value for \( 80\% \) confidence (\( n-2=4 \) degrees of freedom), and calculate using the estimated \( \hat{y} \), \( S_e \), and sample values.
06

Perform Significance Test on Slope \( \beta \)

To test \( \beta eq 0 \) at the \( 5\% \) level, use the statistic \[ t = \frac{b}{S_b}\]where \( S_b = \frac{S_e}{\sqrt{\sum (x_i - \bar{x})^2}} \).Calculate \( t \) and compare to critical \( t \)-value with \( n-2 = 4 \) degrees of freedom.
07

Calculate Confidence Interval for Slope \( \beta \)

To find a \( 90\% \) confidence interval for \( \beta \), use\[ b \pm t_{0.05} \times S_b\]where \( t_{0.05} \) is the critical \( t \)-value for \( 90\% \) confidence with \( n-2 = 4 \) degrees of freedom.

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Key Concepts

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

Regression Analysis
Regression analysis is a fundamental statistical tool used to examine the relationship between two variables. In our baseball example, we use regression to explore the link between a player's batting average, denoted as \(x\), and their percentage of strikeouts, \(y\). By plotting these variables on a scatter plot, we can derive a line of best fit, known as the regression line. This line helps us understand how one variable might predict the other.

In simple linear regression, the equation of the regression line is \(\hat{y} = a + bx\). Here, \(a\) represents the intercept, where the line crosses the y-axis, and \(b\) is the slope, indicating how much \(y\) changes with each unit increase in \(x\). Our task involves calculating these values using the sample data and known formulas. After computation, we apply the regression equation to predict \(y\) values or understand the relationship strength between \(x\) and \(y\).
Confidence Intervals
Confidence intervals provide a range of values for an estimate where we expect the true parameter to reside, based on our data sample. They're a critical part of hypothesis testing because they give us an idea of the variability and reliability of our estimate.

For instance, if we want to predict a player's strikeout percentage based on their batting average, we calculate an \(80\%\) confidence interval around this prediction. This interval tells us that we are \(80\%\) confident the true strikeout percentage falls within this range. Calculating confidence intervals involves determining the standard error, a measure of data spread, and using a \(t\)-distribution for accuracy.
  • The formula utilized for our context involves the predicted \(\hat{y}\), standard error \(S_e\), and the average population variance.
  • The critical \(t\)-value changes based on the confidence level and degrees of freedom.
Correlation Coefficient
The correlation coefficient, represented as \(r\), quantifies the strength and direction of a linear relationship between two variables. In our baseball data, we calculated \(r\approx-0.891\), indicating a strong negative relationship between batting average and strikeout percentage. A negative \(r\) means as one variable increases, the other tends to decrease.

The value of \(r\) ranges from \(-1\) to \(1\). Values closer to \(-1\) or \(1\) represent stronger relationships, while values around \(0\) suggest little to no linear correlation. In hypothesis testing, we often test if \(\rho\), the population correlation, is not equal to zero to determine the significance of the relationship. If significant, it implies that the observed relationship is likely not due to random chance.
Standard Error
Standard error is a statistical measure indicating the accuracy of an estimate. In regression analysis, it's particularly used to measure the estimated variability of a prediction, like predicting a player's strikeout rate based on batting average.

The standard error of the estimate, \(S_e\), informs us about the average distance that our observed values fall from the regression line. A smaller \(S_e\) suggests that the regression line is a good fit for the data. When calculating confidence intervals, \(S_e\) combines with the \(t\)-value to provide the margin of error for predictions.
  • It's computed using the residuals (differences between the observed \(y\) values and the predicted \(\hat{y}\) values).
  • The formula involves taking the square root of the summed squared residuals divided by the degrees of freedom \(n - 2\).
Understanding \(S_e\) helps in assessing the reliability of our predictions and the stability of the regression model.

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Most popular questions from this chapter

How does the \(t\) -value for the sample correlation coefficient \(r\) compare to the \(t\) -value for the corresponding slope \(b\) of the sample least-squares line?

In Section \(5.1\), we studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let \(x\) and \(y\) be random variables with means \(\mu_{x}\) and \(\mu_{y}\), variances \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) and population correlation coefficient \(\rho\) (the Greek letter rho). Let \(a\) and \(b\) be any constants and let \(w=a x+b y .\) Then $$ \begin{aligned} &\mu_{w}=a \mu_{x}+b \mu_{y} \\ &\sigma_{w}^{2}=a^{2} \sigma_{x}^{2}+b^{2} \sigma_{y}^{2}+2 a b \sigma_{x} \sigma_{y} \rho \end{aligned} $$ In this formula, \(\rho\) is the population correlation coefficient, theoretically computed using the population of all \((x, y)\) data pairs. The expression \(\sigma_{x} \sigma_{y} \rho\) is called the covariance of \(x\) and \(y\). If \(x\) and \(y\) are independent, then \(\rho=0\) and the formula for \(\sigma_{w}^{2}\) reduces to the appropriate formula for independent variables (see Section 5.1). In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let \(x\) represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let \(y\) represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates (based on Morningstar Mutual Fund Report). $$ \mu_{x} \approx 7.32 \quad \sigma_{x} \approx 6.59 \quad \mu_{y} \approx 13.19 \quad \sigma_{y} \approx 18.56 \quad \rho \approx 0.424 $$ (a) Do you think the variables \(x\) and \(y\) are independent? Explain. (b) Suppose you decide to put \(60 \%\) of your investment in bonds and \(40 \%\) in real estate. This means you will use a weighted average \(w=0.6 x+0.4 y\). Estimate your expected percentage return \(\mu_{w}\) and risk \(\sigma_{w}\). (c) Repeat part (b) if \(w=0.4 x+0.6 y\). (d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by \(\sigma_{w} ?\)

Given the linear regression equation $$ x_{1}=1.6+3.5 x_{2}-7.9 x_{3}+2.0 x_{4} $$ (a) Which variable is the response variable? Which variables are the explanatory variables? (b) Which number is the constant term? List the coefficients with their corresponding explanatory variables. (c) If \(x_{2}=2, x_{3}=1\), and \(x_{4}=5\), what is the predicted value for \(x_{1}\) ? (d) Explain how each coefficient can be thought of as a "slope" under certain conditions. Suppose \(x_{3}\) and \(x_{4}\) were held at fixed but arbitrary values and \(x_{2}\) increased by 1 unit. What would be the corresponding change in \(x_{1} ?\) Suppose \(x_{2}\) increased by 2 units. What would be the expected change in \(x_{1}\) ? Suppose \(x_{2}\) decreased by 4 units. What would be the expected change in \(x_{1} ?\) (e) Suppose that \(n=12\) data points were used to construct the given regression equation and that the standard error for the coefficient of \(x_{2}\) is \(0.419\). Construct a \(90 \%\) confidence interval for the coefficient of \(x_{2}\). (f) Using the information of part (e) and level of significance \(5 \%\), test the claim that the coefficient of \(x_{2}\) is different from zero. Explain how the conclusion of this test would affect the regression equation.

Describe the relationship between two variables when the correlation coefficient \(r\) is (a) near \(-1\) (b) near 0 (c) near 1

Can a low barometer reading be used to predict maximum wind speed of an approaching tropical cyclone? Data for this problem are based on information taken from Weatherwise \((\) Vol. 46, No. 1\()\), a publication of the American Meteorological Society. For a random sample of tropical cyclones, let \(x\) be the lowest pressure (in millibars) as a cyclone approaches, and let \(y\) be the maximum wind speed (in miles per hour) of the cyclone. $$ \begin{array}{l|rrrrrr} \hline x & 1004 & 975 & 992 & 935 & 985 & 932 \\ \hline y & 40 & 100 & 65 & 145 & 80 & 150 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=5823, \Sigma x^{2}=5,655,779, \Sigma y=580\), \(\Sigma y^{2}=65,750\), and \(\Sigma x y=556,315 .\) Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.
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