91Ó°ÊÓ

The following table gives information on the incomes (in thousands of dollars) and charitable contributions (in hundreds of dollars) for the last year for a random sample of 10 households. $$ \begin{array}{rc} \hline \text { Income } & \text { Charitable Contributions } \\ \hline 76 & 15 \\ 57 & 4 \\ 140 & 42 \\ 97 & 33 \\ 75 & 5 \\ 107 & 32 \\ 65 & 10 \\ 77 & 18 \\ 102 & 28 \\ 53 & 4 \\ \hline \end{array} $$ a. With income as an independent variable and charitable contributions as a dependent variable, compute \(\mathrm{SS}_{x \mathrm{x}}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x v}\) b. Find the regression of charitable contributions on income. c. Briefly explain the meaning of the values of \(a\) and \(b\). d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at the \(1 \%\) significance level whether \(B\) is positive. h. Using the \(1 \%\) significance level, can you conclude that the linear correlation coefficient is different from zero?

Step by step solution

01

- Data Calculation

Firstly, calculation of the sums \(\Sigma X\), \(\Sigma Y\), \(\Sigma X^{2}\), \(\Sigma Y^{2}\), and \(\Sigma XY\) using the given data is needed.

02

- Calculation of \(SS_{xx}\), \(SS_{yy}\), and \(SS_{xy}\)

For \(SS_{xx}\), subtract \(\frac{(\Sigma X)^{2}}{n}\) from \(\Sigma X^{2}\). For \(SS_{yy}\), subtract \(\frac{(\Sigma Y)^{2}}{n}\) from \(\Sigma Y^{2}\). For \(SS_{xy}\), subtract \(\frac{(\Sigma X)(\Sigma Y)}{n}\) from \(\Sigma XY\).

03

- Find the regression

To find the regression line or the least squares line, calculate the slope \(b = SS_{xy} / SS_{xx}\) and the y-intercept \(a = \overline{Y} - b(\overline{X})\).

04

- Interpretation of 'a' and 'b'

'a' is the estimated amount of charitable contribution when income is zero and 'b' is the estimated increase in charitable contributions for a unit increase in income.

05

- Calculation of r and \(r^{2}\)

Calculate the Pearson's correlation coefficient \(r = SS_{xy}/\sqrt{SS_{xx} SS_{yy}}\) and then the coefficient of determination \(r^{2}\).

06

- Interpretation of 'r' and \(r^{2}\)

'r' gives the linear correlation between income and charitable contributions, while \(r^{2}\) gives the proportion of the variance in charitable contributions that is predictable from income.

07

- Calculation of standard deviation of errors

Calculate standard deviation of errors or residuals as \(\sigma_{e} = \sqrt{(SS_{yy} - b(SS_{xy}))/(n - 2)}\).

08

- Construction of 99% confidence interval for 'B'

The confidence interval for 'B' is given by \(B \pm z\sigma_{b}\) where \(z\) is the critical value for the desired confidence level (2.576 for 99%) and \(\sigma_{b} = \sigma_{e}/\sqrt{SS_{xx}}\).

09

- Hypothesis testing for 'B'

Using hypothesis testing, we test the null hypothesis 'H0: B=0' against the alternative 'H1: B > 0'. We calculate the t-value as \(t = B/\sigma_{b}\) and compare it with the critical value. If t > critical value, we reject the null hypothesis.

10

- Testing for correlation coefficient

To check whether the linear correlation coefficient is different from zero, the procedure from Step 9 is repeated but testing 'H0: r=0' against 'H1: r ≠ 0'.

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Key Concepts

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

Correlation Coefficient

The concept of correlation coefficient is crucial in understanding the strength and direction of a relationship between two variables. Here, we are considering income as our independent variable and charitable contributions as our dependent variable. The correlation coefficient, denoted as \( r \), can range from -1 to 1. A value closer to 1 signifies a strong positive correlation, meaning as income increases, charitable contributions tend to increase too. On the other hand, a value closer to -1 indicates a strong negative correlation. If \( r \) is around 0, it suggests no significant linear relationship.

In this exercise, calculating \( r \) involves deriving it from the sums \(SS_{xx}, SS_{yy}, \) and \(SS_{xy}\). Understanding the sign and magnitude of \( r \) helps us actually see how impactful income changes are on charitable contributions.

Least Squares

The least squares method is a popular technique used in linear regression to find the best-fitting line through a set of data points. It aims to minimize the sum of the squares of the vertical distances of the points from the line. This line is then called the least squares line.

To compute this in our exercise, we calculate the slope and y-intercept. The slope \( b \) is found by dividing \( SS_{xy} \) by \( SS_{xx} \), while the y-intercept \( a \) is determined by adjusting the mean of the dependent variable by the product of the slope and the mean of the independent variable: \( a = \overline{Y} - b(\overline{X}) \).

This line enables us to predict charitable contributions based on any given value of income, where \( a \) represents the baseline contribution when income is zero, and \( b \) shows how much contributions are expected to increase with every additional unit of income.

Confidence Interval

Confidence intervals provide a range in which we expect a population parameter to fall, given a certain level of confidence. Here, we're constructing a 99% confidence interval for the regression coefficient \( B \). This coefficient gives us an idea of how much charitable contributions are affected by changes in income across the population.

The calculation involves using the formula \( B \pm z\sigma_{b} \), where \( z \) is the critical value corresponding to the 99% confidence level (2.576), and \( \sigma_{b} \) represents the standard error of the regression coefficient.

A wider interval may suggest we're less certain about our estimate of \( B \), while a narrower one indicates higher confidence. This aspect of linear regression lets us express how precise we feel about our predictive ability.

Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to infer about population parameters based on sample data. In this exercise, we test whether the slope \( B \) is significantly different from zero.

First, we set our null hypothesis as \( H_0: B = 0 \), meaning there's no relationship between income and contributions, against the alternative hypothesis \( H_1: B > 0 \).
Next, we calculate the t-value using the formula \( t = B / \sigma_{b} \) and compare it to the critical threshold for our chosen significance level of 1%. If the calculated t-value exceeds the critical threshold, we can reject the null hypothesis, thus asserting that the positive relationship between income and contributions is statistically significant.

This step confirms whether our observed effect is likely due to real association rather than random chance, giving credence to our regression results.