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Chapter 13: Problem 70

A sample data set produced the following information. $$ \begin{aligned} &n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680 \\ &\Sigma x^{2}=1140, \text { and } \Sigma y^{2}=25,272 \end{aligned} $$ a. Calculate the linear correlation coefficient \(r\). b. Using the \(2 \%\) significance level, can you conclude that \(\rho\) is different from zero?

Short Answer

Expert verified
The Pearson correlation coefficient, \(r\), is approximately 0.978, which indicates a very high linear correlation. For the second part of the question, without specific data for computing the t-value, a conclusion cannot be drawn regarding whether the true population correlation \(\rho\) is significantly different from zero.

Step by step solution

01

Calculation of the Mean

Before calculating the linear correlation coefficient (\(r\)), find the mean of x and y. Given that \(危x=100\) and \(危y=220\) and \(n=10\), the mean of x (\(x虅\)) and the mean of y (\(y虅\)) is calculated as follows: For x: \(x虅 = 危x/n = 100/10 = 10\) For y: \(y虅 = 危y/n = 220/10 = 22\)
02

Calculation of Linear Correlation Coefficient (\(r\))

Let鈥檚 use the formula of \(r\) which is calculated as: \(r = [n(危xy) - (危x)(危y)] / sqrt{[n危x虏 - (危x)虏][n危y虏 - (危y)虏]} Plugging in the given values yields: \(r = [10(3680) - (100)(220)] / sqrt{[10(1140)-(100)虏][10(25272) - (220)虏]} = 0.978\
03

Statistical Significance

We are to carry out a hypothesis test at the 2% significance level, to decide whether the correlation coefficient is significantly different from zero. That is, the null hypothesis H鈧�: 蟻 = 0, and the alternate hypothesis H鈧�: 蟻 鈮� 0. We then use a t-distribution to perform this test. We calculate the value of t with n-2 = 10-2 = 8 degrees of freedom and significance level of 2%. The calculated t value will need to be compared with the critical t value from the t-distribution table for 8 degrees of freedom and the given significance level of 2%.
04

Conclusion

If the calculated t-value is greater than the critical t-value (from the t-distribution table), then we reject the null hypothesis and conclude that the correlation is significantly different from zero. If the calculated t-value is less than or equal to the critical t-value, then we fail to reject the null hypothesis and conclude that the correlation is not significantly different from zero. As the given problem does not provide sufficient specific data to compute the t-value, a specific conclusion for part b cannot be drawn.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a statistical method used to decide if there is enough evidence to reject a null hypothesis in favor of an alternate hypothesis. In the context of the linear correlation coefficient, we test whether the correlation coefficient \(r\) is significantly different from zero. This involves formulating two hypotheses:
  • Null Hypothesis (\(H_0\)): The population correlation coefficient \(\rho = 0\), suggesting no linear relationship between the variables.
  • Alternate Hypothesis (\(H_1\)): The population correlation coefficient \(\rho eq 0\), indicating a linear relationship exists.
In our example, we want to see if the correlation is statistically significant at a 2% significance level. Hypothesis testing helps in making informed decisions about the relationships between datasets, which can be crucial in various fields like finance, medicine, and social sciences.
Statistical Significance
Statistical significance is a measure that helps researchers understand if their results are meaningful. When we say a result is statistically significant, it means the observed result is highly unlikely to have occurred under the null hypothesis. For the linear correlation coefficient, statistical significance helps determine whether or not the observed correlation can be attributed to chance.
In hypothesis testing, we use a pre-determined significance level (\(\alpha\)) to decide what constitutes as statistically significant. In our example, a 2% significance level (\(\alpha = 0.02\)) is chosen. This means we are 98% confident in the results being non-random if the null hypothesis is rejected. A result is considered significant if the calculated test statistic exceeds a critical value which is determined based on the chosen significance level. This ensures that the decision made is as accurate as possible given the data.
t-Distribution
The t-distribution is a probability distribution that helps in statistical analysis, particularly when dealing with small sample sizes. It is a bell-shaped distribution, somewhat similar to the normal distribution but with thicker tails. This makes it ideal for hypothesis testing because it accounts for the added variability expected with smaller sample sizes.
When performing hypothesis tests involving correlation coefficients, especially with small samples, the t-distribution is used to calculate the test statistic called the t-value. \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \]Here, \(n\) represents the sample size, and \(r\) is the correlation coefficient. Once calculated, this t-value is compared to a critical value from the t-distribution table based on degrees of freedom (\(n-2\)) and the chosen significance level.
If the calculated t-value is greater than the critical t-value, the null hypothesis is rejected, suggesting that the correlation is indeed statistically significant. The t-distribution thus plays a crucial role in lending credibility to the conclusions drawn from hypothesis tests.

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Most popular questions from this chapter

Briefly explain the difference between a deterministic and a probabilistic regression model.

The following data give information on the ages (in years) and the numbers of breakdowns during the last month for a sample of seven machines at a large company. $$ \begin{array}{l|lllllll} \hline \text { Age (years) } & 12 & 7 & 2 & 8 & 13 & 9 & 4 \\ \hline \text { Number of breakdowns } & 10 & 5 & 1 & 4 & 12 & 7 & 2 \\ \hline \end{array} $$ a. Taking age as an independent variable and number of breakdowns as a dependent variable, what is your hypothesis about the sign of \(B\) in the regression line? (In other words, do you expect \(B\) to be positive or negative?) b. Find the least squares regression line. Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\mathrm{b}\) d. Compute \(r\) and \(r^{2}\) and explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at the \(2.5 \%\) significance level whether \(B\) is positive. h. At the \(2.5 \%\) significance level, can you conclude that \(\rho\) is positive? Is your conclusion the same as in part g?

The health department of a large city has developed an air pollution index that measures the level of several air pollutants that cause respiratory distress in humans. The accompanying table gives the pollution index (on a scale of 1 to 10 , with 10 being the worst) for 7 randomly selected summer days and the number of patients with acute respiratory problems admitted to the emergency rooms of the city's hospitals. $$ \begin{array}{l|ccccccc} \hline \text { Air pollution index } & 4.5 & 6.7 & 8.2 & 5.0 & 4.6 & 6.1 & 3.0 \\\ \hline \text { Emergency admissions } & 53 & 82 & 102 & 60 & 39 & 42 & 27 \\ \hline \end{array} $$ a. Taking the air pollution index as an independent variable and the number of emergency admissions as a dependent variable, do you expect \(B\) to be positive or negative in the regression model \(y=A+B x+\epsilon ?\) b. Find the least squares regression line. Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Compute \(r\) and \(r^{2}\), and explain what they mean. d. Compute the standard deviation of errors. e. Construct a \(90 \%\) confidence interval for \(B\). f. Test at the \(5 \%\) significance level whether \(B\) is positive. g. Test at the \(5 \%\) significance level whether \(\rho\) is positive. Is your conclusion the same as in part \(\mathrm{f}\) ?

The following table gives information on the limited tread warranties (in thousands of miles) and the prices of 12 randomly selected tires at a national tire retailer as of July 2009. $$ \begin{array}{l|llllllllllll} \hline \text { Warranty (thousands of miles) } & 60 & 70 & 75 & 50 & 80 & 55 & 65 & 65 & 70 & 65 & 60 & 65 \\ \hline \text { Price per tire }(\$) & 95 & 70 & 94 & 90 & 121 & 70 & 84 & 80 & 92 & 79 & 66 & 95 \\ \hline \end{array} $$ a. Taking warranty length as an independent variable and price per tire as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the regression of price per tire on warranty length. c. Briefly explain the meaning of the values of \(a\) and \(b\) calculated in part \(\mathrm{b}\). d. Calculate \(r\) and \(r^{2}\) and explain what they mean. e. Plot the scatter diagram and the regression line. f. Predict the price of a tire with a warranty length of 73,000 miles. g. Compute the standard deviation of errors. h. Construct a \(95 \%\) confidence interval for \(B\). i. Test at the \(5 \%\) significance level if \(B\) is positive. j. Using \(\alpha=.025\), can you conclude that the linear correlation coefficient is positive?

The following table, reproduced from Exercise \(13.26\), gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 varieties of Kellogg's cereal. $$ \begin{array}{l|rrrrrrrrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 & 2 & 7 & 14 & 20 & 3 & 13 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 & 100 & 120 & 190 & 190 & 110 & 120 \\ \hline \end{array} $$ a. Determine the standard deviation of errors. b. Find the coefficient of determination and give a brief interpretation of it.
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