Concept explainers
In Problems 7-12, parts (a) and (b) relate to testing
7. Basketball: Free Throws and Field Goals Let x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of n = 6 professional basketball players gave the following information (Reference: The Official NBA Basketball Encyclopedia, Villard Books):
x | 67 | 65 | 75 | 86 | 73 | 73 |
y | 44 | 42 | 48 | 51 | 44 | 51 |
(a) Verify that
(b) Use a 5% level of significance to test the claim that
(c) Verify that
(d) Find the predicted percentage y of successful field goals for a player with x = 70% successful free throws.
(e) Find a 90% confidence interval for y when x = 70.
(f) Use a 5% level of significance to test the claim that
(a)
To test: The verification of
Answer to Problem 7P
Solution: It is verified that,
Explanation of Solution
Calculation: Let x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season.
The following table shows the
67 | 44 | 4489 | 1936 | 2948 |
65 | 42 | 4225 | 1764 | 2730 |
75 | 48 | 5625 | 2304 | 3600 |
86 | 51 | 7396 | 2601 | 4386 |
73 | 44 | 5329 | 1936 | 3212 |
73 | 51 | 5329 | 2601 | 3723 |
439 | 280 | 32393 | 13142 | 20599 |
In above table, the last column shows the total of corresponding column.
So,
The correlation coefficient, r is calculated as follows:
Conclusion: Hence, it is verified that,
(b)
To test: The claim that
Answer to Problem 7P
Solution: We have sufficient evidence to conclude that the population correlation coefficient between x and y is positive at 5% level of significance.
Explanation of Solution
Calculation: Using the level of significance,
The null hypothesis for testing is defined as,
The alternative hypothesis is defined as,
The sample test statistic is,
The degrees of freedom are
The abovetest is right tailed test, so we can use the one-tail area in the student’s distributiontable (Table 4 of the Appendix). From table, the range of p-value for the sample test statistic
Conclusion: We have sufficient evidence to conclude that the population correlation coefficient between x and y is positive at 5% level of significance.
(c)
To test: The verification of
Answer to Problem 7P
Solution: It is verified that,
Explanation of Solution
Calculation: The results obtained in part (a) are,
Now,
Conclusion: Hence, it is verified that,
(d)
To find: The predicted percentage
Answer to Problem 7P
Solution: The predicted percentage
Explanation of Solution
Calculation: The results obtained in above part are
The regression line equation is
Now to find the predicted
Interpretation: The predicted percentage
(e)
To find: The90% confidence interval for
Answer to Problem 7P
Solution: The90% confidence interval for
Explanation of Solution
Calculation: The find 90% confidence interval for
Step 1: Go to Stat >Regression>Regression > Predict.
Step 2: Select ‘y’ in Response and write 70 in ‘x’ box.
Step 3: Click on Options write 90 in ‘Confidence level’ and select ‘Two-sided’ in Type of interval. Then click on OK.
The 90% confidence interval is obtained as:
Interpretation: The 90% confidence interval for
(f)
To test: The claim that
Answer to Problem 7P
Solution: We have sufficient evidence to conclude that the slopeis positive at 5% level of significance.
Explanation of Solution
Calculation: Using the level of significance,
The null hypothesis for testing is defined as,
The alternative hypothesis is defined as,
The find t statistic and P-valueusing MINITAB software is as:
Step 1: Enter x and y in Minitab worksheet.
Step 2: Go to Stat > Regression > Regression >Fit Regression Model.
Step 2: Select ‘y’ in Response and ‘x’ in ‘Continuous predictors’ box. Then click on OK.
The sample test statistic is
The software gives the P-value for two-tailed test, for finding the p-value for one tailed testwe can divide the obtained P-value by 2.
Since P-value is less than 0.05, hence we can reject the null hypothesis at
Conclusion: We have sufficient evidence to conclude that the slopeis positive at 5% level of significance.
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Chapter 11 Solutions
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- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill