Multiple Choice
In statistics, which value represents the strongest possible linear correlation coefficient?
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11. Correlation
Correlation Coefficient
2:12 minutesProblem 10.1.5
Textbook Question
Interpreting r
In Exercises 5–8, use a significance level of α = 0.05 and refer to the accompanying displays.
Bear Weight and Chest Size Fifty-four wild bears were anesthetized, and then their weights and chest sizes were measured and listed in Data Set 18 “Bear Measurements” in Appendix B; results are shown in the accompanying Statdisk display. Is there sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes? When measuring an anesthetized bear, is it easier to measure chest size than weight? If so, does it appear that a measured chest size can be used to predict the weight?
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with determining if there is sufficient evidence to support the claim of a linear correlation between bear weights and chest sizes. Additionally, we need to assess if chest size can be used to predict weight.
Step 2: Identify the given data. From the image, the correlation coefficient (r) is 0.963141, the critical r value is ±0.2680855, and the p-value (two-tailed) is 0.000. The significance level (α) is 0.05.
Step 3: Compare the correlation coefficient (r) to the critical r value. If the absolute value of r is greater than the critical r, then there is evidence of a significant linear correlation.
Step 4: Evaluate the p-value. If the p-value is less than the significance level (α = 0.05), we reject the null hypothesis and conclude that there is a significant linear correlation.
Step 5: Interpret the results. If there is a significant linear correlation, consider whether chest size is easier to measure than weight and if it can be used as a predictor for weight. This involves practical reasoning based on the context of the problem.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Correlation Coefficient (r)
The correlation coefficient, denoted as 'r', quantifies the strength and direction of a linear relationship between two variables. Values range from -1 to 1, where 1 indicates a perfect positive correlation, -1 a perfect negative correlation, and 0 no correlation. In this case, an r value of 0.963 suggests a very strong positive correlation between bear weights and chest sizes.
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05:43
Correlation Coefficient
P-value
The p-value measures the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. A p-value of 0.000 indicates strong evidence against the null hypothesis, suggesting that the observed correlation is statistically significant. In this context, it implies that there is a significant linear correlation between the weights and chest sizes of bears.
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06:50Step 3: Get P-Value
Significance Level (α)
The significance level, often denoted as α, is the threshold for determining whether a p-value indicates a statistically significant result. In this scenario, α is set at 0.05, meaning that if the p-value is less than 0.05, the null hypothesis can be rejected. Given the p-value of 0.000, the results are significant, supporting the claim of a linear correlation between the two measured variables.
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04:46Step 4: State Conclusion Example 4
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