Exercises 1–10 are based on the following sample data consisti...

Question

Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

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Change in Scale Exercise 1 stated that for the given paired data, r = 0.846. How does that value change if all of the amounts of dinners are left unchanged but all of the tips are expressed in cents instead of dollars?

Accepted Answer

Hello everyone. Let's take a look at this question together. A study collected data on taxi fares in dollars and the corresponding tips in dollars left by passengers. The correlation coefficient for the given paired data is or is equal to 0.912. How does this value change if all the taxi fares remain unchanged, but all the tips are converted from dollars to cents? Is it answer choice A, it becomes zero. Answer choice B, it increases to 91.2. Answer choice C, it decreases to 0.00912, or answer choice D, it remains the same at 0.912. So in order to solve this question, we have to recall what we have learned about the correlation coefficient to determine how the value of R equals 0.912 changes if all of the taxi fares remain unchanged. But all of the tips are converted from dollars to cents. Well, we can recall that we know the linear correlation coefficient R measures the strength and direction of the linear relationship between two variables, which this linear correlation coefficient R is unitless, which means that it is not affected by changes in. such as multiplication by a positive constant. And in the scenario that we are given, we know we are converting dollars to cents, which the conversion of dollars to cents involves each value being multiplied by 100, which the multiplication by 100 is a multiplication by a positive constant, and since the linear correlation efficient R is not affected by changes in scale, the correlation remains the same, so the value of R remains at 0.912. And looking at our answer choices, we can identify that answer choice D is the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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Key Concepts

Correlation Coefficient (r)

Scale Invariance of Correlation

Linear Transformation

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