Question

question 31 of 39
the table presents four sets of data prepared by the statistician frank anscombe to illustrate the dangers of calculating without first plotting the data.
four data sets for exploring correlation and regression
data set a
x ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
y 8.04 6.95 8.81 8.33 9.96 7.24 4.26 10.84 4.82
data set b
x ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
y 9.14 8.14 8.77 9.26 8.10 6.13 3.10 7.26
data set c
x ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
y 7.46 6.77 12.74 7.11 7.81 8.84 6.08 5.39 6.42
data set d
x ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
y 6.58 5.76 8.84 8.47 7.04 5.25 5.56 6.89 12.50
click to download the data in your preferred format.
side-by-side
without making scatterplots, find the correlation for all four data sets. give your answers to three decimal places.
r_set a =
r_set b =
r_set c =
r_set d =
without making scatterplots, find the least-squares regression line for all four data sets. what do you notice based on the obtained values?
○ the least-squares regression lines for data sets a and c are the same, but different from data sets b and d.
○ the least-squares regression lines are approximately the same for all four data sets.
○ the least-squares regression line for each data set is markedly different.
○ the least-squares regression lines for data sets b and c are the same but different

Explanation:

Step1: Recall Correlation Formula

The correlation coefficient \( r \) is calculated using the formula:

$$ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

where \( n \) is the number of data points. For Anscombe's quartet, each data set has \( n = 11 \) (standard Anscombe's data: 11 points per set).

Step2: Analyze Data Sets (Anscombe's Quartet)

Anscombe’s quartet has four data sets with identical summary statistics (mean \( \bar{x} \), mean \( \bar{y} \), variance of \( x \), variance of \( y \), and correlation \( r \)) but different distributions. For each set:

• Data Set A: \( x \)-values: \( 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \); \( y \)-values: \( 8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68 \)
• Data Set B: \( x \)-values same as A; \( y \)-values follow a curve.
• Data Set C: \( x \)-values same as A; one outlier in \( y \).
• Data Set D: \( x \)-values have one outlier; \( y \)-values clustered.

Despite distribution differences, the summary stats (including \( r \)) are identical.

Step3: Calculate \( r \) for Each Set

Using the formula (or known result for Anscombe’s quartet), the correlation \( r \) for each data set is approximately \( 0.816 \) (to three decimal places).

Answer:

\( r_{\text{Set A}} = 0.816 \), \( r_{\text{Set B}} = 0.816 \), \( r_{\text{Set C}} = 0.816 \), \( r_{\text{Set D}} = 0.816 \)