Statistics for The Behavioral Sciences (MindTap Course List)
Statistics for The Behavioral Sciences (MindTap Course List)
10th Edition
ISBN: 9781305504912
Author: Frederick J Gravetter, Larry B. Wallnau
Publisher: Cengage Learning
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Chapter 15, Problem 12P

In the Chapter Preview we discussed a study by Judge and Cable (2010) demonstrating a negative relationship between weight and income for a group of women professionals. The following are data similar to those obtained in the study. To simplify the weight variable, the women are classified into five categories that measure actual weight relative to height, from = thinnest to 5 = heaviest. Income figures are annual income (in thousands), rounded to the nearest $1,000.

a. Calculate the Pearson correlation for these data.

b. Is the correlation statistically significant? Use a two-tailed test with α = .05 .

    Weight (X) Income (Y)
    1 115
    1 78
    4 53
    3 63
    5 37
    2 84
    5 41
    3 51
    1 94
    5 44

Expert Solution & Answer
Check Mark
To determine
  1. The Pearson’s correlation for the given data.
  2. If the correlation is statistically significant using a two tailed test with α=.05 .

Answer to Problem 12P

Solution:

a.

    Weight(X)Income(Y)XYX2Y2
    1115115113225
    1787816084
    453212162809
    36318993969
    537185251369
    28416847056
    541205251681
    35115392601
    1949418836
    544220251936
    X= 30 Y= 660 XY= 1619 X2= 116 Y2= 49566

Therefore, the Pearson’s correlation between the weight(X)and the income(Y)is 0.9135, which strong negative correlation is.

b. Since the test statistic is beyond the left tail of the critical value 2.306 , therefore we reject H0. That is, the sample of n=10 pairs of observation indicates that the correlation is statistically significant.

Explanation of Solution

Given:

The Given data:

    Weight (X) Income (Y)
    1115
    178
    453
    363
    537
    284
    541
    351
    194
    544

Formula Used:

  r=NXY( X )( Y)[N X 2 ( X ) 2 ][N Y 2 ( Y ) 2]

Calculations:

a. Now to compute the Pearson’s correlation between the weight (X) and the income (Y) ,we will use the below formula of correlation:

  r=NXY( X )( Y)[N X 2 ( X ) 2 ][N Y 2 ( Y ) 2]

Where:

  N = Number of pairs of observations.

  X= Sum of X observations

  Y=Sum of Y observations

  XY=Sum of the products of paired observations 

  X2=Sum of squared X observations

  Y2=Sum of squared Y observations

    Weight(X)Income(Y)XYX2Y2
    1115115113225
    1787816084
    453212162809
    36318993969
    537185251369
    28416847056
    541205251681
    35115392601
    1949418836
    544220251936
    X= 30 Y= 660 XY= 1619 X2= 116 Y2= 49566

  r=10×161930×660 [ 10×116 30 2 ][ 10×49566 660 2 ]=1619019800 (1160900)(495660435600)=3610 260×60060=36103951.658=0.9135

Therefore, the Pearson’s correlation between the weight(X)and the income(Y)is0.9135, which is strong negative correlation.

b. Now to find if the correlation is statistically significant, we will follow the 4-step procedure of hypothesis testing as:

Step 1: The first step is to set up the hypothesis and determine the significance level

The null and alternative hypotheses are given below:

  H0:ρ=0 i.e., the observed sample correlation coefficient is not significant of any correlation in the population.

  Ha:ρ0 i.e., the observed sample correlation coefficient is significant of correlation in the population.

Also the significance level is α=.05 .

Step 2: The second step is to set the criteria for a decision.

Here n=10 and the two tailed critical value for n2=102=8 df and .05 significance level is

  t0.05,8=±2.306

The Critical tvalue

  =±2.306 and the criteria for decision is, if the test statistic is less than the critical value then we fail to reject the null hypothesis else we reject the null hypothesis.

Step 3: The third step is to compute the test statistic. Now we have to calculate the test statistic. The test statistic is given below:

  t=r (n2) (1 r 2 )=0.9135 (102) (1 (0.91354) 2 )=0.9135×2.8284 10.8345=2.58380.4068=6.35  (rounded to two decimals)

Step 4: The fourth step is to make a decision.

Since the test statistic is beyond the left tail of the critical value 2.306 , therefore we reject H0. That is, the sample of n=10 pairs of observation indicates that the correlation is statistically significant.

Conclusion:

  1. The correlation between the weight (X) and Income (Y) is 0.9135 .
  2. Using a two-tailed test with α=.05, we clearly see that the correlation is statistically significant.

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