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The Correlation Coefficient at a Particular Significance Level

1. A math teacher believes there is a correlation between a student's Exam I score (x) and the Final Exam score (y). Below is the test data (although Jean's scores were blotted out by accident) and the summations from all the students' scores in the class. Determine the appropriate correlation coefficient, r, and whether there is a correlation at a particular significance level.

# Student x y xy x2 y2

1 Eric 92 89 8188 8464 7921
2 Tina 92 89 8188 8464 7921
3 Robert 89 89 7921 7921 7921
4 Michael 89 82 7298 7921 6724
5 Mary Beth84 94 7896 7056 8836
6 Christy 85 94 7990 7225 8836
7 Lauren 94 89 8366 8836 7921
8 Giovanni 77 82 6314 5929 6724
9 Jean 81 77 6237 6561 5929
10 Ryan 68 61 4148 4624 3721

Sum 851 846 72,546 73,001 72,454

2. A researcher developed a model that predicts that the eye color of the first-born child will be brown 84% of the time, blue 10% of the time, and green 6% of the time. From a sample of 150 first-born children, 116 had brown eyes, 24 had blue eyes, and 10 had green eyes. Use the  2 distribution to determine the test statistic and test the claim that the observed frequencies correspond to the predicted frequencies at the 0.05 significance level. The data is tabulated below.

Eye color Observed frequency
O Expected probability Expected frequency
E

Brown 116 0.840 126.0
Blue 24 0.100 15.0
Green 10 0.060 9.0

Sum 150.0 1.000 150.0

Solution Preview

1. The correlation coefficient is calculated as

r = [NΣxy - Σx Σy] / √[nΣx^2 - (Σx)^2] [nΣy^2 - (Σy)^2]

r = [10*72456 - 851*846] / [730010 - (851)^2] [ 724540 - (846)^2]

r = 0.770164

Testing correlation coefficient:

The test statistic is

t =[ ...

Solution Summary

Tutorial are provided for two different problems on correlation coefficients and frequency.

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