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12.4: the regression equation

13 . r ( s y s x ) = 0.73 ( 9.6 4.0 ) = 1.752 1.75

14 . a = y ¯ b x ¯ = 141.6 1.752 ( 68.4 ) = 21.7632 21.76

15 . y ^ = 21.76 + 1.75 ( 68 ) = 140.76

12.5: correlation coefficient and coefficient of determination

16 . The coefficient of determination is the square of the correlation, or r 2 .
For this data, r 2 = (–0.56)2 = 0.3136 ≈ 0.31 or 31%. This means that 31 percent of the variation in fuel efficiency can be explained by the bodyweight of the automobile.

17 . The coefficient of determination = 0.32 2 = 0.1024. This is the amount of variation in freshman college GPA that can be explained by high school GPA. The amount that cannot be explained is 1 – 0.1024 = 0.8976 ≈ 0.90. So about 90 percent of variance in freshman college GPA in this data is not explained by high school GPA.

18 . r = r 2
0.5 = 0.707106781 0.71
You need a correlation of 0.71 or higher to have a coefficient of determination of at least 0.5.

12.6: testing the significance of the correlation coefficient

19 . H 0 : ρ = 0
H a : ρ ≠ 0

20 . t = r n 2 1 r 2 = 0.33 30 2 1 0.33 2 = 1.85
The critical value for α = 0.05 for a two-tailed test using the t 29 distribution is 2.045. Your value is less than this, so you fail to reject the null hypothesis and conclude that the study produced no evidence that the variables are significantly correlated.
Using the calculator function tcdf, the p -value is 2tcdf(1.85, 10^99, 29) = 0.0373. Do not reject the null hypothesis and conclude that the study produced no evidence that the variables are significantly correlated.

21 . t = r n 2 1 r 2 = 0.45 25 2 1 0.45 2 = 2.417
The critical value for α = 0.05 for a two-tailed test using the t 24 distribution is 2.064. Your value is greater than this, so you reject the null hypothesis and conclude that the study produced evidence that the variables are significantly correlated.
Using the calculator function tcdf, the p-value is 2tcdf(2.417, 10^99, 24) = 0.0118. Reject the null hypothesis and conclude that the study produced evidence that the variables are significantly correlated.

12.7: prediction

22 . y ^ = 25 + 16 ( 5 ) = 105

23 . Because the intercept appears in both predicted values, you can ignore it in calculating a predicted difference score. The difference in grams of fiber per serving is 6 – 3 = 3 and the predicted difference in grams of potassium per serving is (16)(3) = 48.

12.8: outliers

24 . An outlier is an observed value that is far from the least squares regression line. A rule of thumb is that a point more than two standard deviations of the residuals from its predicted value on the least squares regression line is an outlier.

25 . An influential point is an observed value in a data set that is far from other points in the data set, in a horizontal direction. Unlike an outlier, an influential point is determined by its relationship with other values in the data set, not by its relationship to the regression line.

26 . The predicted value for y is: y ^ = 5 + 0.3 x = 5.6 . The value of 6.2 is less than two standard deviations from the predicted value, so it does not qualify as an outlier.
Residual for (2, 6.2): 6.2 – 5.6 = 0.6 (0.6<2(0.4))

Questions & Answers

limitations of simple random sampling
pratyaksh Reply
how can l calculate G. M from the following size 125 133 141 173 182 frequency 7 5 4 1 3
Stancy Reply
how they find mean population
Joy Reply
parts of statistics
Edwin Reply
what is a mean?
Onele Reply
given the sequence 128,64,32 find the 12th term of the sequence
Shehu Reply
12th number is 0.0625
Thangarajan
why do we use summation notation to represent set of observations
MICHAEL Reply
what is the potential outlier ?
Anik Reply
A pharmaceutical company claims that their pain reliever capsule is 70% effective. But a clinical test on this capsule showed 65 out of 100 effectiveness
jelly Reply
Part of statistics
charls Reply
how to find mean population
Dawit Reply
what is data value
Ravneet Reply
what is relative frequency
Adeyemi Reply
liner regression analysis
Swathy Reply
Proper definition of outlier?
Sumbal Reply

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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