Question 6.7 (a) Estimate whether the following pairs of scores for X and Y reflect a
positive relationship, a negative relationship, or no relationship.
|
X |
Y |
|
64 |
66 |
|
40 |
79 |
|
30 |
98 |
|
71 |
65 |
|
55 |
76 |
|
31 |
83 |
|
61 |
68 |
|
42 |
80 |
|
57 |
72 |
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The score pairs have a slightly positive relationship considering the value of Y is almost always higher than that of X.
b. Construct a scatterplot for X and Y. Verify that the scatterplot does not describe a pronounced curvilinear trend.
c. Calculate r using the computation formula.
r = SPxy/(SSx*SSy) (^ 1/2)
SPxy = X*Y-( X* Y)/n
=33, 304 - (451*687)/9
= - 1, 122.33
SSx = X^2 - ( X) ^ 2/n
= 24, 357-(451) ^ 2/9
= 1, 756.89
SSy = Y^2 - ( Y) ^ 2/n
= 53, 299-(687) ^ 2/9
= 858
Therefore, r = -1,122.33/ (1,756.89*858) ^ (1/2) = - 0.914
6.10. On the basis of an extensive survey, the California Department of Education reported an r of -0.32 for the relationship between the amount of time spent watching TV and the achievement test scores of school children. Each of the following statements represents a possible interpretation of this finding. Indicate whether each is true or false.
a. Every child who watches a lot of TV will perform poorly on the achievement tests. False
b. Extensive TV viewing causes a decline in test scores. False
c. Children who watch little TV will tend to perform well on the tests. True
d. Children who perform well on the tests tend to watch little TV. True
e. If Gretchen’s TV viewing is reduced by one half, we can expect a substantial improvement in her test scores. False
f. TV viewing could not possibly cause a decline in test scores. False
6.11 Assume that an r of 0.80 describes the relationship between daily food intake, measured in ounces, and body weight, measured in pounds, for a group of adults. Would a shift in the units of measurement from ounces to grams and from pounds to kilograms change the value of r? Conversion of weight to kg instead of lb would not change the value of r since it is has no units and is a mere measure of the extent of linear association between variables.
7.8 Each of the following pairs represents the number of licensed drivers (X) and the number of cars (Y) for seven houses in my neighborhood.
|
X |
Y |
|
5 |
4 |
|
5 |
3 |
|
2 |
2 |
|
2 |
2 |
|
3 |
2 |
|
1 |
1 |
|
2 |
2 |
Construct a scatterplot to verify the lack of pronounced curvilinearity.
Determine the least squares equation for this data. (Remember you will first have to calculate r, SSy, and SSx).
Y’= b*X + a
b = r* (SSy/ SSx) ^ 0.5 = 0.558
a = Ybar -b* Xbar = 0.692
SSy = Y^2 - (( Y) ^ 2)/n = 5.429
SSx = X ^ 2 - (( X) ^ 2)/n = 14.857
r = SPxy/ ((SSx*SSy) ^ 0.5) = 0.923
SPxy = X*Y - ( X* Y)/n = 8.286
Y’ = 0.558*X + 0.692
Determine the standard error of estimate, Sy|x, given that n=7
Sy|x = ((SSy (1-r^2))/ (n-2)) ^ 0.5 = 0.402
Predict the number of cars for each of two new families with two and five drivers.
Y’ = b*X + a
For x = 2; Y’= 0.558*2 + 0.692 = 1.81 ≈ 2
For x = 5; Y’= 0.558*2 + 0.692 = 3.48 ≈ 4
7.10 Assume that r^2 equals 0.5 for the relationship between height and weight for adults. Indicate whether the following statements are true or false.
a. Fifty percent of the variability in heights can be explained by the variability in weights. False
b. There is a cause-effect relationship between height and weight. False
c. The heights of 50% of adults can be predicted exactly from their weight. True
d. Fifty percent of the variability in weights is predictable from heights. True
7.13 In the original study of regression toward the mean, Sir Francis Galton noted a tendency for offspring of both tall and short parents to drift toward the mean height for offspring and referred to this tendency as “regression toward mediocrity.” What is wrong with the conclusion that eventually all heights will be close to their mean?
The problem with such a conclusion is its failure to consider the element of lack which influences regression towards the mean. The conclusion is thus a regression fallacy.
11.19 How should a projected hypothesis test be modified if you’re particularly concerned about
(a) the type I error? The error can be reduced by decreasing significance level or increasing sample size.
(b) the type II error? The error can be reduced by increasing significance level or increasing sample size
12.7 In Question 10.5 on page 231, it was concluded that the mean salary among the population of female members of the American Psychological Association is less than that ($82,500) for all comparable members who have a doctorate and teach full time.
(a) Given a population standard deviation of $6,000 and a sample mean salary of $80,100 for a random sample of 100 female members, construct a 99 percent confidence interval for the mean salary for all female members.
6000/ (√100) = 600
± ( z conf ) ( ) = 80, 100 ± (2.58) (600) = 78, 552 or 81, 648
(b) Given this confidence interval, is there any consistent evidence that the mean salary for all female members falls below $82,500, the mean salary for all members?
Yes. This is because with the initial confidence level, it can be inferred that given a ninety-nine percent confidence, the population mean would be between the values calculated above which are both below the mean for males.
12.8 In Review Question 11.12 on page 263, instead of testing a hypothesis, you might prefer to construct a confidence interval for the mean weight of all 2-pound boxes of candy during a recent production shift.
(a) Given a population standard deviation of .30 ounce and a sample mean weight of 33.09 ounces for a random sample of 36 candy boxes, construct a 95 percent confidence interval.
33.09 ± (1.96) (.05) = 33.09 ± 0.098 = 33.19 or 32.99
(b) Interpret this interval, given the manufacturer’s desire to produce boxes of candy that on the average exceed 32 ounces.
This confidence level is indicative of the fact that the manufacturer can probably produce boxes exceeding the required 32 ounces.
12.10 Imagine that one of the following 95 percent confidence intervals estimates the effect of vitamin C on IQ scores:
(a) Which one most strongly supports the conclusion that vitamin C increases IQ scores? 3
(b) Which one implies the largest sample size? 1
(c) Which one most strongly supports the conclusion that vitamin C decreases IQ scores? 5
(d) Which one would most likely stimulate the investigator to conduct an additional experiment? 4
