Stat 322 – HW 5 Solutions

 

1) problem 1 (p. 453)

(a) F = MSA/SSA = (30.6/4)/(59.2/4 × 3) =  1.55

Compare this to F(4, 12).

            From Table: 1.55 < 2.48 so p-value > .100  (F > 3.26 so p-value > .05)

            From Minitab: 1-.7499 = .2501

F distribution with 4 DF in numerator and 12 DF in denominator

   x  P( X <= x )

1.55     0.749909

With such a large p-value, we fail to reject the null hypothesis and conclude that the true average tire lifetimes do not differ across the makes of cars.

 

(b) F = MSB/SSE = (44.1/3)/(59.2/12) = 2.98

Compare this to F(3,12, .05) = 3.49

Since our F-statistic is less than 3.49, this test is not significant at the.05 level.  We conclude that there is no difference in true average tire lifetime due to different brands of tires.

 

2) problem 18 (p. 464), feel free to use Minitab

Note: You can get Minitab to store the fitted values and residuals automatically!

Using Stat > ANOVA > Two-way:

Two-way ANOVA: Yield versus Speed, Formulat

 

Source       DF       SS       MS       F      P

Speed         2   230.81   115.41   19.27  0.000

Formulat      1  2253.44  2253.44  376.27  0.000

Interaction   2    18.58     9.29    1.55  0.252

Error        12    71.87     5.99

Total        17  2574.70

 

S = 2.447   R-Sq = 97.21%   R-Sq(adj) = 96.05%

 

(a) F = (18.58/(3-1)×(2-1))/(71.87/12) = 9.29/5.99 = 1.55

            We could compare to F(2,12,.05) = 3.89

            Since our F-statistic < 3.89, this is not significant at the 5% level.

            Minitab gives us a large p-value.

Fail to reject H0: no interaction between speed and formulation.

 

(b) It appears that yield depends on both formulation (F = 376.27, p-value < .001) and speed (F = 19.27, p-value < .001).

 

(c) The main effects are just the group means (the row and column totals in bold).  The fitted values are just the cell means (the average of the 3 values inside the cell).

 

 

 

speed

 

 

 

 

60

70

80

 

formulation

1

189.47

180.6

191.03

187.03

 

2

166.20

161.03

166.73

164.66

 

 

177.83

170.82

178.88

175.85

 

To find the estimated interaction effects, take the cell average, subtract each main effect for that cell and add the overall mean:

                        g11-hat = 189.47 – 187.03-177.83 + 175.85 = .46

 

The first residual is 189.7 – 189.47 = .233 etc.

 

3) A 1989 study (Capron & Duyme found in Ramsey and Schafer) investigated the effect of heredity and environment on intelligence.  From adoption registers in France, researchers elected samples of adopted children whose biological parameters and adoptive parameters came from either the very highest or the very lowest socioeconomic status (SES) categories (based on years of education and occupation).  They attempted to obtain samples of size 10 from each combination: (1) high adoptive SES and high biological SES, (2) high adoptive SES and low biological SES, (3) low adoptive SES and high biological SES, and (4) low SES for both parents.  They 38 selected children were given intelligence quotient (IQ) tests.

SES of Adoptive

SES of biological

IQ scores of adopted children

High

High

136

99

121

33

125

131

103

115

116

117

High

Low

94

103

99

125

111

93

101

94

125

91

Low

High

98

99

91

124

100

116

113

119

 

 

Low

Low

92

91

98

83

99

68

76

115

86

116

(a) Identify the observational units, explanatory variable(s), and response variable in this study.  Identify each variable as quantitative or qualitative.  Is this an observational study or an experiment? 

observational units = adopted children

explanatory variable 1 = SES of adoptive parents (categorized)

explanatory variable 2 = SES of biological parents (categorized)

response variable = IQ score (Quantitative)

This is an observational study since the explanatory variables were observed, not manipulated (imposed) by the researchers.

 

(b) Enter these data into a blank Minitab worksheet, being careful to treat each column as a different variable.  How many rows should you end up with?

There should be 38 rows, one for each child.

 

(c) Run a one-way ANOVA to see if the mean IQ scores differ significantly depending on the SES of the adoptive parents.  Report the F statistic, p-value, and your conclusion.

Analysis of Variance for IQ

 

Source        DF       SS     MS     F      P

SES adoptive   1    531.3  531.3  1.42  0.241

Error         36  13454.6  373.7

Total         37  13985.9

 

F = 1.42, p-value = .241.  From this large p-value (> .05), we would fail to reject the null hypothesis that the average IQ of the children is the same for the High SES and Low SES groups of the adoptive parents.

 

(d) Run a two-sample t-test to decide whether there is a difference in  the mean IQ scores for the high and low SES adoptive parents.  How does your p-value compare to that in (d)?  What technical condition is required by the procedure in (c) but not in (d).

Two-sample T for IQ

 

SES

adoptive   N   Mean  StDev  SE Mean

high      20  106.6   22.1      4.9

low       18   99.1   15.6      3.7

 

 

Difference = mu (high) - mu (low)

Estimate for difference:  7.48889

95% CI for difference:  (-5.04656, 20.02434)

T-Test of difference = 0 (vs not =): T-Value = 1.21  P-Value = 0.233  DF = 34

 

The p-value is similar.  They would be identical if we had used a pooled t-test but that (and ANOVA) require equal population variances.

 

(e) What happens if you try to run Stat > ANOVA > Two-Way, using the IQ scores as the response, the SES of the adoptive parents as one factor and the SES of the biological parents as the other?

Since we are missing two of the low, high children, we do not have a balanced design so this command will not work.

 

(f) Instead, choose Stat > ANOVA > General Linear Model, entering the IQ scores as the response, and using the two SES variables in the Model box.  Click OK.

 

Analysis of Variance for IQ, using Adjusted SS for Tests

 

Source          DF   Seq SS   Adj SS  Adj MS     F      P

SES adoptive     1    531.3    452.0   452.0  1.27  0.267

SES biological   1    998.5    998.5   998.5  2.81  0.103

Error           35  12456.0  12456.0   355.9

Total           37  13985.9

 

- What is the p-value for the SES adoptive factor?  What hypotheses are being tested?

p-value = .267, we fail to reject that the mean IQ is the same for the high SES adoptive and low SES adoptive populations (H0: mAdopt low = madopt high vs. Ha: madopt low ≠ madopt ­high or equivalently H0: a1 = a2 = 0 vs. Ha: at least one a ≠ 0)

 

- What is the p-value for the SES biological factor? What hypotheses are being tested?

p-value = .103, we fail to reject that the mean IQ is the same for the high SES adoptive and low SES biological populations. (H0: mbio low = mbio high vs. Ha: mbio low ≠ mbio ­high or equivalently H0: a1 = a2 = 0 vs. Ha: at least one a ≠ 0)

 

(g)

- What does the interactions plot reveal about the overall effect of the adoptive parents’ SES?

The children’s IQs tend to be higher when the adoptive parents’ SES is higher.

 

- What does the interactions plot reveal about the overall effect of the biological parents’ SES?

The children’s IQs tend to be higher when the biological parents’ SES is higher.

 

- Does the difference in mean scores for those with high and low SES biological parents depend on whether the adoptive parents were high or low SES?

Yes, the difference (in mean IQ for high and low SES biological parents) is larger for the children of low SES adoptive parents than for children of high SES adoptive parents.

 

Analysis of Variance for IQ, using Adjusted SS for Tests

 

Source                       DF   Seq SS   Adj SS  Adj MS     F      P

SES adoptive                  1    531.3    416.2   416.2  1.15  0.290

SES biological                1    998.5   1047.6  1047.6  2.90  0.097

SES adoptive*SES biological   1    194.8    194.8   194.8  0.54  0.467

Error                        34  12261.2  12261.2   360.6

Total                        37  13985.9

 

(h) Return the Session window and report the p-value for the Interaction effect.  Does the size of this value make sense based on the graph?

The p-value is rather large, indicating that the interaction effects are not statistically significant.  This makes sense since the lines are not that far from parallel.

 

(i) If there is no interaction (your answer to the last question in (f) is no), then how much is the mean IQ score affected by the SES of adoptive parents, and how much is it affected by the SES of the biological parents? Is one of these effects larger than the other? Hint: Use multiple comparison procedures to help answer this question.

 

Tukey 95.0% Simultaneous Confidence Intervals

Response Variable IQ

All Pairwise Comparisons among Levels of SES adoptive

SES adoptive = high  subtracted from:

 

SES

adoptive   Lower  Center  Upper  -------+---------+---------+---------

low       -19.23  -6.650  5.930  (-----------------*----------------)

                                 -------+---------+---------+---------

                                    -14.0      -7.0       0.0

Tukey Simultaneous Tests

Response Variable IQ

All Pairwise Comparisons among Levels of SES adoptive

SES adoptive = high  subtracted from:

 

SES       Difference       SE of           Adjusted

adoptive    of Means  Difference  T-Value   P-Value

low           -6.650       6.190   -1.074    0.2903

 

 

Tukey 95.0% Simultaneous Confidence Intervals

Response Variable IQ

All Pairwise Comparisons among Levels of SES biological

SES biological = high  subtracted from:

 

SES biological   Lower  Center  Upper   ---+---------+---------+---------+---

low             -23.13  -10.55  2.030   (-----------------*-----------------)

                                        ---+---------+---------+---------+---

                                       -21.0     -14.0      -7.0       0.0

Tukey Simultaneous Tests

Response Variable IQ

All Pairwise Comparisons among Levels of SES biological

SES biological = high  subtracted from:

 

                Difference       SE of           Adjusted

SES biological    of Means  Difference  T-Value   P-Value

low                 -10.55       6.190   -1.704    0.0974

 

The mean IQ score is up to 19.23 points higher in the high SES adoptive group compared to the low SES adoptive group though zero is included in the confidence interval.

The mean IQ score is up to 23.13 points higher in the high SES biological group compared to the low SES biological group, though again zero is included in the interval.  However, comparing the two, this effect appears to be larger.