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ECON 227D2
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Economics (Arts)

ECON 227D2

Kenneth Mac Kenzie

Winter

Description

1
1. Data have been collected on nutrition for a suitably random sample of twelve
popular dry cereals. For a standard serving of each cereal measurements were
made of the numbers of units of protein, carbohydrates, fat, vitaminA, and
calories.
Brand Protein Carbo Fat Calories VitA
Life 6 19 1 110 0
Grape Nuts 3 23 0 100 25
Super Sugar Crisp 2 26 0 110 25
Special K 6 21 0 110 25
Rice Krispies 2 25 0 110 25
Raisin Bran 3 28 1 120 25
Product 19 2 24 0 110 100
Wheaties 3 23 1 110 25
Total 3 23 1 110 100
Puffed Rice 1 13 0 50 0
Sugar Corn Pops 1 26 0 110 25
Sugar Smacks 2 25 0 110 25
a) Predict the number of calories of another cereal with 5 units of protein, 23
carbohydrate units, 1 fat unit, and 25 units of vitamin A. Use MODEL I.
111.2875
b) What proportion of the variation in calories is explained by the regression
relationship in i) MODEL I ? ii) MODEL II ?
MODEL I: 0.777403 MODEL II: 0.956654
2
c) Is the increase in R significant from MODEL I to MODEL II ? Justify
your answer numerically from the printouts.
The ‘protein’ p-value is listed as 0.001. Since this is less than 0.05, the
increase is significant.
d) Give a 90% confidence interval for the marginal contribution of a unit of
carbohydrates based on the results in MODEL I.
t DF = error DF = 8 t = 1.860
answer: 3.6630 ± 1.860 × 0.7889 = 3.663 ±1.467
e) Find the missing numbers in the analysis-of-variance table for MODEL II. 2
Analysis of Variance
Source DF SS MS F
Regression 4 3348.289 837.07225 38.62
Residual Error 7 151.711 21.673
Total 11 3500.000
f) Find Pearson’s correlation coefficient for the Carbo and Fat data, and
determine whether these two variables are significantly correlated.
r = 0.04669296
H 0 ρ = 0
H a: ρ ≠ 0
critical t with DF = 10 2.228
test statistic =0.0466296 = 0.1478
2
1 − 0.0466296
10
Do not reject the null hypothesis. Carbo and Fat data are not significantly
correlated.
MODEL I
The regression equation is
Calories = 16.71 + 3.663 Carbo + 9.726 Fat + 0.0241 VitA
Predictor Coef StDev T P
Constant 16.71 17.74 0.94 0.374
Carbo 3.6630 0.7889 4.64 0.002
Fat 9.726 6.071 1.60 0.148
VitA 0.02410 0.09612 0.25 0.808
S = 9.868 R-Sq = 77.7403%
Analysis of Variance
Source DF SS MS F P
Regression 3 2720.91 906.97 9.31 0.005
Residual Error 8 779.09 97.39
Total 11 3500.00
MODEL II 3
The regression equation is
Calories= -4.082+ 3.9805Carbo+ 5.215Protein+ 2.208Fat+0.06052VitA
Predictor Coef StDev T P
Constant -4.082 9.217 -0.44 0.671
Carbo 3.9805 0.3768 10.56 0.000
Protein 5.2150 0.9693 5.38 0.001
Fat 2.208 3.187 0.69 0.511
VitA 0.06052 0.04585 1.32 0.228
S = 4.655 R-Sq = 95.6654%
Analysis of Variance
Source DF SS MS F P
Regression * ******* ****** 38.62 0.000
Residual Error * ****** *****
Total ** *******
ECON 227
2. Regress Calories on VitA either using statistics functions on the
calculator or using formulas and the following partial
computations:
∑ x = 400 ∑ y = 1260 ∑ x = 25000 ∑ xy = 44000 ∑ y = 135800 .
a) Find the OLS line equation
y = 99 .2857 + 0.1714 x
b) Test whether the number of units of vitamin A is
significant for predicting the number of calories in a
standard serving.
Method I t test
H : ρ = 0
0
H a: ρ ≠ 0
critical t with DF = 102.228
0.312984319
test statistic = 2 = 1.042
1 − 0.312984319
10
Method II: ANOVA with the same null and alternative hypotheses.
Analysis of Variance 4
Source DF SS MS F
Regression 1 342.857 342.857 1.086
Residual Error 10 3157.143 315.714
Total 11 3500.000
Critical 1,10 4.965
Do not reject the null hypothesis. The number of units of
vitamin A is not significant for predicting the number of
calories in a standard serving.
c) Regardless of your conclusion in b) form a 95% prediction
interval for the number of calories in standard servings
with 25 units of vitamin A. Use the simple-regression
model of this question.
Note the change in the question.
1 (25 − 33.3)
103.57 ± 2.228 315.714 1+ + 2 = 103.57 ± 41.32
12 400
25000 −
12
d) What is the standard error of the estimate in this model?
MSE ≈ 17 .77
e) Show that the point (x , ) satisfies the OLS equation.
3. a) Test whether the variances of Carbo and VitA are
significantly different.
2
H : σ1 = 1
0 σ 2
2
σ 2
H a 1 ≠ 1
σ 2
critical F11,11,0.025omewhere between 3.4 and 3.8
2
32.56694736
test statistic = 2 = 67.82945736
3.954284213
Reject the null hypothesis. The variances of Carbo and VitA
are significantly different.
b) Test whether the mean of VitA is significantly greater than
the mean of Carbo. 5
H : μ ≤ μ
0 V C
H a μ V μ C
2 2
s2 s 2 32.5669 2 3.9543 2
1 + 2 +
n1 n 2 12 12
df = 2 2 = 2 2
s2 s 2 32.5669 2 3.9543 2
1 2
n1 n 2 12 12
+ +
n1−1 n 21 11 11
=11.3 Truncate to 11
critical t : 1.796
33.3 − 23
test statistic = =1.09
32.5669 2 3.9543 2
+
12 12
Do not reject the null hypothesis. The mean of VitA is not
significantly greater than the mean of Carbo.
c) What are the technical requirements for performing the
tests you used in parts a) and b) ?
Bo

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