課程名稱︰統計學下
課程性質︰工管系必修
課程教師︰蔣明晃
開課系所︰工管系
考試時間︰95/05/01 14:00 ~ 17:00
試題 :
I.Multiple Choice Question
1. In one-way ANOVA,the amount of total variation that is unexplained is
measured by the:
a. sum of squares for treatments
b. sum of squares for error
c. totat sum of squares
d. degrees of freedom
2. Which of the following is not true of the F-distribution ?
a. Mean and median are equal
b. It is skewed to the right
c. Its values are always positive
d. It is used in ANOVA test
3. A survey will be conducted to compare the grade point averages of high
school students from four different school districts. Students are to be
randomly selected from each of the four districts and their grade point
averages recorded. The ANOVA model most likely to fit this situation is:
a. one-way ANOVA
b. two-way ANOVA
c. randomized block design
d. complete 4*4 factorial design
4. When the effect of a level for one factor depends on which level of another
factor is present,the most appropriate ANOVA design to use in the situation
is the:
a. one-way ANOVA
b. two-way ANOVA
c. randomized block design
d. matched pairs design
5. One-way ANOVA is performed on indeoendent samples taken from three normally
distributed populations with equal variances. The following summary
statistics were calculated:
_
n1 = 7 , x1 = 65 , s1 = 4.2
_
n2 = 8 , x2 = 65 , s2 = 4.9
_
n3 = 9 , x3 = 65 , s3 = 4.6
The value of the test ststistics,F,equals:
a. 65
b. 24
c. 137
d. 0
6. One-way ANOVA is applied to independent samples taken from three normally
distributed populations with equal variances. The folowing summary
statistics were calculated:
_
n1 = 8 , x1 = 15 , s1 = 2
_
n2 = 10 , x2 = 18 , s2 = 3
_
n3 = 8 , x3 = 20 , s3 = 2
The within-treatments variation equals:
a. 137
b. 460
c. 154
d. 60
7. If four confidence interval estimates for the population means were
simultaneously constructed with 95% confidence for four independent
treatments,the probability that all four intervals would contain the
population means would be:
a. 0.857
b. 0.815
c. 0.903
d. 0.950
8. Consider the following partial ANOVA table:
Source of Variation SS df MS F
Treatments 75 * 25 6.67
Error 60 * 3.75
Total 135 19
The numerator and denominator degree of freedom (identified by asterisks)
are:
a. 4 and 15
b. 3 and 16
c. 15 and 4
d. 16 and 3
9. One-way ANOVA is applied to independent samples taken from three normally
distributed populations with equal variances. The folowing summary
statistics were calculated:
_
n1 = 10 , x1 = 40 , s1 = 5
_
n2 = 10 , x2 = 48 , s2 = 6
_
n3 = 10 , x3 = 50 , s3 = 4
The between-treatments variation equals:
a. 460
b. 688
c. 560
d. 183
10. Given that
variance of x = 500 , variance of y = 750 , cov(x,y) = 100 , n = 6
The standard error of estimate is:
a. 12.247
b. 24.933
c. 30.2076
d. 11.180
11. If the coefficinet of correlation is 0.90,the percentage of the variation
in the dependent variable y that is explained by the variation in the
independents variable x is:
a. 90%
b. 81%
c. 0.90%
d. 0.81%
12. If the coefficinet of correlation between x and y is close to 1.0,this
indicates that
a. y causes x to happen
b. x causes y to happen
c. Both a and b are correct answers
d. there may or may not be any causal relationship between x and y
13. In regression analysis,if the coefficinet of correlation is -1.0,then:
a. the sum of squares for error is -1.0
b. the sum of squares for regression is 1.0
c. SSE and SSR are equal
d. SSR and total variation in y are total
14. Given the following data points:(x,y) = (3,3),(4,4),(5,5),(6,6),(7,7),the
least squares estimates of the y-intercept the slope are respectively:
a. 0 and 1
b. -1 and 0
c. 5 and 5
d. 5 and 0
15. When all the actual values of y and the predicted values of y are equal,
the standard error of estimate will be:
a. 1.0
b. -1.0
c. 0.0
d. 2.0
16. For the values of coefficient of determination listed below,which one
implied the greatest value of the sum of squares for regression given that
the total variation in y is 1800:
a. 0.69
b. 0.96
c. 96
d. -100
17. Which of the following statistics and precedures can be used to determine
whether a linear model should be employed ?
a. The standard error of estimate
b. The coefficient of determination
c. The t-test of the slope
d. All of the above are correct answers
18. If we are interested in determining whether two vatiables are linear
related,it is necessary to:
a. perform the t-test of the slope β1
b. perform the t-test of the coefficient of correlation ρ
c. either a or b since they are identical
d. calculate the standard error or estimate Sε
19. The least squares method requires that the variance of error variable
is a constant no matter what the value of x is. When this reguirement is
violated,the condition is called:
a. non-independence of ε
b. homoscedasticity
c. heteroscedasticity
d. influential observation
20. In a regression problem the following pairs of (x,y) are given:(3,1),
(3,-1),(3,0),(3,-2),(3,2). That indicates that the:
a. correlation coefficient is -1
b. correlation coefficient is 0
c. correlation coefficient is 1
II. True-False Question: If the answer is FALSE,correct the sentence to get a
full credir
1. We don't need the t-test of μ1-μ2,since the analysis of variance can be
used to test the difference between the two population means.
2. In employing the randomized block design,the primary interest lies in
reducing sum of squares for block (SSB).
3. The Bonferroni adjustment to Fisher's Least Significant Difference (LSD)
multiple comparison method is made by dividing the specified experimentwise
Type I error rate by the number of combinations of pairs of population
means.
4. The sum of squares for treatments,SST,achieves its smallestvalue (zero)
when all the sample sizes are equal.
5. The calculated value of F in a one-way analysis is 7.88. The numerator
degree of freedom and denominator degree of freedom are 3 and 9,
respectively. The most accurate statement to be made about the p-value is
that p-value < 0.01.
6. If all the valuesof an independent variable x are equal,then regressing a
dependent variable y on x will result in a coefficient of determination of
zero.
7. In a simple linear regression model,testing whether the slope β1 of the
population regressing line could be zero is the same as testing whether or
not the population coefficient of correlation ρ equals zero.
8. If the coefficient of correlation is -0.81,then the percentage of the
variation in y that is explained by the regression line is 81%.
9. The value of the sum of squares for regression SSR can never be larger than
the value of sum of squares for error SSE.
10.The regression line y-hot = 2 + 3x has been fitted to the data points
(4,11),(2,7),and(1,5). The residual sum of squares will be 10.0.
III.Calculations:
1. In a completely randomized design,12 experimental unit were assigned to the
first treatment,15 units to the second treatment,18 umits to the third
treatment. A partial ANOVA table is shown below:
Source of Variation SS df MS F
Treatments * * * 9
Error * * 35
Total * *
a.Fill in the blanks (identified by asterisks) in the above ANOVA table.
b.Test at the 5% significance level to determine if diffrernces exist among
the three treatment means.
2. A professor of statistics is trying to determine which of three
statistical software is the best for his students. He believes that the
time (in hours) it takes a student to master particular software may be
influenced by gender. A 3*2 factorial experiment with three replicates was
designed,as shown below:
Gender
Software male female
1 29 26
24 32
20 30
2 32 23
26 31
21 25
3 18 27
20 22
25 30
Extra information:F(α= 0.1,1,12) = 3.177
F(α= 0.1,2,12) = 2.807
a.Fill in the blanks (identified by asterisks) in the above ANOVA table.
Source of Variation SS df MS F
Software * * * 0.978
Gender * * *
Interaction * * 13.389
Error * * 17.778
Total 328.278
b.Is there sufficient evidence at the 10% significance level to infer that
the time it takes a student to master software and the gender of the
student interact ?
c.Test at the 10% significance level to determine if differences exist
among male and female student.
3. In order to examine the differences in ages of teachers among five school
districts,an educational statistician took random samples of six teachers'
ages in each district. The data listed below:
Ages of Teachers among Five School District
1 2 3 4 5
41 39 36 45 53
53 48 28 37 55
28 41 29 46 49
45 51 33 48 56
40 49 27 51 48
59 50 26 49 61
The partial ANOVA table is given as follows:
Source of Variation SS df MS
Treatment
Error 40.993
Total 2846.967
a.Test at the 5% significance level to determine if differences in
teachers' ages exist among the five districts.
b.Use Tukey's multiple comparison method to determine which means differ.
c.Use Fisher's LCD procedure with α = .05 to determine which population
means differ.
4. A scatter disgram includes the following data points:
x│ 3 2 5 4 5
─┼──────────
y│ 8 6 12 10 14
Two regression models are proposed:
Model 1 : y-hot = 1.2 + 2.5x
Model 2 : y-hot = 5.5 + 4.0x
Using the least squares method,which of these regression models provide the
better fit to the data ? Why ?
5. Quality of oil is measured in API gravity degree - the higher the degree
API,the higher the quality. The table shown below is produced by an expert
in the field who believes thats there is a relationship between quality and
price per barrel.
Oil degree API Price per barrel(in $)
27.0 12.02
28.5 12.04
30.8 12.32
31.3 12.27
31.9 12.49
34.5 12.70
34.0 12.80
34.7 13.00
37.0 13.00
41.0 13.17
41.0 13.19
38.8 13.22
39.3 13.27
A partial computer output follows:
-----------------------------------------------------------------------------
Descriptive Statistics
Variable N Mean Standard Deviation
Degrees 13 34.60 4.613
Price 13 12.730 0.457
-----------------------------------------------------------------------------
-----------------------------------------------------------------------------
Covariance
Degree Price
Degree 21.281667
Price 2.026750 0.208833
P.S. 21.281667 is variance of degree
0.208833 is variance of price
-----------------------------------------------------------------------------
-----------------------------------------------------------------------------
Regression Analysis
coefficients Standard Error
Intercept 9.4349 0.2867
Degrees 0.095235 0.008220
-----------------------------------------------------------------------------
-----------------------------------------------------------------------------
Analysis of Variance
Source DF SS
Regression 1 2.3162
Residual Error 11 0.1898
Total 12 2.5060
-----------------------------------------------------------------------------
a. Determine the coefficient of determination and discuss what its value tells
you about the two variables.
b. Conduct a test of the population coefficient of correlation to determine
at the 5% significance level whether a linear relationship exists between
the quality of oil and price per barrel.
c. Predict with 95% confidence the oil price per barrel for an API degree of 35
d. Identify possible outliers.
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