In each of the multiple choice questions ONE of the answers is CORRECT, except where stated otherwise (and then ONE is WRONG).

Which of these statements is wrong?

Q1. Analysis of Variance,

has the prime aim of comparing variances
has the prime aim of comparing means
is based on a linear model
removes systematic variation in a set of data
compares a set of experimental treatments.

Q2. Analysis of Variance

splits up a total sum of squares
is carried out on frequency data
assumes a normal distribution of residuals
helps to fit a line to a set of points on a graph
estimates the variance of residuals.

Q3. The following expressions can each be transformed into a “linear model” for regression analysis, none of the constants a, b, c, k, α being known:

y = bx + cz^α
y = ae^bx
y² = cx^α
e^y = kx
y = ae^{(cx + kz)}

Now continue looking for the correct answers!

Q4. A mean square is

the square of a treatment mean
a sum of squares divided by its degrees of freedom
the smallest sum of squares in an ANOVA table
the average of the squares of all the observations
a corrected sum of squares.

Q5. One-way Analysis of Variance (ANOVA)

is used to fit a straight line to data
removes one source of systematic variation from the data
requires equal replication of all experimental treatments
tests the Null Hypothesis that treatment means are different
can be used when treatments have different variances

 

Q6. Two-way ANOVA

compares two means
is used to fit a quadratic curve to a set of data
assumes treatments have been allocated to units       completely at random
is used when there are two explanatory variables in regression
removes two sources of systematic variation from the data

Q7. Four different schemes J, K, L, M for carrying out an inspection of a large computer installation were compared, the times in minutes to complete the inspection being recorded. The schemes were carried out in random order over a series of inspections with these results:

J 5 inspections total time 77.9
K 6 inspections total time 119.6
L 4 inspections total time 52.1
M 4 inspections total time 68.1.

The corrected total sum of squares was 210.21. An F-test to compare the mean squares for schemes and residual was carried out, and schemes J and K were compared using a t-test. The values of F and t were:

5.69, 2.11
3.52, 2.06
7.11, 2.99
5.33, 1.02
7.59, 3.54

Q8. Six people I - VI take part in an experiment to measure their speed of response to a visual warning signal. Times are measured in seconds and four types of signal P, Q, R, S are used. Totals for signals are P, 25; Q, 23; R, 18; S, 30. Totals for people are I, 13; II, 24; III, 7; IV, 23; V, 8; VI, 21. A partial two-way Analysis of Variance table is

Source of Variation	Degrees of Freedom	Sum of Squares
People (subjects)		73.00
Treatments (signals)		12.33
Residual		16.67
TOTAL

The least significant difference between two means at the 5% significance level is d and the residual mean square is s². The values of s² and d are:

0.833, 1.30
0.794, 1.19
0.833, 1.12
1.111, 1.12
1.111, 1.30

Q9. In a linear regression of y on x, y = a + bx,

both variables have the same variance
a fitted straight line goes through (0, 0)
the variance of x is constant for all values of y
x increases by b units for every unit increase in y
x is the explanatory variable.

Q10. In an Analysis of Variance for linear regression of y on x using n pairs of observations

degrees of freedom for residual are (n – 1)
r² (or R²) is near to 100% when the regression line is       almost straight
the sum of squares for regression has 2 degrees of freedom
a confidence interval for b may be calculated from the residual sum of squares
the F-test examines the hypothesis b ≠ 0.

Q11. In multiple linear regression

a response variable is explained using two or more        explanatory variables
explanatory variables must all be measured in the same        units
explanatory variables must not be correlated with one another
stepwise fitting methods should be used
a “best subset” of explanatory variables is bound to have the greatest possible value        of R².

Q12. A simple linear regression of y on x using 11 pairs of data gave corrected sums of squares for regression and total whose values were 783.77 and 1580.73 respectively. The estimates of α and ß were 26.52 and 0.4315. The F-test of the Null Hypothesis β = 0” was carried out and the value predicted for y when x = 50 was calculated. The correct values of F and y are

8.81, 48.0
9.83, 44.0
9.40, 49.6
8.85, 48.1
9.84, 50.0

Q13. Two predictor variables x₁ and x₂ were used in a regression equation to predict a response y. Twelve sets of data (y, x₁, x₂) were available. Corrected sums of squares for regression and residual were respectively 187.421 and 82.895. Other relevant computer output was:                  T     P
Const  60.49   4.46  0.002
X₁      0.6159  2.50  0.034
X₂      0.9895  1.21  0.258

The significant terms in the full model and the value of y predicted by it when x₁ = 30 and x₂ = 10 were

const, x₁, 88.9
const, x₁, x₂, 88.9
const, x₁, 79.0
x₁, x₂, 28.4
const, x₁, x₂, 79.0

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