THE UNIVERSITY OF NEW SOUTH WALES
DEPARTMENT OF STATISTICS
Exercises for MATH3811, Statistical Inference
Goodness of Fit Tests
1. Of 64 offsprings of a certain cross between guinea pigs 34 were red, 10 were black and 20 were white.
According to the genetic model these numbers should be in the ratio 9:3:4. Are the data consistent
with the model?
2. It is hypothesized that X, the number of books withdrawn from a library by a randomly selected
borrower has the geometric probability distribution
P (X = x) = 0.4(0.6)x , x = 0, 1, 2, . . .
A sample of 100 borrowers had the following frequency distribution:
Number of books withdrawn:
0
1
2 3
Number of borrowers: 37 26 17 5
≥4
15
Carry out a Chi-square goodness of fit test of the above hypothesis at the 0.05 level of significance.
3. The progeny of a certain mating were classified by a physical attribute into three groups, the numbers
being 10, 53 and 46. According to a genetic model the frequencies should be in the ratio θ2 : 2θ(1−θ) :
(1 − θ)2 . Are the data consistent with the model at the 5 percent level of significance?
4. For the purposes of stocking, the management wishes to find the probability distribution of time
intervals between replacements of standard machine parts which are in constant use. It is thought
that such time intervals T, measured in weeks, may be exponentially distributed with density function
f (t; θ) = θe−θt , t ≥ 0, θ > 0. Fifty observations of T gave a minimum chi-square estimate of θ of 0.1
and the following frequency distribution of observations:
t: 0-2 2-5 5-9 9-16 > 16
Frequency:
7
12
14
8
9
Carry out a chi-square goodness of fit test at 5% level of significance, for the hypothesis that T is
exponentially distributed.
5. A random sample of four observations on a continuous variate X gave the following results:
0.06, 0.82, 0.81, 0.46.
Carry out a Kolmogorov-Smirnov test of the hypothesis that the density of X is

 2/3 0 ≤ x < 1,
1/3 1 ≤ x ≤ 2,
f (x) =

0 elsewhere
6. An experiment involving measurements of settling velocity of a sediment (in cm/s) was reported.
Seven samples were investigated and the results were: 12.9, 13.7, 14.5, 13.3, 12.8, 13.8, 13.4. We
wish to test if the data are from a normal population with mean 14 and standard deviation 2, i.e.
the hypothesized distribution is N (14, 22 ). Test the above hypothesis using Kolmogorov’s test.
7. Let Fn (x) denote the sample cumulative distribution function for a random sample of size n from a
continuous distribution with the cumulative distribution function F (x). Prove that
Cov(Fn (s), Fn (t)) =
1
(F (min(s, t)) − F (s)F (t)).
n
1
8. Use the asymptotic distribution of the Kolmogorov-Smirnov statistic Dn to verify that
1.07
1.22
limn→∞ P (Dn > √ ) ≈ 0.20, limn→∞ P (Dn > √ ) ≈ 0.10,
n
n
1.36
1.63
limn→∞ P (Dn > √ ) ≈ 0.05, limn→∞ P (Dn > √ ) ≈ 0.01.
n
n
These approximations are very accurate for practical applications as long as n exceeds 35.
9. The resistance of road pavements to rapidly varying and concentrated loads when measured under
laboratory conditions leads to the results shown below, where for two pavements the times to reach
a stage judged unserviceable are recorded.
Pavement A: 36.1 36.4 30.5 43.1
Pavement B: 19.6 35.4 17.8 40.7 27.3
Carry out a Smirnov test of the hypothesis that the two samples come from populations having the
same distribution function.
Answers
1. df = 2, Q = 1.44 < 5.99; Accept H0 .
2.df = 4, Q = 2.72 < 9.49; Accept H0 .
3. θ̂ = .3349, df = 1, Q = .9135 < 3.84; Accept H0 .
4. df = 3, Q = 2.86 < 7.82; Accept H0 .
5. Dn = .4533, n = 4. Accept H0 .
6. Dn = .4013, n = 7. Accept H0 .
9. K-S Statistic Dm,n = 0.6, p-value=.2857. Accept H0 .
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