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Cork Institute of Technology
Master of Science in Computing in Education
National Framework of Qualifications – Level 9
February 2005
Quantitative Analysis
(Time: 2 Hours)
Instructions Answer any FOUR questions. All questions carry equal marks. A booklet of statistical tables and formulae is provided.
Examiners: Mr. K. J. Kelly Dr. H. Gibbons
1. (a) For the following data construct an ordered stem and leaf plot and hence compute the
median and the quartiles.
(b) A set of raw data has been condensed into a frequency distribution as follows:
Value (€) Frequency
Less than 10 25
10 but less than 20 38
20 but less than 30 52
30 but less than 50 65
For this distribution estimate the median and the mode.
(c) A box contains five balls - two red balls and three black balls. Balls are drawn from the
box, one at a time, without replacement, until both red balls have been drawn. Find the
probability distribution of the number of drawings required to obtain the two red balls.
Find also the expected value of the resulting distribution.
Contd…/
(d) The publisher of “Statistics Monthly” claims that 30% of its subscribers are from social
class A, 40% from social class B, 20 % from social class C and the remainder from
other social classes. A sample of 300 subscribers was tabulated by social class as
follows:
Social Class Number of
Subscribers
A 80
B 130
C 50
Other 40
Does the sample data support the publisher’s claim? Use a standard test at the 5% level
of significance.
2. (a) It is estimated that fifteen percent of postal questionnaires are returned. If
a pilot study involved sending out 25 questionnaires find the probability
that less than two are returned.
If in the actual study 400 postal questionnaires were sent out find the
probability that more than 50 were returned.
(b) Component alpha produced by a company has a target weight of 550g but the actual
weights of the alphas produced are normally distributed with mean 550g and standard
deviation 20g. Find the percentage of alphas with weights between 530 and 565g.
The company produces component beta also. These components have weights normally
distributed with mean 390g and standard deviation 15g. Assemblies are produced by
joining alphas and betas together. Find the percentage of assemblies with weights
greater than 980g.
Assemblies weighing more than 980g are regarded as unacceptable and scrapped. Find
the median (middle) weight of acceptable assemblies. Contd…/
(c) A simple project has three activities A, B and C. Activity B cannot start until activity A
has been completed but activity C can be completed in parallel with either A or B. The
activity durations are not fixed but have the probability distribution as follows.
A Duration (Days) 7 8
Probability 0.3 0.
B Duration (Days) 4 5
Probability 0.6 0.
C Duration (Days) 11 12 13
Probability 0.2 0.6 0.
Obtain the exact distribution for total completion time for this project (assuming all
activities begin as early as possible) and that the activity durations are independent
random variables.
4. (a) What is meant by the sampling distribution of the mean? What can be said about the
sampling distribution of the mean if the parent population is normal?
A random sample of five observations from a normal distribution is given below.
(i) Calculate the sample mean and the sample standard deviation for these
observations.
(ii) Compute a 95% confidence interval for the population mean.
(iii) Compute a 99% confidence interval for the population standard deviation.
Contd…/
2
(^13)
A
B
C
(b) A market research company wishes to estimate the proportion of households in the area
that use the product KLENFAST. What sample size should be used if the proportion is
to be estimated to within one percentage point with 99% confidence? Take into account
that there are 5,000 households in the area and the proportion is certainly less than
(c) A company claims that the average strength of its ropes is 250 kg. A random
sample of 80 ropes was taken and the average strength was 246.7 kg with a
sample standard deviation of 9.52 kg. Test the company's assertion using an
appropriate test of hypothesis.
Suppose the company's claim is incorrect and that the true population mean
strength of its ropes is 248 kg with a population standard deviation σ of 9.52 kg.
What is the probability that, in a test of hypothesis testing that the true mean is
250 kg using a sample of size 80 ropes, the null hypothesis "Ho: μ = 250" will be
accepted? Relate your answer to the concept of beta risk.
Explain what is meant by the operating characteristic curve in this context.
5. (a) A training organisation is interested in predicting performance on its course by using an
aptitude test prior to the course. For a random sample of 10 course graduates the
following pairs of scores are available relating final course score (y) to aptitude test
score (x).
Aptitude Score x^20 25 26 28 39 45 55 68 79
Final Score y 66 95 85 110 121 118 130 147 155 151
∑ =^470 ∑ x =^27 ,^146 ∑ xy=61,240 ∑ y=1,178 ∑ y =^146 ,^606.
2 2
x
(i) Calculate the equation of the regression line of Final Score on Aptitude Score.
(ii) For the regression above compute the total variation in y and the explained and
unexplained variation. Hence or otherwise compute R
2
Contd…/
Minitab Output for Question 5
Input Data (some rows are omitted) Row GPA SATM SATV 1 3.065 700 640 2 2.760 640 530
…………………………………………………………………… 13 3.175 750 610 14 2.675 650 460
Regression Analysis 1: GPA versus SATM The regression equation is GPA = 0.587 + 0.00338 SATM
Predictor Coef SE Coef T P Constant 0.5873 0.2354 A 0. SATM 0.0033820 0.0003787 8.93 0.
S = D R-Sq = 86.9%
Analysis of Variance Source DF SS MS F P Regression 1 1.2914 1.2914 C 0. Residual Error 12 B 0. Total 13 1.
Regression Analysis 2: GPA versus SATV
The regression equation is GPA = 0.977 + 0.00314 SATV
Predictor Coef SE Coef T P Constant 0.9774 0.5107 1.91 0. SATV 0.0031431 0.0009413 3.34 0.
S = 0.2533 R-Sq = 48.2% Analysis of Variance Source DF SS MS F P Regression 1 0.71557 0.71557 11.15 0. Residual Error 12 0.77012 0. Total 13 1. Regression Analysis 3: GPA versus SATM, SATV The regression equation is GPA = - 0.0007 + 0.00283 SATM + 0.00173 SATV
Predictor Coef SE Coef T P Constant -0.00067 0.07859 -0.01 0. SATM 0.0028271 0.0001106 25.56 0. SATV 0.0017282 0.0001381 12.51 0.
S = 0.03405 R-Sq = 99.1%
Analysis of Variance
Source DF SS MS F P Regression 2 1.47293 0.73647 635.06 0. Residual Error 11 0.01276 0. Total 13 1.
