L1025 INTRODUCTION TO STATISTICS 2018
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L1025
BA AND BSC FIRST YEAR EXAMINATION
May/June (A2) 2018
INTRODUCTION TO STATISTICS
Assessment period: May/June (A2) 2018
SECTION A
Please answer Question 1 [30 marks]
1.
a. 10 moneylenders were surveyed and asked to report the interest
rate they charged for a short term loan. The reported interest rates in percentages (%) are displayed below:
Interest rate (%) |
10 |
5 |
8 |
12 |
20 |
5 |
11 |
6 |
5 |
8 |
i. Calculate the mean, median and mode for the interest rates charged [5 marks]
ii. Using class intervals of 0-4.99, 5-9.99, 10-14.99 etc.,
construct a table showing frequencies and relative
frequencies of interest rates charged. Draw a diagram
showing the distribution of interest rates charged based on your table. [10 marks]
b. The height (in centimetres) of boys aged five in a developing country is a random variable (X) that follows a Normal distribution with a
mean of 110cm and a standard deviation of 4cm. According to
Unicef, a child is stunted if its height is below 2 standard deviations of the mean.
Find the probability that a child is stunted. Sketch a graph to support your answer. [5 marks]
c. The weight (in kilograms) of boys aged five in a high-income country is a random variable (X). The distribution of X is not known.
According to the World Health Organization, a child is underweight if its weight is below 2 standard deviations of the mean and overweight if its weight is above 2 standard deviations of the mean.
Find the maximum percentage of boys aged five who are over-
weight or under-weight. Explain your reasoning carefully and sketch a graph to support your answer. [10 marks]
SECTION B
Please answer TWO questions. Each question carries 35 marks.
2.
a. The following pairs of scores were obtained from a random sample of 6 students who took the Macroeconomics (Mac) and
Microeconomics (Mic) exams. The first score is the Mac grade and the second the Mic grade.
(47, 57) (40, 56) (41, 40) (32, 44) (69, 67) (77, 70)
i. Compute and interpret the Spearman Rank Order Correlation Coefficient for the Mac and Mic scores. [5 marks]
ii. Using the answer you obtained in (i) above and a significance level of 0.05 test whether there is a significant correlation in ranks across the exam scores for the two subjects. [5 marks]
iii. Discuss under what circumstances the Spearman correlation coefficient would be preferred over the Pearson correlation coefficient. [5 marks]
b. The average daily outputs of 10 workers in company A are
compared to those of 10 workers in company B. The following data are obtained:
x̅A = 420 x̅B = 408
SA = 25 SB = 20
i. Conduct a hypothesis test to determine whether there is a difference in variances between the two groups using a 0.05 level of significance. What do you conclude? [10 marks]
ii. Using a 0.05 level of significance, undertake a hypothesis test to determine whether on average workers from company A
produce more per day than workers from company B. What do you conclude? [10 marks]
3.
a. It is claimed that an average child spends 15 hours per week
watching television. A survey of 100 children finds an average of 14.5 hours per week, with a standard deviation of 8 hours.
Undertake a hypothesis test of the claim using a 0.05 level of significance. [8 marks]
b. From a sample of 150 Economics students at Manchester
university, 67% were found to have passed their Statistics test. From a sample of 120 Economics students at Sussex university, 62%
were found to have passed their Statistics test. Estimate the 99% confidence interval for the true difference between the proportions passing the exam. [8 marks]
c. The daily output produced in three different factories (A, B, and C) is observed and reported to be as follows:
A |
44 |
35 |
60 |
28 |
B |
41 |
36 |
58 |
|
C |
48 |
37 |
61 |
37 |
Is there a significant difference in output produced across the three factories at the 5% significance level? [10 marks]
d. A random sample of 1000 people were asked to report their age and preferred music category. The information is summarised in the table below.
Age |
Music category |
Totals |
||
Pop |
Classical |
Jazz |
||
16 - 29 years |
200 |
50 |
50 |
300 |
30 – 49 years |
60 |
40 |
100 |
200 |
50 – 69 years |
120 |
300 |
80 |
500 |
Totals |
380 |
390 |
230 |
1000 |
Is there evidence at the 1% significance level that there is an
association between preferred music category and an individual's age? [9 marks]
4. To study the determinants of the demand for margarine the following
regression equation was estimated using Ordinary Least Squares (OLS):
Margconsi = α + β1Margpricei + β2Butterpricei + β3Incomei + ui
where the variables are defined as follows:
Margcons represents the consumption of margarine in ounces per person per week;
Margprice represents the price of margarine in pence per 8 ounce pack;
Butterprice represents the price of butter (a likely substitute for margarine) in pence per 8 ounce pack;
Income represents real income in £s;
ui is an error term.
The regression results are presented below:
Regression Statistics
Multiple R |
0.94 |
|
|
|
R Square |
0.89 |
|
|
|
Adjusted R Square |
0.86 |
|
|
|
Standard Error |
0.20 |
|
|
|
Observations |
19 |
|
|
|
ANOVA |
||||
|
df |
SS |
MS |
F |
Regression |
3 |
4.53 |
1.51 |
39.182 |
Residual |
15 |
0.58 |
0.04 |
|
Total |
18 |
5.11 |
|
|
|
Coefficients |
Standard Error |
t Stat |
Intercept |
3.78 |
1.43 |
2.65 |
Margprice |
-0.02 |
0.005 |
-4.79 |
Butterprice |
0.02 |
0.004 |
4.32 |
Income |
0.003 |
0.008 |
0.39 |
a. Interpret the estimated coefficients of β1 , β2 , and β3 and test the statistical significance of each individual coefficient at the 5% level. State clearly the null and alternative hypotheses in each case. [10 marks]
b. Test the overall significance of the regression at the 5% level. State the null and alternative hypotheses clearly. [10 marks]
c. The following are the means of each variable:
Margcons 3.55
Margprice 110.62
Butterprice 110.57
Income 89.15
Calculate and interpret the price elasticity of demand, the income elasticity and the cross-price elasticity. [10 marks]
d. What could you do to improve the model specified above? [5 marks]
2023-08-04