Table of Contents
KHYBER PAKHTOON KHWA, PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR THE POSTS OF PROVINCIAL MANAGEMENT SERVICE (BPS-17)
STATISTICS
Time Allowed: 3 Hours
Maximum Marks: 100
NOTE: (i) Attempt any FIVE questions in all. All the questions carry equal marks.
(ii) Statistical table may be provided on request.
(iii) The use of calculator is ALLOWED.
Q-1 (a) Write the applications of statistics in daily life with examples
(b) A study of credit card fraud was conducted by a crime researcher. According to collected data, it was found that 243 cases were of stolen cards, 85 of counterfeit card, 52 of mail orders and 46 of both stolen card and mail orders. Find the probability that a randomly selected case
(i) belongs to counterfeit card
(ii) belongs to stolen card or mail order fraud
(c) Suppose that the probability of success in an oral interview for the civil services is 0.32. If 10 candidates are being interviewed,
(i) What is the probability that none of these will be succeeded?
(ii) What is the probability that at least half of the candidates will be succeeded?
(07+06+07=20)
Q-2 (a) Define the term “probability”, using different approaches.
(b) It has been studied that the average number of road accidents occurring at night on the G.T. road is 5. If the number of accidents follows a Poisson distribution, then find out the probability that last night
(i) one accident occurred,
(ii) no accident occurred.
(c) A drug supplied by a pharmaceutical company is administered to 3 patients, and the random variable X represents the number of cures that occur. The following table lists the probability distribution of X.
| X | 0 | 1 | 2 | 3 |
| P(X=x) | 0.10 | 0.20 | 0.40 | 0.30 |
Find the expected value of X and its variance.
(05+07+08=20)
Q-3 (a) What is Statistical Inference? Explain its importance with examples.
(b) Suppose average marks (out of 10) of 35 randomly selected candidates appeared in an interview for civil services, are 6.6. The marks are normally distributed with a standard deviation of 2. Calculate 95% confidence interval for mean marks of all the candidates.
(c) Let X denote the time taken to run a road race. Suppose X is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race
(i) in less than 160 minutes?
(ii) in 215 to 245 minutes?
(05+05+10=20)
Q-4 (a) Differentiate between
(i) Null hypothesis and alternative hypothesis
(ii) One-tailed and Two-tailed tests
(b) The manufacturer of a certain brand of auto batteries claims that the mean life of these batteries is 45 months. A consumer protection agency took a random sample of 24 such batteries and found that the mean life for this sample is 43.05 months and standard deviation 4.3 months. Should the agency reject the claim of the manufacturer at 5% level of significance?
(P.T.O)
(c) An exercise physiologist wants to decide whether a certain type of running program will reduce heart rates. He measures the heart rates of 10 randomly selected people who are then placed on the running program. One month later the exercise physiologist again measures the heart rates of the 10 people. The heart rates, both before and after the running program, are displayed as below:
Before Program: 68, 76, 74, 71, 71, 72, 75, 83, 75, 74;
After Program: 67, 77, 71, 70, 69, 70, 71, 77, 71, 74.
Do the data provide sufficient evidence to conclude that the running program will reduce heart rates? (Use 5% level of significance.)
(04+06+10=20)
Q-5 (a) Suppose we want to compare the mean protein intake of all people with income below the poverty level to that of all people with incomes above the poverty level. The data given below display the protein intakes, in grams, of a 24-hours period for independent random samples of 10 people with income below the poverty level and 8 people with income above the poverty level.
Below poverty level: 51.4, 76.7, 73.7, 66.2, 65.5, 49.7, 65.8, 62.1, 75.8, 62.0
Above poverty level: 86.0, 59.7, 68.6, 98.6, 87.7, 69.0, 80.2, 78.1
Do the data provide sufficient evidence to conclude that the mean protein intake of the people with above poverty level is higher than those with below poverty level? (Use 5% level of significance.)
(b) For randomly selected homes recently sold in a county, the living areas (in 100 square feet: X) are listed along with the annual taxes (in 1000 USD: Y). Find out how the living area affects the annual taxes with fitting a regression line.
X: 15 38 23 16 16 13 20 24
Y: 1.9 3 1.4 1.4 1.5 1.8 2.4 4
(10+10=20)
Q-6 (a) Suppose we take a sample of seven households from a small city and collect information on their incomes (X) and food expenditures (Y) for the last month. The information obtained (in hundreds of dollars) is given below:
X: 55 83 38 61 33 49 67
Y: 14 24 13 16 9 15 17
Compute the coefficient of correlation between income and food expenditure and interpret it.
(b) The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 10 varieties of Kellogg’s cereal.
X(Sugar): 4 15 12 11 8 6 7 2 7 14
Y(Calories): 120 200 140 110 120 80 190 100 120 190
Fit a regression line to predict calories counts. What will be the number of calories if sugar in the cereal is 10 grams?
(10+10=20)
Q-7 (a) What are the advantages and disadvantages of sampling?
(b) Where do we use the stratified random sampling? Write a general procedure to draw a stratified random sample.
(c) What do you understand by sampling distribution? For a population of with elements 2, 4, 6, 8, 10, draw all possible random sample of size 2 without replacement and compute the sample means and compare the results with the population mean.
(05+05+10=20)
Q-8 Write note on any FOUR of the following:
(a) Independent events and conditional probability
(b) Applications of the normal distribution
(c) Interval estimation
(d) Use of Analysis of Variance (ANOVA)
(e) Partial and multiple correlation
(f) Systematic sampling
(g) Applications of Statistics in Public Health
(05+05+05+05=20)