CPT Section D Quantitative Aptitude Chapter 15 Prof. Bharat Koshti

Theory of estimation :

It deals with estimating unknown values of the Population parameters.

There are two types of estimation techniques :

1.Point Estimation

2. Interval Estimation

1. Point Estimation: When a single value is proposed to estimate, it is called as Point estimation.

1. Suppose the unknown parameter is ϴ.

(e.g. ϴ = Population Mean)

2. We take a sample of size ‘n’ from the population of

size ‘N’ at random. So we get sample observations

( x1, x2, x3, ……………… xn )

3. Now based on these sample observations we

construct statistic T (e.g. Sample Mean) which will estimate ϴ

T is a single value obtained from the sample & is known as point estimator.

The point estimator of Population Mean(μ), Population Standard Deviation(σ) & Population Proportion(P) are the corresponding sample Mean(x̅),Sample Standard Deviation(s) & Sample Proportion(p).

Ex1. Consider sample observations 14,15,6,17,28 from population of 100 units. Find an estimate of population mean.

Estimate of μ is given by X̅

X̅ = (14+15+6+17+28)/5

X̅ = 16

Hence, estimate of μ is 16.

Ex2. A random sample of 150 articles taken from a large batch of articles contains 15 defective articles. Find an estimate of the proportion of defective articles in the entire batch? Also find its Standard error(SE).

Estimate of p = P i.e. Sample Proportion estimates Population Proportion.

p = 15/150 = 0.10

SE = SQRT{p(1-p)/n} = SQRT{ 0.1(1-0.1)/150} = 0.0245

(since N is very large fpc is ignored)

Interval estimation specifies two values that contain unknown parameter ϴ on the basis of random sample drawn from the population. Thus interval (T1,T2) is likely to contain parameter ϴ. T1 is called Upper Confidence Limit(UCL) & T2 is called Lower Confidence Limit(LCL).

The probability that the confidence interval contains the parameter is called confidence co-efficient and it is denoted as (1-α)%.

If α is 0.05 then (1- α)% would be (1-0.05)% = 95%. It shows the amount of confidence.

If α is 0.01 then (1- α)% would be (1-0.01)% = 99%.

Assuming we have drawn random sample from the Normal Population with mean μ and standard deviation σ. Following Z values are important for various confidence Level. Confidence Level (α) 99% 95% 90% Confidence Co-efficient(Z) 2.58 1.96 1.64

From a certain college of 3000 students a sample of 200 students is taken and their weights are recorded. The average weight of 200 students was found to be 65kg with the standard deviation of 10kg. Construct a 99% confidence interval for the population mean. 99 % confidence interval = ( x̅ - 2.58 σ / √n ̅, x̅ + 2.58 σ / √n ) = ( 65 -2.58X 10/ √200 , 65 +2.58X 10/ √200) = (63.17 , 66.82)

A manufacturing industry which manufactures electronics IC’s. From the daily production process of 10000 IC’s a sample of 150 IC’s is taken at random and 5% IC’s were found to be defective. Estimate the no. of defective IC’s that can be produced in daily production process. Use 95% level of significance. 95 % confidence interval = (p - √pq/n x sqrt(N-n/N-1), p + √pq/n x sqrt (N-n/N-1)) Here p = 0.05, N = 10,000, n = 200 q = 1-p = 0.95 = ( 0.0323,0.0676) Hence (0.0323x10000, 0.0676x10000) = (323, 676)

FORMULA: Sample size = n = (σ Z / E)2

Where, E = Admissible error while estimating the parameter μ. σ = SD & Z = Table Value for Normal Distribution. Example: Given σ = 10 , what should be size of the sample in order to be 99% confident that error estimate of mean would not exceed 2? Here E = 2, σ = 10 Z = 2.58 n = (σ Z / E)2 = (10x2.58 /2)2

= 166

MCQ’s

(a) Only one

(b) Two

(c) Three

(d) Many

Answer: D

MCQ.2: The most commonly used confidence interval is

(a) 95 percent

(b) 90 percent

(c) 94 percent

(d) 98 percent

Answer: A

MCQ.3:It is known that the population standard deviation in waiting of getting PAN card is 13 days. How large a sample should be taken to be 99% confident that the waiting time is within 8 days of true average? (use z = 2.58)

(a) 18 days

(b) 13 days

(c) 19 days

(d) 14 days

Answer: A

MCQ:4 • It is known that x̅ = 55 for sample of 64

units & SE(x̅)=1.5, calculate confidence interval at 99%.(use z = 2.58 )

(a) (51.13,58.87)

(b) (51.13,51.87)

(c) ( 51.87,58.13)

(d) (51.31,51.78)

Answer: A

MCQ.5:The estimate of the parameter is stated as an interval with a specified degree of

(a) confidence

(b) interval

(c ) class

(d ) None

Answer: A

MCQ.6: The standard deviation in the sampling is called

(a) Standard error

(b) Absolute error

(c ) Relative error

(d ) None

Answer: A

MCQ.7: The standard deviation in the sampling is called

(a) Standard error

(b) Absolute error

(c ) Relative error

(d ) None

Answer: A

MCQ.8:The ratio of no. of elements possessing a characteristic to the total no. of elements in a sample is known as -

(a) Sample Proportion

(b) Population Proportion

(c ) Sample size

(d ) None

Answer: A

9. The Confidence limits are the upper & lower limits of the

(a) Point estimate

(b) Interval estimate

(c ) Confidence interval

(d ) None

Answer: C

MCQ.10: Different types of estimates about a population parameter are -

(a) Two

(b) Three

(c ) Four

(d ) Five

Answer: A

Thank you