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Sampling Distribution of Sample Proportion

Consider a lot with proportion of defectives P. If a random sample of size n with proportion of detectives p is drawn from this population then the sampling distribution of p is approximately normal distribution with mean = P and S.D. = S.E. of sample proportion =  where, Q = 1 - P and the sample size n is sufficiently large. If the random sample is drawn from a finite population without replacement then we have to multiply a correction factor to the S.D. formula.

If p1 and p2 denote the proportions from independent samples of sizes n1 and n2 drawn from two populations with proportions P1 and P2 respectively then

where, P1 + Q1= 1  and P 2 + Q2 = 1.

 

Example 5. It has been found that 3% of the tools produced by a certain machine are defective.

What is the probability that in a shipment of 450 such tools, 2% or more will be defective ?

Solution. Since the sample size n = 450 is large, the sample proportion (p) is approximately normally distributed with mean = P = 3% = 0.03.

=0.008

Required probability = P[p > 0.02]

= P[z > -1.25] = 0.5 + φ (1.25)

= 0.5 + 0.3944 = 0.8944.

 

 

PROBLEMS

  1. A population consists of 5 numbers (2, 3, 6, 8, 11). Consider all possible samples of size two which can be drawn with replacement from this population. Calculate the S.E. of sample means.
  2. When we sample from an infinite population, what happens to the standard error of the mean if the sample size is (a) increased from 30 to 270, (b) decreased from 256 to 16 ?
  3. A random sample of size 400 is taken from an infinite population having the mean µ = 86 and the variance of σ2 = 625. What is the probability that X will be greater than 90 ?
  4. The number of letters that a department receives each day can be modeled by a distribution having mean 25 and standard deviation 4. For a random sample of 30 days, what will be the probability that the sample mean will be less than 26 ?
  5. A random sample of 400 mangoes was taken from a large consignment and 30 were found to be bad. Find the S.E. of the population of bad ones in a sample of this size.
  6. From a population of large number of men with a S.D. 5, a sample is drawn and the standard error is found to be 0.5, what is the sample size·?
  7. A population consists of 20 elements, has mean 9 and S.D. 3 and a sample of 5 elements is taken without replacement. Find the mean and S.D. of the sampling distribution of the mean. What will be the S.D. for samples of size 10 ?
  8. A machine produces a component for a transistor set of the total produce, 6 percent are defective. A random sample of 5 components is taken for examination from (i) a very large lot of produce, a box of 10 components. Find the mean and S.D. of the average number of defectives found among the 5 components taken for examination.
  9. A population consists of five numbers 2, 3, 6, 8, 11. Consider all possible samples of size two which can be drawn without replacement from the population. Find

(a)    The mean of the population

(b)   Standard deviation of the population

(c)    The mean of the sampling distribution of means

(d)   The standard deviation of the sampling distribution of means.

 

 

ANSWERS

1. 2.32

2. (a) It is divided by 3 (b) It is multiplied by 4

3. 0.0007

4. 0.9147

5. 0.013

6. 100

7. For sample of 5 elements, sampling mean = 8, S.D. =

For sample of 10 elements, sampling mean = 8, S.D. =

8. Mean= 0.06, S.D.= 0.106

9. (a) 6, (b) 3.29, (c) 6, (d) 2.12.