In most statistical studies population parameters are unknown, and must be estimated through sampling. The primary goal of a sampling activity is to make an inference about something using the least amount of information possible. It is possible to draw valid conclusion about the population parameter from sampling distributions. The problem is to construct a sample quantity that will serve to estimate the unknown parameter. Such a sample quantity is called the estimator.
(i.e.) Any sample statistics that is used to estimate an unknown population parameter is called an “ESTIMATOR”.
‘ẍ’ Sample mean can be an estimator of the population mean ‘µ’
‘P1’- sample proportion can be an estimator of the population proportion ‘P’.
We can make 2 types of estimates about a population.
Is a single number that is used to estimate an unknown population parameter, in other words, the estimate of a population parameter given by a single number is called the point estimate of the parameter.
For example, if a firm takes a sample of 50 salesman and finds out that the average amount of time that the each salesman spends with his customer is 80 minutes. If this figure is used for an estimate of all the salesman employed by the firm it is referred to as a point estimate because we are using one value to obtain the population value.
Often a point estimate is insufficient as it is either right or wrong. If the estimate is wrong we cannot be certain how wrong the estimate is. Therefore a point estimate is useful if it’s accompanied by the estimate of error that might be involved.
A good estimator is one which is close to the population parameter being estimated. The desirable properties of an estimator are
An estimator is a random variable as it is always a function of sample values. Then if the average of these sample values is equal to the population parameter then it is unbiased estimate.
As the sample size increases the difference between the sample statistic and population parameter should become smaller and smaller. If the difference continues to become smaller and smaller as the sample size becomes larger, the sample statistics is said to be consistent estimator of the parameter.
If the variance of the estimator is small, the distribution of the estimator will be better in that the value to be closer to the parameter value.
A sufficient estimator is one that uses all information about the population parameter contained in the sample. A sufficient estimator ensures that all information that a sample can furnish with respect to the estimation of a parameter is being utilized.
If an estimate of a population parameter is given by 2 distinct numbers between which the population parameter may be expected to lie then the estimate is called an interval or confidence interval or confidence limits.
It is a range of values with in which, with a known probability or to a known degree of reliance, the value of the population parameter is expected to lie.
ESTIMATION AND DETERMINATION OF SAMPLE SIZE
ESTIMATION OF POPULATION MEAN WHEN STANDARD DEVIATION IS KNOWN
1. In order to introduce some incentives for higher balance in savisngs account a random sample of size 64 savings account at a banks branch was studied to estimate the average monthly balance in savings bank accounts. The mean and S.D. of the 64 savings account were found to be Rs.8,500 and Rs.2000 respectively. Find (i) 90% (ii) 95% (iii) 99% confidence interval for the population mean. Ans. 90%-(8911.25, 8088.75)- 95%-(8990, 8010)-99% (9145, 7855)
2. The shopping bills of customers of a departmental stores are known to follow normal distribution with mean Rs.2000 and variance Rs.2,50,000. One day the first hundred customers bills are found to have an average of Rs.2,200. Can the first 100 customers be regarded as truely representative are random sample of the population of all customers.
3. The Quality department of a wire manufacturing company periodically selects a sample of wire specimens in order to test for breaking strength. Past experience has shown that the breaking strength of a certain type of wire are normally distributed with S.D. of 200 kg. A random sample of 64 specimens gave a mean breaking strength of 6,200 kg. The quality control supervisor wanted a 95% confidence interval for the mean breaking strength of the population. Ans.6249 kg, 6151 kg.
4. The average number of customers coming to a banks branch is found be 20/ hour. Determine 95% confidence limits for the number of customers in an hour. Ans. 23.92, 16.08.
5. In order to improve the quality of items produced by a production process, sample of items are inspected and no. of defects in each item is recorded. One such example is the no. Of missing rivets on the body of a bus. If the average no. Of missing rivets in a sample of 4 bus bodies is found to be 25 then find 95% confidence limit for the no. Of missing rivets in a bus body.
6. A manager wants an estimate of average sales of salesman in company. A random sample of 100 out of 500 salesman is selected and the average sales is found to be Rs.7,50 thousand. If population standard deviation is Rs. 150 thousands, manager specifies a 98% level of confidence. What is the interval estimate for the average sales of salesman? Ans. 781.289, 718.710
7. The Human Resource Director of a large organization wanted to know what proportion of all persons who had ever been interviewed for a job with his organization had been hired. He was willing to settle for 95% confidence interval. A random sample of 500 interview records revealed that 76 of the persons in the sample had been hired. Estimate the population proportion @ 95% confidence interval. Ans. 0.183, 0.121.
8. Out of 20000 customer ledger account, a sample of 600 accounts was taken to test the accuracy of posting and balancing where in 45 mistakes were found. Assign limits with in which the no. Of defective cases can be expected at 5% level of significance. Ans. 0.097, 0.053
9. In an attempt to control the quality of output for a manufactured parts. A sample of parts is chosen randomly and examine in order to estimate the population proportion of part that are defective. The manufacturing process operated continuously unless it must be stopped for inspection or adjustment. In the latest sample of 90 parts, 15 detectives are found. Determine (a) point estimate (b) 98% interval estimate. a). Ans.0.167 b). Ans. 0.259, 0.075.
10. Suppose we want to estimate the proportion of families in a town which has 2 or more children. A random sample of 144 families shows that 48 families have 2 or more children, Construct a 95% confidence interval.
11. A ball pen manufacturer makes a lot of 10,000 refills the procedure desire some control over these lots so that no lots will contain an excess number of defective refills. He decides to take a random sample of 400 refills for inspection from a lot of 10,000 and finds 9 defectives obtain a 90% confidence interval for the number of defectives in the entire lot. Ans. 347, 103
12. A sample of 150 items from machine A had an average life of 1400 hours a similar sample of 100 items from machine B has a mean life of 1200 hours. Past records indicate that the S.D of the items produced by machine A is 120 hrs.and by machine B is 80 hrs. Find 95% confidence limits on the difference in the average life time of the population of the items produced by the two machines. Ans. 224.696, 175.304
CONFIDENCE INTERVAL FOR MEAN WHEN POPULATION STANDARD DEVIATION IS UNKNOWN
13. For assessing the number of monthly transactions in credit cards issued by a bank, transactions in 25 cards were analyzed. The analysis revealed an average of 7.4 transactions and sample S.D of 2.25 transactions. Find the confidence limits for the monthly number of transactions by all the credit card holders of the bank. Ans. 8.34, 6.45
