Bstat7 - Lessons in Business Statistics Prepared By P.K....

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Unformatted text preview: Lessons in Business Statistics Prepared By P.K. Viswanathan Chapter 7: Estimation Introduction Marketing Manager in an organization needs to estimate the likely market share his company can achieve in the market place. Quality Assurance Manager may be interested in estimating the proportion defective of the finished product before shipment to the customer. Manager of the credit department needs to estimate the average collection period for collecting dues from the customers. How confident are they in their estimates? This chapter provides some insights into point estimation and interval estimation that are essential in business planning. Please remember that the three components-point estimation, interval estimation, and hypothesis testing together constitute the all-important inferential statistics. 1) Point Estimation Point Estimation deals with the task of selecting a specific sample value as an estimate for a population parameter. 1) Point Estimation-Continues A point estimate is a specific value of a sample statistic that is used to estimate a population parameter. 1)Point Estimation Population Mean The sample mean X is an unbiased estimator of the population mean μ . An unbiased estimator is one whose expected value is equal to the population parameter. That is, E( X ) is equal to μ . Of course the samples drawn must be independent random samples from the population of interest. 1)Point Estimation Population Proportion Sample proportion is an unbiased estimator of the population proportion. That is p is an unbiased estimator of P. So E(p) = P. A particular value of p based on a sample survey becomes a point estimate. 2) Interval Estimation A point estimate cannot be expected to coincide exactly with the population parameter. Suppose in a survey you find that the average income of a household is Rs. 300000 per year. Is it that the income of every household is Rs.300000 per year? Certainly not. Some households may have more than Rs. 300000 and some may have less than this amount. In other words point estimate will not coincide with the population parameter. How to cope with this problem? Interval Estimation comes to our help in this regard. Interval Estimation establishes an interval consisting of a lower limit and an upper limit in which the true value of the population parameter is expected to fall. This interval is called “ Confidence Interval ” in the parlance of inferential statistics. 2) Interval Estimation Picture 3) Confidence Interval for Population Mean-Large Sample (1- Confidence Interval for the ) Population Mean is given by The 3) Confidence Interval for Population Mean-Large Sample-Example A machine produces components, which have a standard deviation of 1.6 cm in length. A random sample of 64 parts is selected from the output and this sample has a mean length of 90 cm. The customer will reject the part if it is either less than 88cm or more than 92cm. Does the 95% confidence interval for the true mean length of all the components produced ensure acceptance by the customer? 3) Confidence Interval for Population Mean-Solution to the Example To answer the question of acceptance by the customer, you should first work out the 95% confidence interval for the population mean μ (Here μ is the mean length of the components in the population). When you want 95% confidence level, the Z value is 1.96 from the standard normal distribution. See diagram in the next slide. 3) Confidence Interval for Population MeanLarge Sample- Solution to the Example Picture Showing Z value for 95% confidence level 3) Confidence Interval for Population MeanLarge Sample- Solution to the Example Picture Showing for 95% confidence level for μby using the formula X 1.96 σ n μ X 1.96 σ n 3) Confidence Interval for Population Proportion-Large Sample The (1- Confidence Interval for the Population ) Proportion P is given by p(1- p) p(1 - p) p Z PpZ n n Where P is the population proportion, p is the sample proportion In particular, 95% confidence interval for the population proportion is given by p(1 - p) p(1 - p) p 1.96 P p 1.96 n n 3) Confidence Interval for Population Proportion-Large Sample-Example In a health survey involving a random sample of 75 patients who developed a particular illness, 70% of them are cured of this illness by a new drug. Establish the 95% confidence interval for the population proportion of all the patients who will be cured by the new drug. This would help assess the market potential for this new drug by a pharmaceutical company. 3) Confidence Interval for Population Proportion-Large Sample-Solution to the Example Applying the formula for p 1.96 0.70 1.96 p(1 - p) p(1 - p) P p 1.96 n n = 0.70(1- 0.70) 0.70(1 - .70) P 0.70 1.96 75 75 0.5963 8037 P 0. = 4) Confidence Interval for Population Mean-Small Sample-t Distribution 4) Confidence Interval for Population Mean-Small Sample The 1-confidence interval for the population mean is given by X n t1 t n 1 S S n μX t1 n n is the value of the t distribution with n -1 degrees of freedom for an area of in both /2 the tails of the distribution. 4) Confidence Interval for Population Mean-Small Sample-Example The average travel time taken based on a random sample of 10 people working in a company to reach the office is 40 minutes with a standard deviation of 10 minutes. Establish the 95% confidence interval for the mean travel time of everyone in the company? This will help the company redesign the working hours. 4) Confidence Interval for Population Mean-Small Sample-Solution Substituting in the formula below the values of X 40 S 10 .16 3 n 10 X n t1 S S n μX t1 n n t9 is the value of the t distribution with 9 degrees of freedom for an area of 0.025 in both the tails of the distribution. It is = 2.26 4) Confidence Interval for Population Mean-Small Sample-Solution On simplification, we have 32.86 .14 μ 47 The probability that the true mean travel time of everyone in the office will fall in this interval of 32.86 .14 is 95% μ 47 5) How to Determine Sample Size using Confidence Interval Sample Size -Population Mean 2 Z2 σ n 2 E E Sampling Error Xμ Sample Size –Population Proportion Z 2 p(1 ) p n E2 E Sampling Error - P p 5) How to Determine Sample Size using Confidence Interval Example 1 A marketing manager of a fast food restaurant in a city wishes to estimate the average yearly amount that families spend on fast food restaurants. He wants the estimate to be within Rs 100 with a confidence level of 99%. It is known from an earlier pilot study that the standard deviation of the family expenditure on fast food restaurant is Rs 500. How many families must be chosen for this problem? 5) How to Determine Sample Size using Confidence Interval Example 1-Solution 2 Z2 σ , we have Applying the formula, n E2 2.58 2 (5002 ) n 100 2 . Simplifying we find n = 166 5) How to Determine Sample Size using Confidence Interval Example 2 A company manufacturing sports goods wants to estimate the proportion of cricket players among high school students in India. The company wants the estimate to be within 0.03 with a confidence level of 99%. A pilot study done earlier reveals that out of 80 high school students, 36 students play cricket. What should be the sample size for this study? 5) How to Determine Sample Size using Confidence Interval Example 2-Solution p=36/80 =0.45 Applying the formula for calculating the sample size Z 2 p(1 ) p n . Substituting, we have 2 E 2.582 (0.45)(1 ) 0.45 n 0.032 n = 1831 . Simplifying, we have ...
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