BUSINESS MATHEMATICS & STATISTICS

Estimating from Samples: Inference LECTURE 42 Estimating from Samples: Inference Part 2

EXAMPLE In a factory 25% workforce is women How likely is it that a random sample of 80 workers contains 25 or more women? Mean = 25% STEP = [25(100 – 25)/80]^1/2 = [(25 x 75)/80]^1/2 = 4.84% % women in sample = (25/80) x 100 = 31.25% z = (31.25 – 25)/4.84 = 1.29 P(sample contains 25 women) : (Look for p against z = 1.29) = 0.985 or about 10%

What is the probability that such a sample will arise? APPLICATIONS OF STEP What is the probability that such a sample will arise? Estimate the percentage P from information obtained from a single sample How large a sample will be required in order to estimate a population percentage with a given degree of accuracy?

CONFIDENCE LIMITS A market researcher wishes to conduct a survey to determine % consumers buying company’s products He selects a sample of 400 consumers at random He finds that 280 of these are purchasers of the product What can he conclude about % of all consumers buying the product?

CONFIDENCE LIMITS First let us decide some limits It is common to use 95% confidence limits These will be symmetrically placed around the 70% buyers In a normal sampling distribution 2.5% corresponds to a z-value of 1.96 on either side of 70% Now the sample percentage of 70% can be used as an approximation for population percentage P STEP = [(70(100 – 70)/400]^1/2 = 2.29%

CONFIDENCE LIMITS Estimate for population percentage = 70 +/- 1.96 x STEP Or 70 +/- 1.96 x 2.29 = 65.515 and 74.49% as the two limits = 85% confidence interval We can round off 1.96 to 2 Then with 95% confidence we estimate the population percentage with that characteristic as lying in the interval P +/- 2 x STEP

A sample of 60 students contains 12 who are left handed or 20% EXAMPLE A sample of 60 students contains 12 who are left handed or 20% Find the range with 95% confidence in which the entire left handed students fall Range = 20 +/- 2 x STEP = 20 +/- 2 x [(20 x 80)/60]^1/2 = 9.67% and 30.33%

ESTIMATING PROCESS SUMMARY Identify n and P (the sample size and percentag in the sample Calculate STEP using these values The 95% confidence interval is approximately P +/- 2 STEP 99% confidence Z-value = 2.58

FINDING A SAMPLE SIZE To satisfy 95% confidence 2 x STEP = 5 Pilot survey value of P = 30% STEP = [(30 x 70)/n]^1/2 = 2.5 Solving n = 336 We must interview 336 persons to be 95% confident that our estimate is within 5% of the true answer

Distribution OF SAMPLE MEANS The standard deviation of the sampling distribution of means is called STandard Error of the Mean STEM STEM = s.d/(n)^1/2 s.d denotes standard deviation of the population n is the size of the sample

EXAMPLE What is the probability that if we take a random sample of 64 children from a population whose mean IQ is 100 with a StDev of 15, the mean IQ of the sample will be below 95? s = 15; n= 64; population mean = 100 STEM = 15/(64)^1/2 = 15/(64)^1/2 = 15/8 = 1.875 z = 100 – 95 /STEM = 5/1.875 = 2.67 This gives a probability of 0.0038 So the chance that the average IQ of the sample is below 95 is very small

BUSINESS MATHEMATICS & STATISTICS