1. BUSINESS MATHEMATICS
&
STATISTICS

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

3. 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)
= or about 10%

4. 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?

5. 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?

6. 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%

7. CONFIDENCE LIMITS Estimate for population percentage =
70 +/ x STEP Or
70 +/ x 2.29
= 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

8. 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%

9. 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

10. 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

11. 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

12. 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
So the chance that the average IQ of the sample is below 95 is very small

13. BUSINESS MATHEMATICS
&
STATISTICS