1. Introduction to Marketing Research
CHAPTERS 13 : SAMPLE SIZE SELECTION AND BASIC MEASURES OF CENTRAL TENDENCY
Idil Yaveroglu Lecture Notes

2. What do Statistics Mean?
Descriptive statistics
Number of people
Trends in employment
Data
Inferential statistics
Make an inference about a population from a sample
Determining Sample Size

3. Measures of Central Tendency
Mean - arithmetic average
Median - midpoint of the sorted distribution
Mode - the value that occurs most often
Determining Sample Size

4. Measures of Central Tendency
Number of
Salesperson Sales calls
Mike
Patty
Billie
Bob
John
Frank
Chuck
Samantha 26

5. In the previous example measures of central tendency are calculated as follows…
Sample mean
Xi n
Xi = X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 =
= = 26
Xi 26 =
= 3.25 n 8
Sample median = 3
Mode = 3

6. In the previous example measures of dispersion are calculated as follows…
Range = largest value – smallest value = 5 – 1= 4
Interquartile Range = middle 50 percent of observations
= 4.5 – 2.5 = 2
Deviation scores
- Variance = S2 =
(Xi-X)2 =
n - 1
- Standard Deviation = = =

7. Calculating the Standard Deviation and Variance

8. Low Dispersion 5 4 3 2 Frequency 1 Value on Variable
Determining Sample Size

9. High dispersion 5 4 3 2 1
Frequency
Value on Variable

10. The Normal Distribution
Normal curve
Bell-shaped
Almost all of its values are within plus or minus 3 standard deviations
IQ is an example
Determining Sample Size

11. Standardized Normal Distribution
SYMMETRICAL ABOUT ITS MEAN
MEAN IDENTIFIES HIGHEST POINT
INFINITE NUMBER OF CASES - A CONTINUOUS DISTRIBUTION
AREA UNDER CURVE HAS A PROBABILITY DENSITY = 1.0
MEAN OF ZERO, STANDARD DEVIATION OF 1

12. Standardized Normal Distribution
Pr(Z) -3 -2 -1 1 2 3 Z

13. An Example of the Distribution of Intelligence Quotient (IQ) Scores
13.59%
13.59%
34.13%
34.13%
2.14%
2.14% 70 85
100
115
130 IQ

14. Definitions and Symbols
POPULATION DISTRIBUTION – frequency distribution of the elements of the population
Parameter: A parameter is a summary description of a fixed characteristic or measure of the target population. A parameter denotes the true value which would be obtained if a census rather than a sample was undertaken.
SAMPLE DISTRIBUTION – A frequency distribution of the sample
Statistic: A statistic is a summary description of a characteristic or measure of the sample. The sample statistic is used as an estimate of the population parameter.
Random sampling error: The error when the sample selected is an imperfect representation of the population of interest.
SAMPLING DISTRIBUTION – A theoretical probability distribution of sample means for all possible samples of a certain size drawn from a particular population
Standard error of the mean – The standard deviation of the sampling distribution

15. Symbols for Population and Sample Variables
Mean μ
Proportion π p
Variance
Standard deviation σ s
Size N n
Standard error of the mean
Standard error of the proportion
Standardized variate (z)

16. Central Limit Theorem
The central limit theorem states that as a sample size increases, the distribution of sample means of size n, randomly selected approaches a normal distribution

18. Definitions and Symbols
Precision level: When estimating a population parameter by using a sample statistic, the precision level is the desired size of the estimating interval. This is the maximum permissible difference between the sample statistic and the population parameter.
Confidence interval: The confidence interval is the range into which the true population parameter will fall, assuming a given level of confidence.
Confidence level: The confidence level is the probability that a confidence interval will include the population parameter.

19. The Confidence Interval Approachz

20. The Confidence Interval Approach (Cont.)

21. 95% Confidence Interval
0.475

22. Area for 95%=
0.5-(.05/2)=.475
Z=1.96

23. The Confidence Interval Approach (Cont.)
The confidence interval is given by
We can now set a 95% confidence interval around the sample mean of $182. Assume that sample size is 300, and σ=55.
The 95% confidence interval is given by
= (3.18) =
Thus the 95% confidence interval ranges from $ to $

24. Confidence Interval
Suppose you plan to open a sporting good store to cater to working women who golf. In a survey of 100 women you find that:
Construct a 95% confidence interval. (z-value for 95% is 1.96)

25. Example: Determining CI for Population Mean
We are interested in the mean starting income of Koc University MBAs so we go out and obtain a SRS of 100 MBAs. The mean starting income in our sample is TL115,000 with standard deviation (s.d.) equal to TL20,000. Construct a 95% confidence interval (CI) of the true population mean starting income of all Koc MBAs.
Answer: = 115,000, = 20,000, n = 100, = 1.96
= ($11,1080 , $11,8920)

26. Confidence Interval for proportion p
A proportion is a mean where each observation is a 0 or 1, so we can compute a CI for a population
We can construct a 100(1- q)% CI for population mean p by computing
where
= the sample proportion
n = the sample size
= , 1.96, for a 90%, 95% and 99% confidence interval

27. Example: Determining CI for Proportion
We are interested in the proportion of Koc students who have tried a new soup. We go out and obtain an SRS of 400 students and ask them if they have tried the new soup. Suppose 20% of the 400 students say they have tried it. Construct a 90% confidence interval (CI) of the true population market share.
Answer:
(0.1671, )

28. Sample Size Three factors are required to determine sample size
VARIANCE (STANDARD DEVIATION)
MAGNITUDE OF ERROR
CONFIDENCE LEVEL

29. The Confidence Interval Approach and Determining Sample Size
Means
Proportions

30. The Size of the Sample (using Mean)
What sample size do I need so that if I construct a 100(1-q)% confidence interval it will be an interval with width D ?
Compute Size of the Sample
Determine q the desired CI you want, e.g., = is a 90% CI
Determine the interval width you want, e.g.,
Compute an estimate of the standard deviation of the population of interest, denoted by s.
Compute the necessary sample size

31. Where do you get ‘s’ from?
Historical data
Predicted worst case scenario (largest possible)
Pre-tests / previous surveys

32. Ex.: Size of the sample (using Mean)
Suppose you want a 95% CI for mean MBA starting salaries such that the interval will have a width of $5000. Using the results of the previous example where s = $20,000, how many MBAs do I need to obtain in my SRS to obtain this desired width?
Answer:
Ideally, you need to do this for
every question in your survey
and then take the maximum
 61.5

33. Sample Size Determination for Means and Proportions
Steps
Means
Proportions
1. Specify the level of precision.
2. Specify the confidence level (CL).
CL = 95%
3. Determine the z value associated with the CL.
Z value is 1.96

34. Sample Size Determination for Means and Proportions
Steps
Means
Proportions
4. Determine the standard deviation of the population
Estimate σ
σ = 55
Estimate π
π = 0.64
5. Determine the sample size using the formula for the standard error
6. If necessary, reestimate the confidence interval by employing s to estimate σ

35. A sample size of 400 is enough to represent China’s more than 1
A sample size of 400 is enough to represent China’s more than 1.3 billion people or the more than 300 million American people.
The sample size is independent of the population size for large populations.

36. Adjusting the Statistically Determined Sample Size
Incidence rate refers to the rate of occurrence or the percentage of persons eligible to participate in the study.
In general, if there are c qualifying factors with an incidence of Q1, Q2, Q3, ...QC, each expressed as a proportion,
Incidence rate = Q1 x Q2 x Q3....x QC
Initial sample size = Final sample size
Incidence rate x Completion rate