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1 Business Research Methods 9e Zikmund Babin Carr Griffin 17 Determination of Sample Size A Review of Statistical Theory Chapter 17 Determination of Sample Size: A Review of Statistical Theory 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2 LEARNING OUTCOMES 1. Understand basic statistical terminology 2. Interpret frequency distributions, proportions, and measures of central tendency and dispersion 3. Distinguish among population, sample, and sampling distributions 4. Explain the central-limit theorem 5. Summarize the use of confidence interval estimates 6. Discuss major issues in specifying sample size 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17-2

3 Federal Reserve Finds Cards are Replacing Cash Payment options have gone high-tech and research supports that. The Federal Reserve conducted surveys of depository institutions, asking them to report the number of each type of payment the institutions processed. A sample of 2,700 was drawn from the 14,117 institutions in the population, with a 95% confidence interval and accuracy of ±5 percent Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17-3

4 Introduction Descriptive Statistics Describe characteristics of populations or samples. Inferential Statistics Make inferences about whole populations from a sample. Sample Statistics Variables in a sample or measures computed from sample data. Population Parameters Variables in a population or measured characteristics of the population Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17 4

5 Making Data Usable Frequency Distribution A set of data organized by summarizing the number of times a particular value of a variable occurs. Percentage Distribution A frequency distribution organized into a table (or graph) that summarizes percentage values associated with particular values of a variable. Probability The long-run relative frequency with which an event will occur Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17 5

6 EXHIBIT 17.1 Frequency Distribution of Deposits 17 6

7 EXHIBIT 17.2 Percentage Distribution of Deposits 17 7

8 EXHIBIT 17.3 Probability Distribution of Deposits 17 8

9 The Well-Chosen Average Average pay could mean there s a highly paid executive and many lowpaid employees. Median may be more informative Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17-9

10 Making Data Usable (cont d) Proportion The percentage of elements that meet some criterion Measures of Central Tendency Mean: the arithmetic average. Median: the midpoint; the value below which half the values in a distribution fall. Mode: the value that occurs most often. Population Mean Sample Mean 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

11 EXHIBIT 17.4 Number of Sales Calls Per Day by Salesperson 17 11

12 Measures of Dispersion The Range The distance between the smallest and the largest values of a frequency distribution Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

13 EXHIBIT 17.5 Sales Levels for Two Products with Identical Average Sales 17 13

14 EXHIBIT 17.6 Low Dispersion versus High Dispersion 17 14

15 Measures of Dispersion (cont d) Why Use the Standard Deviation? Variance A measure of variability or dispersion. Its square root is the standard deviation. Standard deviation A quantitative index of a distribution s spread, or variability; the square root of the variance for a distribution. The average of the amount of variance for a distribution. Used to calculate the likelihood (probability) of an event occurring Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

16 Calculating Deviation Standard Deviation = 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

17 EXHIBIT 17.7 Calculating a Standard Deviation 17 17

18 The Normal Distribution Normal Distribution A symmetrical, bell-shaped distribution (normal curve) that describes the expected probability distribution of many chance occurrences. 99% of its values are within ± 3 standard deviations from its mean. Example: IQ scores Standardized Normal Distribution A purely theoretical probability distribution that reflects a specific normal curve for the standardized value, z Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

19 The Normal Distribution (cont d) Characteristics of a Standardized Normal Distribution 1. It is symmetrical about its mean. 2. The mode identifies the normal curve s highest point, which is also the mean and median, and the vertical line about which this normal curve is symmetrical. 3. The normal curve has an infinite number of cases (it is a continuous distribution), and the area under the curve has a probability density equal to The standardized normal distribution has a mean of 0 and a standard deviation of Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

20 EXHIBIT 17.8 Normal Distribution 17 20

21 The Normal Distribution (cont d) Standardized Values Used to compare an individual value to the population mean in units of the standard deviation 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

22 EXHIBIT 17.9 Standardized Normal Distribution 17 22

23 EXHIBIT Standardized Values Can Be Computed from Flat or Peaked Distributions Resulting in a Standardized Normal Curve 17 23

24 EXHIBIT Standardized Distribution Curve 17 24

25 Population Distribution, Sample Distribution, and Sampling Distribution Population Distribution A frequency distribution of the elements of a population. Sample Distribution A frequency distribution of a sample. 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 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

26 EXHIBIT Fundamental Types of Distributions 17 26

27 Central-limit Theorem Central-limit Theorem The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

28 EXHIBIT Distribution of Sample Means for Samples of Various Sizes and Population Distributions 17 28

29 Estimation of Parameters and Confidence Intervals Point Estimates An estimate of the population mean in the form of a single value, usually the sample mean. Gives no information about the possible magnitude of random sampling error. Confidence Interval Estimate A specified range of numbers within which a population mean is expected to lie. An estimate of the population mean based on the knowledge that it will be equal to the sample mean plus or minus a small sampling error Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

30 Sampling the World Gallup s WorldView generates a comprehensive snapshot of the opinions of people from across the globe. Collects data from over 150 countries, generally through samples of 1,000 individuals in each. Provides valuable information on poorer and more rural populations Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

31 Confidence Intervals Confidence Level A percentage or decimal value that tells how confident a researcher can be about being correct. It states the long-run percentage of confidence intervals that will include the true population mean. The crux of the problem for a researcher is to determine how much random sampling error to tolerate. Traditionally, researchers have used the 95% confidence level (a 5% tolerance for error) Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

32 Calculating a Confidence Interval Approximate location (value) of the population mean Estimation of the sampling error 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

33 Calculating a Confidence Interval (cont d) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

34 Target and Walmart Shoppers Really Are Different 40% of respondents named both Target and Walmart as places where they shop. But 30% shopped at Walmart and not Target and 12% shopped at Target but not Walmart. Target shoppers who shun Walmart shop at more upscale stores, while Walmart shoppers who shun Target shop at discounters Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

35 Sample Size Random Error and Sample Size Random sampling error varies with samples of different sizes. Increases in sample size reduce sampling error at a decreasing rate. Diminishing returns - random sampling error is inversely proportional to the square root of n Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

36 EXHIBIT Relationship between Sample Size and Error 17 36

37 Factors of Concern in Choosing Sample Size Variance (or Heterogeneity) A heterogeneous population has more variance (a larger standard deviation) which will require a larger sample. A homogeneous population has less variance (a smaller standard deviation) which permits a smaller sample. Magnitude of Error (Confidence Interval) How precise must the estimate be? Confidence Level How much error will be tolerated? 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

38 Estimating Sample Size for Questions Involving Means Sequential Sampling Conducting a pilot study to estimate the population parameters so that another, larger sample of the appropriate sample size may be drawn. Estimating sample size: 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

39 Sample Size Example Suppose a survey researcher, studying expenditures on lipstick, wishes to have a 95 percent confident level (Z) and a range of error (E) of less than \$2.00. The estimate of the standard deviation is \$ What is the calculated sample size? 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

40 Sample Size Example Suppose, in the same example as the one before, the range of error (E) is acceptable at \$4.00. Sample size is reduced Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

41 Calculating Sample Size at the 99 Percent Confidence Level 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

42 Determining Sample Size for Proportions 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

43 Determining Sample Size for Proportions (cont d) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

44 Calculating Example Sample Size at the 95 Percent Confidence Level p q = = n = (1. 96 ) 2 (. 035 (. ) 6 )(. 2 4 ) = = = ( )( ) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part

45 EXHIBIT 17.6 Low Dispersion versus High Dispersion 17 45

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