Week 3

Chapter 17

Determination of Sample Size: A Review of Statistical Theory

AT-A-GLANCE

I. Introductions

A. Descriptive and inferential statistics

B. Sample statistics and population parameters

II. Making Data Usable

A. Frequency distributions

B. Proportions

C. Measures of central tendency

· The mean

· The median

· The mode

D. Measures of dispersion

· The range

· Why use the standard deviation?

· Variance

· Standard deviation

III. The Normal Distribution

IV. Population Distribution, Sample Distribution, and Sampling Distribution

V. Central-Limit Theorem

VI. Estimation of Parameters

A. Point estimates

B. Confidence intervals

VII. Sample Size

A. Random error and sample size

B. Factors in determining sample size for questions involving means

C. Estimating sample size for questions involving means

D. The influence of population size on sample size

E. Factors in determining sample size for proportions

F. Calculating sample size for sample proportions

G. Determining sample size on the basis of judgment

H. Determining sample size for stratified and other probability samples

I. Determining level of precision after data collection

VIII. A Reminder About Statistics

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

CHAPTER VIGNETTE: Federal Reserve Finds Cards Are Replacing Cash

Payment options have gone high tech, and today’s spenders are more likely to pay with a debit or credit card or through a variety of methods for electronic transfer of funds. Researchers at the Federal Reserve conducted surveys of depository institutions (i.e., banks, savings and loans, credit unions) asking them to report the number of each type of payment the institutions processed. The Fed’s carefully designed the sample, including the number of institutions to contact. The total number of institutions was known—14,117—and the researchers had to select enough from this population to be confident that the answers would be representative of transactions nationwide. A stratified random sample was used so that each type of institution would be included. They determined that 2,700 institutions would need to be sampled so that they could say that the results, with 95 percent confidence, were accurate to within (5 percent of the responses. 1,500 institutions responded, and their responses confirmed earlier analysis showing that the number of checks is declining while the number of electronic payments is increasing.

SURVEY THIS!

Students are asked to consider the question on the survey that asks if the respondent is employed:

1. What percentage of respondents do you think will answer “yes”?

2. Based on your estimate, how many respondents would you need to be 95 percent confident your responses are ±5 percent of the population proportion?

3. Look at the data from your class. At the 95 percent confidence level, how precise is the measure regarding employment status?

Consider the question that asks how “interesting” or “boring” they feel their life is:

4. Review the “rule of thumb” provided in the chapter regarding estimating the value of the standard deviation of a scale. How many respondents would you need to be 95 percent confident your responses are ±5 percent of the population proportion?

5. What if you want to be 99 percent confident—what would the sample size have to be?

6. Look at the data from your class. At the 95 percent confidence level, how precise is this measure?

RESEARCH SNAPSHOTS

· The Well-Chosen Average

The average pay of the people who work in an establishment may mean something or it may not. If the average is a median that means half the employees make more than that and half make less. But if it is a mean, you may be getting nothing more than the average of one large income (i.e., the proprietor’s) and the salaries of a crew of underpaid workers. A simple example is given where the “average wage” is $57,000. However, the mode, which is the most common and occurs most frequently, is $20,000, and the median is $30,000, which means that half the people earn more than this and half earn less. Do politicians use statistics to lie or do the statistics lie when they claim that the “rich do not pay taxes”? Two websites are given to look at the tax figures.

· Sampling the World

The Gallup organization’s WorldView generates a comprehensive snapshot of the opinions of people from across the globe. The company collects data from over 150 countries on various topics using representative samples from 98 percent of the world’s adult population, with samples generally consisting of 1,000 individuals. Gallup’s WorldView program provides extremely valuable information on poorer and more rural populations. To accurately reflect the world’s sentiments and opinions, you must have an accurate sample of the world’s population.

· Target and Wal-Mart Shoppers Really Are Different

Scarborough Research sampled over 200,000 adults to compare consumers who shop exclusively at either Target or Wal-Mart. The largest share (40 percent) named both stores when asked to identify the stores at which they had shopped during the preceding three months. However, 31 percent shopped at Wal-Mart but not Target, and 12 percent shopped at Target but not Wal-Mart. Target-only shoppers were younger and more likely to have a high household income, were more likely to shop at more upscale stores, and to visit many different stores than the Wal-Mart-only shoppers. The Wal-Mart shoppers were more likely to shop at discounters (i.e., Dollar General and Kmart) and were more likely to be at least 50 years old. Given a U.S. adult population of approximately 220 million, do you think the sample size was adequate to make these comparisons?

OUTLINE

I. INTRODUCTION

· Descriptive and Inferential Statistics

· There are two applications of statistics: (1) to describe characteristics of the population or sample (descriptive statistics) and (2) to generalize from the sample to the population (inferential statistics).

· Sample Statistics and Population Parameters

· Sample statistics are variables in the sample or measures computed from the sample data.

· Population parameters are variables or measured characteristics of the population.

· We will generally use Greek lowercase letters to denote population parameters (e.g., ( or () and English letters to denote sample statistics (e.g., X or S).

II. MAKING THE DATA USABLE

· To make data usable, this information must be organized and summarized.

· Methods for doing this include:

· frequency distributions

· proportions

· measures of central tendency and dispersion

· Frequency Distributions

· Constructing a frequency table or frequency distribution is one of the most common means of summarizing a set of data.

· The frequency of a value is the number of times a particular value of a variable occurs.

· Exhibit 17.1 represents a frequency distribution of respondents’ answers to a question asking how much customers had deposited in the savings and loan.

· It is also quite simple to construct a distribution of relative frequency, or a percentage distribution, which is developed by dividing the frequency of each value by the total number of observations, and multiply the result by 100 (see Exhibit 17.2).

· Probability is the long-run relative frequency with which an event will occur.

· Inferential statistics uses the concept of a probability distribution, which is conceptually the same as a percentage distribution except that the data are converted into probabilities (see Exhibit 17.3).

· Proportions

· A proportion indicates the percentage of population elements that successfully meet some standard on the particular characteristic.

· May be expressed as a percentage, a fraction, or a decimal number.

· Measures of Central Tendency

· There are three ways to measure the central tendency, and each has a different meaning.

· Mean

· The mean is simply the arithmetic average, and it is a common measure of central tendency.

· It is the sum of all the observations divided by the number of observations.

· Often we will not have enough data to calculate the population mean,

m

, so we will calculate a sample mean,

image2.wmf

X

(read as “X bar”).

· Median

· The median is the midpoint of the distribution, or the 50th percentile.

· In other words, the median is the value below which half the values in the sample fall.

· Mode

· The mode is the measure of central tendency that merely identifies the value that occurs most often.

· Determined by listing each possible value and noting the number of times each value occurs.

· Measures of Dispersion

· Accurate analysis of data also requires knowing the tendency of observations to depart from the central tendency.

· Thus, another way to summarize the data is to calculate the dispersion of the data, or how the observations vary from the mean.

· There are several measures of dispersion discussed below.

· Range

· The simplest measure of dispersion.

· It is the distance between the smallest and largest values of a frequency distribution.

· Does not take into account all the observations; it merely tells us about the extreme values of the distribution.

· While we do not expect all observations to be exactly like the mean, in a skinny distribution they will be a short distance from the mean, while in a fat distribution they will be spread out (see Exhibit 17.6).

· The interquartile range is the range encompassing the middle 50 percent of the observations (i.e., the range between the bottom quartile and the top quartile).

· Why Use the Standard Deviation?

· It is perhaps the most valuable index of spread, or dispersion.

· Learning about the standard deviation will be easier if we present several other measures of dispersion that may be used. Each of these has certain limitations that the standard deviation does not.

· Deviation—a method of calculating how far any observation is from the mean.

· Average deviation – determined by calculating the deviation score of each observation value (i.e., its difference from the mean) and summing up each score; then we divide by the sample size (n).

· While this means of calculating a measure of spread seems of interest, it is never used because the positive deviation scores are always canceled out by the negative deviation scores, thus leaving an average deviation value of zero.

· One might correct the disadvantage of the average deviation by computing the absolute values of the deviations. We could ignore all the positive and negative signs and utilize only the absolute values of each deviation to give us the mean absolute deviation, but there are some technical mathematical problems that make the mean absolute deviation less valuable than some other measures.

· Variance

· This is another means of eliminating the sign problem caused by the negative deviations canceling out the positive deviations.

· Useful for describing the sample variability.

· This procedure is to square the deviation scores and divide by the number of observations. (In actual fact n - 1 rather than n is used in most pragmatic business research problems.)

· A very good index of the degree of dispersion.

· The variance, S2, will be equal to zero if—and only if—each and every observation in the distribution is the same as the mean.

· Standard Deviation

· The variance does have one major drawback—it reflects a unit of measurement that has been squared.

· Because of this, statisticians have taken the square root of the variance.

· The square root of the variance for distribution is called the standard deviation.

· Exhibit 17.7 illustrates the calculation of a standard deviation.

· S is the symbol for the sample standard deviation, while

s

is the symbol for the population standard deviation.

III. THE NORMAL DISTRIBUTION

· One of the most common probability distributions in statistics is the normal distribution, commonly represented by the normal curve.

· Bell shaped and almost all (99 percent) of its values are within ( 3 standard deviations from its mean.

· The standardized normal distribution is a specific normal curve that has several characteristics

1. It is symmetrical about its mean.

2. The mode identifies its highest point, which is also the mean and median, and vertical line about which this 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 1.0.

4. The standardized normal distribution has a mean of 0 and a standard deviation of 1.

· Exhibit 17.9 illustrates these properties, and Exhibit 17.10 is a summary version of the typical standardized normal table found at the end of most statistics textbooks.

· The standardized normal distribution is a purely theoretical probability distribution, but it is the most useful distribution in inferential statistics.

· The standardized normal distribution is extremely valuable because we can translate or transform any normal variable, X, into the standardized value, Z.

· This has many pragmatic implications for the business researcher.

· The standardized normal table in the back of most statistics and research books allows us to evaluate the probability of the occurrence of certain events without any difficulty.

· Computing the standardized value, Z, of any measurement expressed in original units is simple:

· Subtract the mean from the value to be transformed, and divide by the standard deviation (all expressed in original units).

· In the formula note that

s

, the population standard deviation, is used for calculation:

Z

=

X

-

m

s

where ( = hypothesized or expected value of the mean

· Example: Suppose that a toy manufacturer has experienced mean sales,

m

, of 9,000 units and a standard deviation,

image7.wmf

s

, of 500 units during the month of September. The production manager wishes to know if wholesalers will demand between 7,500 and 9,625 units during the month of September in the upcoming year. Because there are no tables in the back of our textbook showing the distribution for a mean of 9,000 and a standard deviation of 500, we must transform our distribution of toy sales, X, into the standardized form utilizing our simple formula.

Z

=

7

,

500

-

9

,

000

500

=

-

3

.

00

Z

=

9

,

625

-

9

,

000

500

=

1

.

25

When Z = 3.00, the area under the curve (probability) equals .499

When Z = 1.25, the area under the curve (probability) equals .394

Thus, the total area under the curve is .499 + .394 = .893

The area under the curve portraying this computation is the shaded area in Exhibit 17.12. Thus, the sales manager knows there is a .893 probability that sales will be between 7,500 and 9,625.

IV. POPULATION DISTRIBUTION, SAMPLE DISTRIBUTION, AND SAMPLING DISTRIBUTION

· Three additional types of distribution must be defined:

1. population distribution

2. sample distribution

3. sampling distribution

· A frequency distribution of the population elements is called a population distribution.

· The population distribution has its mean and standard deviation represented by the Greek letters

m

and

image11.wmf

s

.

· A frequency distribution of a sample is called the sample distribution.

· The sample mean is designated with

X

, and the sample standard deviation is designated S.

· However, we must now introduce another distribution: the sampling distribution of the sample mean.

· A sampling distribution is a theoretical probability that shows the functional relation between the possible values of some summary characteristic of n cases drawn at random and the probability (density) associated with each value over all possible samples of size n from a particular population.

· The sampling distribution’s mean is called the expected value of the statistic.

· The expected value of the mean of the sampling distribution is equal to

m

.

· The standard deviation of the sampling distribution is called the standard error of the mean (

S

X

) and is approximately equal to

image15.wmf

s

n

.

· Exhibit 17.13 shows the relationship among a population distribution, the sample distribution, and three sampling distributions of varying sample size.

· As sample size increases, the spread of the sample mean around ( decreases (i.e., larger samples will have a skinnier sampling distribution).

V. CENTRAL-LIMIT THEOREM

· The central-limit theorem states: As the sample size, n, increases, the distribution of the mean,

X

, of a random sample taken from practically any population approaches a normal distribution (with a mean

image17.wmf

m

and a standard deviation,

image18.wmf

s

n

).

· The central-limit theorem works regardless of the shape of the original population distribution (see Exhibit 17.14).

· This theoretical knowledge about distributions can be used to solve two very practical business research problems:

1. estimating parameters

2. determining sample size

VI. ESTIMATION OF PARAMETERS

· Point Estimates

· Our goal in using statistics is to make an estimate about the population parameters.

· The population mean,

m

, and standard deviation,

image20.wmf

s

, are constants, but in most instances of business research they are unknown.

· To estimate the population values, we are required to sample.

· Example: To estimate the average number of people participating in racquetball in one week we may take a sample of 300 racquetball players throughout the area where our researcher is thinking of building club facilities. If the sample mean,

X

, equals 2.6 days per week, we may use this figure as a point estimate.

· This single value, 2.6, is the best estimate of the population mean.

· However, we would be extremely lucky if the sample estimate were exactly the same as the population value.

· A less risky alternative would be to calculate a confidence interval.

· Confidence Intervals

· A confidence interval estimate is based on the knowledge that

m

=

X

±

a small sampling error.

· After calculating an interval estimate, we can determine how probable it is that the population mean will fall within this range of statistical values.

· In the racquetball example the researcher, after setting up a confidence interval, would be able to make a statement such as “with 95 percent confidence, I think that the average number of days played per week is between 2.3 and 2.9.”

· The researcher has a certain confidence that the interval contains the true value of the population mean.

· The crux of the problem for the researcher is to determine how much random sampling error to tolerate.

· The confidence level is a percentage or decimal that indicates the long-run probability that the results will be correct.

· Traditionally, researchers have utilized the 95 percent confidence level.

· The confidence interval gives the estimated value of the population parameter, plus or minus an estimate of the error:

m

=

X

±

a small sampling error (E)

where E =

Z

c

.

1

.

S

X

Z

c

.

1

.

= the value of Z, our standardized normal variable at a specified confidence level.

S

X

= the standard error of the mean.

· The following is a step-by-step procedure for calculating confidence intervals:

1. Calculate

X

from the sample.

2. Assuming

s

is unknown, estimate the population standard deviation by finding S, the sample deviation.

3. Estimate the standard error of the mean, using the following formula:

S

X

=

S

n

.

4. Determine the Z-value associated with the confidence level desired. The confidence level should be divided by 2 to determine what percentage of the area under the curve must be included on each side of the mean.

5. Calculate the confidence interval.

· Sample statistics, such as the sample means,