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Index to Module 5 Notes
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Module Notes 5.1 covered the basics of descriptive statistics for analysis of categorical data.
Pause
and Reflect
We describe data for
categorical variables by counting the observations of interest within the class
or event divisions of the categorical variable. We can convert the count into a
probability (also called long term relative frequency, proportion or percent)
by dividing the number of observations in the event of interest by the total
observations from all the events in the sample space. When we are interested in
studying the relationship between two categorical variables, we can exam joint
and marginal events; and joint, marginal and conditional probabilities as well.
When the data collected for a
categorical variable represents a sample, from a survey for example, we are
often interested in making an inference from the sample to the population.
Recall that the tools of inferential statistics are confidence intervals and
tests of hypothesis.
One of the most popular applications of confidence intervals for the population
proportion is the manner in which public opinion polls are reported. Often
following a major political news story, the media sponsors public opinion polls
via the telephone. Typically 1,000 to 2,000 registered voters are called across
the country and asked questions like "do you agree or disagree with _____
on ______." At the 11p.m. news, we then hear something like "55% of
registered votes in the US agree with ___ on ____." The really good polling
organizations then follow the statement with, "the margin of error is
3%." Guess what - that's a confidence interval for a proportion!!
Confidence
Interval Estimation for the Proportion
We are interested in
using a sample proportion to estimate the population proportion. We do this
similar to any confidence interval.
Pause
and Reflect
A confidence interval
for the true population parameter being estimated is the sample estimate of
that parameter plus/minus the sampling error. The sampling error is a function
of the confidence level, the variation in the data, and the sample size.
The population parameters of
interest up to now were the mean, median and variance of numerical variables, and
the slope and dependent variable of regression relationships. The population
parameter of interest to us now is the population proportion, p. The sample
statistic used to estimate p is the sample proportion, ps. The
sample proportion is determined by dividing the count of observations in the
event class of interest by the total observations in all event classes in the
sample space.
For example, suppose we surveyed 1,000 shoppers at Kmart and asked them to rate
their shopping experience as excellent, good or poor. Here, there are three
event classes in the categorical variable called "rating." Further
suppose that of the 1,000 shoppers, 272 rated the experience as excellent.
Equation 5.2.1 shows the calculation for the sample proportion:
Eq. 5.2.1: ps = X / n = 272 / 1,000 = 0.272 or 27.2%
where X = the count of observations within the event class of interest
Technically, when a categorical variable of interest has only two outcomes, such as success or failure, male or female, success or failure, excellent rating or not excellent rating, that categorical variable follows a distribution called the binomial distribution. Any sampling distribution used in estimating a population parameter has a mean and a standard error. For a sampling distribution that follows the binomial distribution, the mean is ps and the standard error is:
Eq.
5.1.2: Standard Error of ps
= Square Root [ ps * ( 1-ps) / n]
= Sq Rt [ 0.272 * (1 - 0.272) / 1,000] = 0.014
As long as the sample size is
assumed to be sufficiently large, the binomial distribution is
approximated by the normal distribution which means that we can use the Z
statistic to form the sampling error portion of the confidence interval. The
test for "large" is that both nps and n(1-ps)
are at least 5. For this sample, we meet the "large" test since
nps = 1,000 * 0.272 = 272 and n * (1-ps) = 1,000 * (1-
0.272) = 728: 272 and 728 are both greater than 5.
Now we can put together the confidence interval for the population proportion.
The 95% confidence interval for the proportion is:
Eq. 5.2.3: 95% Confidence Interval for the True Population Proportion:
=
ps + Z * Std Error of ps
= ps + 1.96 * Sq Rt [ ps * (1 - ps) / n
]
= 0.272 + 1.96 * Sq Rt [0.272 * (1 - 0.272) / 1,000]
= 0.272 + 1.96 * 0.14
= 0.272 + 0.028
or 0.244 < p < 0.300
The interpretation: we are
95% confident that between 24.4% and 30.0% of this population rate the shopping
experience at Kmart as excellent. Although the math is fairly easy for a
confidence interval for a proportion, we can set up an Excel Worksheet template
as shown in Worksheet 5.2.1:
Worksheet 5.2.1
|
Row 1 |
Col I |
J |
K |
|
2 |
Confidence Interval for Proportion |
||
|
3 |
n |
1000 |
|
|
4 |
Successes |
272 |
|
|
5 |
Failures |
728 |
= J3 - J2 |
|
6 |
Confidence Level |
0.95 |
|
|
7 |
|||
|
8 |
Sample Proportion |
0.272 |
= J4 / J3 |
|
9 |
Z |
1.96 |
= NORMSINV(0.5+(J6/2)) |
|
10 |
Standard Error |
0.014 |
= SQRT((J8*(1-J8))/J3) |
|
11 |
Half-Width of C.I. |
0.028 |
= J9 * J10 |
|
12 |
Lower Limit of C.I. |
0.244 |
= J8 - J11 |
|
13 |
Upper Limit of C.I. |
0.300 |
= J8 + J11 |
Do you remember how the
analyst can make the sampling error portion of the confidence interval tighter
(smaller)? That's right - increase the sample size. What happens when we use a
sample size of 10,000:
Eq. 5.2.4: 95% Confidence Interval for the True Population Proportion:
=
ps + Z * Std Error of ps
= ps + 1.96 * Sq Rt [ ps * (1 - ps) / n
]
= 0.272 + 1.96 * Sq Rt [ 0.272 * (1 - 0.272) / 10,000 ]
= 0.272 + 1.96 * 0.0045
= 0.272 + 0.009
or 0.263 < p < 0.281
We have certainly come closer
to capturing the true population proportion - of course we have, we sampled
10,000 people! That can be expensive. So, most of the time, you hear polling
companies say, the sampling error (they actually say margin of error most of
the time) is + 3%. Exactly how large a sample do you need to get a
margin of error of exactly 3%?
Computational Formula for the Sample Size for Estimating the Proportion
To find this out, we solve the following sample size formula:
Eq. 5.2.5: n = [ Z2 * p * (1-p) ] / e2
where
e = the desired sampling error,
and p = true population proportion.
Often (most of the time) we do not know the true population proportion - if we did, we would not need to find the confidence interval! So, a conservative approach is to let p = 0.50, which results in the largest value for n. If we want to be 95% confident, and want a sampling error of 3%, the formula becomes:
Eq. 5.2.6: n = [1.962 * (0.50) * (1 - 0.50) ] / 0.032 = 1,067.
As I stated earlier, this is
the figure most often used by polling companies. Of course, to get 1,067
responses, you have to call more than 1,067 citizens to account for the non
responses.
That is how one estimates a population proportion using a confidence interval.
Let me close by noting what to do if you have a finite population.
Finite Population Correction Factor
Most populations we deal with are really finite - not infinite. Technically
speaking, the confidence interval formula for estimating the population
proportion applies to infinite populations. However, in reality, the formula
works just fine as long as the population is not too small compared to the
sample size. The rule of thumb is that as long as the sample size is less than
5% of the population size (Sincich, p. 412), there is no need to apply the
finite population correction factor (Levine, 2000).
For example. What if the Kmart store used in these notes is located in Cape
Coral, with an approximate population of 100,000. If the sample size is 1,000,
then the ratio of sample size to population is 1,000/100,000 or 1% which passes
the above rule of thumb. On the other hand, if you planned to survey the 100
winter residents of Upper Captiva on some issue, and your sample size was 50, then
the ratio of sample size to population size is 50% and the finite population
correction factor should be applied.
Any time we have to adjust for a finite population, the confidence interval
formula becomes:
Eq. 5.2.7: 95% Confidence Interval for the True Population Proportion with Finite Population Correction Factor:
= ps + Z * Std Error of ps * Sq Rt [ (N -
n) / (N - 1)]
= ps + 1.96 * Sq Rt [ps * (1 - ps) / n
] * Sq Rt [ (N - n) / (N - 1)]
The sample size formula becomes:
Eq. 5.2.8: n = no * N / [ no + (N - 1 ) ] where no is the sample size determined from Eq. 5.2.5
The other tool of inferential statistics is the test of hypothesis, this time
for the true population proportion.
Test
of Hypothesis for the Proportion
Suppose we wanted to
test if the true population proportion of Kmart shoppers rating the shopping
experience at Kmart as excellent was equal to 32% versus not equal to 32%. We
would proceed as with any hypothesis test, starting with the hypotheses:
H0:
p = 0.32 (null hypothesis)
Ha: p =/= 0.32 (alternative hypothesis)
We would gather our data, compute the sample statistics, and then the test statistic. Using the above example, the sample proportion for a sample size of 1,000 shoppers came out to be 0.272 or 27.2%. The test statistic for large sample test of hypothesis for the population proportion is the Z:
Eq. 5.2.9: Z = (ps - p) / Sq Rt [ p * (1 - p) / n ] where p is hypothesized p.
Z = (0.272 - 0.32 ) / Sq Rt [ 0.32 * (1- 0.32) / 1000] = - 3.254
The probability of us getting
a sample proportion of 27.2%, given a hypothesized proportion of 32% is the
p-value, obtained by the Excel function =NORMSDIST(-3.254). This function
returns a p-value of 0.0005. Since this is a two tailed probability, and since
the =NORMSDIST function returns a lower tail p-value, we have to multiply the
p-value by 2 to get the full p-value. This gives 0.001.
The conclusion is that since the p-value is less than a specified value of
alpha of 0.05, we reject the null hypothesis and conclude that the population proportion
is not equal to 32%. From the sample proportion, we would further conclude that
the proportion is less than 32%, and from the confidence interval computed in
Equation 5.2.3, we are confident that the true proportion lies between 24.4%
and 30.0%. Note that the confidence interval confirms the result of the test of
hypothesis, which will always be the case with a not equal, two-tailed
alternative hypothesis.
Worksheet 5.2.2 provides a template for test of hypothesis for a proportion.
Worksheet 5.2.2
|
1 |
Column R |
S |
T |
|
2 |
Hypothesis Test for the Proportion |
||
|
3 |
n |
1000 |
|
|
4 |
Successes |
272 |
|
|
5 |
Sample Proportion |
0.272 |
= S4 / S3 |
|
6 |
Null Hypothesis p = |
0.32 |
|
|
7 |
Standard Error |
0.015 |
= SQRT((S6*(1-S6))/S3) |
|
8 |
Alpha |
0.050 |
|
|
9 |
Z Test Statistic |
-3.254 |
= (S5 - S6) / S7 |
|
10 |
p-value (1-tail) |
0.0006 |
= (1-NORMSDIST(ABS(S9))) |
|
11 |
p-value (2-tail) |
0.0011 |
= 2 * S10 |
|
12 |
This template is designed
so that the user need only insert information in cells S3, S4, S6 and S8. The
computation formulas in the remaining cells in Column S are shown as notes in
Column T.
Material covered in this module note can be used to answer questions one and
two of the Airline Satisfaction Survey Assignment in Main Module 5 Overview.
Our last Module Note 5.3 covers methods of comparing multiple samples of
categorical data.
References:
Anderson, D.,
Sweeney, D., & Williams, T. (2010). Essential of Modern Business Statistics
with Microsoft Excel. Cincinnati, OH: South-Western, Chapter 7 (Section 7.6)
and Chapter 8 (Section 8.4).
Ken Black. Business Statistics for
Contemporary Decision Making. Fourth Edition, Wiley. Chapter 4
& 12
D.
Groebner, P. Shannon, P. Fry & K. Smith.
Business Statistics: A Decision Making Approach, Fifth Edition,
Prentice Hall,
Chapter
4 and 14
Levine,
D., Berenson, M. & Stephan, D. (1999). Statistics for Managers Using
Microsoft Excel (2nd. ed.). Upper Saddle River, NJ: Prentice-Hall, Chapters 7 and 8.
Mason, R., Lind, D. & Marchal, W. (1999). Statistical Techniques in
Business and Economics (10th. ed.). Boston: Irwin McGraw Hill,
Chapters 8 and 9.
Sincich, T. (1992). Business
Statistics by Example (4th ed.). New York: Dellen/ Macmillan Publishing Company