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Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer. Chapter 15 The Analysis of Qualitative Data. Doing Statistics for Business. Chapter 15 Objectives In this chapter you will learn to use chi- square tests for: - PowerPoint PPT Presentation
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DoingDoing Statistics for BusinessStatistics for Business Data, Inference, and Decision Making
Marilyn K. PelosiTheresa M. Sandifer
Chapter 15Chapter 15The Analysis of The Analysis of Qualitative DataQualitative Data
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Doing Statistics for Business
Chapter 15 Objectives
In this chapter you will learn to use chi-square tests for:Testing whether a particular probability
model fits a set of data (Goodness of Fit test)
Testing equality of proportions for more than 2 populations
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Doing Statistics for Business
Chapter 15 Objectives (con’t)
Testing whether 2 qualitative variables are dependent or independent
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Doing Statistics for Business
The chi-square goodness of fit test checksto see how well a set of data fit the modelfor a particular probability distribution.
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Doing Statistics for Business
The observed frequencies are the actualnumber of observations that fall into eachclass in a frequency distribution or histogram.
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Doing Statistics for Business
The expected frequencies are the numberof observations that should fall into each class in a frequency distribution under the hypothesized probability distribution.
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Doing Statistics for Business
A uniform distribution is one in which eachoutcome or class of outcomes is equally likelyto occur.
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Doing Statistics for Business
TRY IT NOW!Seat-Belt Usage
Setting Up the Hypotheses for theGoodness of Fit TestAnalysts for insurance companies assume that the number of drivers
who wear seat belts is a binomial random variable with = 0.70. To
test this assumption they decide to set up checkpoints and sample 10
drivers every 2 hours. Set up the hypotheses to perform an appropriate
chi-square goodness of fit test.
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Doing Statistics for Business
TRY IT NOW!Seat-Belt Usage
Calculating the Expected Frequencies
The insurance analysts collect data for 1000 samples of 10 drivers
and obtain the frequency distribution shown in the following table.
Find the expected frequency distribution for the data if the distribution
is really binomial with n = 10 and = 0.70.
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Doing Statistics for Business
TRY IT NOW!Seat-Belt UsageCalculating the Expected Frequencies(con’t) # Wearing
Seatbelts (x)ObservedFrequency
p(x) ExpectedFrequency
0 01 02 13 64 335 1166 2137 2758 2169 119
10 21Total 1000
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Doing Statistics for Business
TRY IT NOW!Seat-Belt Usage
Calculating the Expected Frequencies(con’t)Create frequency histograms for both the observed and the expected
frequency distributions. At this point, does it appear that the observed
data conform to the binomial distribution with n = 10 and = 0.70.?
Why or why not?
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Doing Statistics for Business
TRY IT NOW!Seat-Belt UsageCalculating the Chi-Square StatisticFill in the following table to calculate the value of thechi-square statistic for the data obtained by the insurance analysts.
# WearingSeatbelts
ObservedFrequency
p(x) ExpectedFrequency
(o-e) (o -e)2/e
0 0 0.000 01 0 0.000 02 1 0.001 13 6 0.009 94 33 0.037 375 116 0.103 1036 213 0.200 2007 275 0.267 2678 216 0.233 2339 119 0.121 121
10 21 0.028 28Total 1000 1 1000
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Doing Statistics for Business
k - p - 1
Figure 15.1 The Upper Tail of a Chi-Square Distribution
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Doing Statistics for Business
TRY IT NOW!Seat-Belt UsageFinding the Critical Value and Performingthe Test
The insurance analysts decide that they want to test the goodness of fithypothesis at the 0.01 level of significance.
How many degrees of freedom will the critical value for the test have?
Find the critical value for the test.
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Doing Statistics for Business
TRY IT NOW!Seat-Belt Usage
Finding the Critical Value and Performingthe Test (con’t)
Based on the chi-square test statistic and the critical value, what can you
conclude about the distribution of the number of people in a sample size
of 10 that wear seat belts?
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Doing Statistics for Business
TRY IT NOW!Technical Support
Setting Up the Chi-Square Test forProportions
A company that sells computer software has three different locationsset up to provide customers with technical support for their products. The support representatives keep a log for each call to technical support,and as part of that log, they record whether the problem was resolvedsuccessfully.
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Doing Statistics for Business
TRY IT NOW!Technical Support
Setting Up the Chi-Square Test forProportions (con’t)
The company analysts are interested in knowing whether the percentageof calls that are successfully resolved is the same for each location. Theyrandomly select logs from each location and collect data on the numberof calls that result in a successful resolution of the problem. The dataare summarized in the following table.
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Doing Statistics for Business
TRY IT NOW!Technical SupportSetting Up the Chi-Square Test forProportions (con’t)
Set up the hypotheses for the software company.
Calculate the proportion of successfully resolved calls for each location. Based solely on these numbers, do you think that the proportion ofsuccessfully resolved calls for all three locations is the same?
LocationNumber of 1 2 3 TotalsSuccessful Calls 257 264 283 804Unsuccessful Calls 43 86 97 226Totals 300 350 380 1030
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Doing Statistics for Business
TRY IT NOW!Technical SupportCalculating the Expected FrequenciesThe data for the computer software company interested in itstechnical support locations are
LocationNumber of 1 2 3 TotalsSuccessful Calls 257 264 283 804Unsuccessful Calls 43 86 97 226Totals 300 350 380 1030
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Doing Statistics for Business
TRY IT NOW!Technical SupportCalculating the Expected Frequencies
Calculate , the percentage of calls that are resolved successfully, assuming that the three locations are the same.
Use the overall proportion of successful calls to find the expectedfrequency of successful calls for each location.
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Doing Statistics for Business
Non-Binge Infrequent Frequent All
Never 77 39 31 14 58.25 40.22 48.54 147.0
Once or 7 19 3965 More 25.75 17.78 21.46 65.0
All 84 58 70 21 84.00 58.00 70.00 212.0
Chi-Square = 40.484, DF = 2, P-Value = 0.00
Cell Contents -- Count Exp Freq
Figure 15.2 Minitab Output from Chi-Square Analysis
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Doing Statistics for Business
TRY IT NOW!Technical Support
Performing the Chi-Square Test forProportions
The computer software company with the different technical support
location wants to complete the test to determine whether the percentage
of successfully resolved calls is the same at all three locations. It wants
to test at the 0.01 level of significance.
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Doing Statistics for Business
TRY IT NOW!Technical Support
Performing the Chi-Square Test forProportions (con’t)
Calculate the value of the chi-square test statistic and complete the test.
Is the proportion of successfully resolved calls the same at each location?
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Doing Statistics for Business
TRY IT NOW!Drinking Survey
Setting Up the Contingency Table for aTest for Independence
The Public Health student who did the study on drinking also collected
data on the number of times that the student drove while intoxicated in the
last two weeks (coded). The contingency table for the usable responses is
given on the following slide.
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Doing Statistics for Business
TRY IT NOW!Drinking Survey
Setting Up the Contingency Table for aTest for Independence (con’t)
Number of Times Drive While IntoxicatedClass Not at All Once Twice or More Total
Freshman 72 5 9 86
Sophomore 19 8 9 36
Junior 16 8 6 30
Senior 8 4 7 19
Total 115 25 31 171
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Doing Statistics for Business
TRY IT NOW!Drinking Survey
Setting Up the Contingency Table for aTest for Independence (con’t)
The university is interested in knowing if the number of times a student
drove while intoxicated is related to his or her class in school. It feels
that this information will help target student audiences for programs on
drinking and driving.
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Doing Statistics for Business
TRY IT NOW!Drinking SurveySetting Up the Contingency Table for aTest for Independence (con’t)
Set up the hypotheses that the university should test.
Calculate the expected frequencies for each cell and put them in theappropriate location in the table.
Are any of the expected frequencies less than 5? If so, can you suggest alogical way to combine categories to avoid this problem?
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Doing Statistics for Business
TRY IT NOW!Drinking SurveyPerforming the Chi-Square Test forIndependenceThe university that is looking at the relationship between class year
and drinking and driving wants to perform the test at the 0.05 level of significance.
Calculate the value of the chi-square statistic for the data.
Find the critical value and perform the test.
Are class and drinking and driving independent?
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Doing Statistics for Business
The Chi-Square Test for Independence in Excel - Creating the Contingency Table
1. From the Data menu, select Pivot Table Report and follow the steps of the pivot table wizard to create the table.
2. At step 4 of 4, after you indicate where you want the table to be placed, do not select Finish. Instead, click Options.
3. Make sure that the checkbox next to For empty cells, show is checked and in the textbox next to it, type “0”. (Excel will not accept empty cells in the contingency table.)
4. Click OK and then Finish.
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Doing Statistics for Business
Figure 15.4 Pivot Table Options Dialog Box
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Doing Statistics for Business
Figure 15.5 Contingency Table for Binge Drinking Data
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Doing Statistics for Business
The Chi-Square Test for Independence In Kadd
Select Hypothesis Testing>Chi Square Test from the Kadd menu
The dialog box opens
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Doing Statistics for Business
The Chi-Square Test for Independence Dialog Box
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Doing Statistics for Business
The Chi-Square Test for Independence
Complete the Dialog Box:1. The input range should be the location of
the contingency table for the data. Make sure you do not select the column and row that contain the totals.
2. Indicate where you want the output located.
3. Click OK.
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Doing Statistics for Business
The Chi-Square Test for Independence
Chi-square test statistic = 15.3884 Number of:
p-value = 0.0174 rows = 4columns = 3
Actual frequencies Expected frequenciesVariable B
Non-binge Infrequent Frequent TotalsVariable A Residence Hall or Dormitory 35 25 46 106
Fraternity or Sorority 0 1 0 1Other University Housing 0 2 1 3
Off Campus House or Apartment 49 30 24 103Totals 84 58 71 213
Chi-square calculationsVariable B
Non-binge Infrequent FrequentVariable A Residence Hall or Dormitory 1.1071 0.5172 3.2201
Fraternity or Sorority 0.3944 1.9447 0.3333Other University Housing 1.1831 1.7135 0.0000
Off Campus House or Apartment 1.7289 0.1360 3.1100
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Doing Statistics for Business
Chapter 15 Summary
In this chapter you have learned: The chi-square test involves comparing observed
and expected frequencies for different classes of data.
The chi-square test is quite versatile and can be used to test:
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Doing Statistics for Business
Chapter 15 Summary (con’t)
Goodness of Fit
Equality of Proportions for more than 2 Populations
Independence of Qualitative Variables
The results of a chi-square test do not solve a problems, but simply point out when further action is indicated and when it is not.