Chapter 20 • Data Analysis: • Examining Differences Source:
Selecting Statistical Tests • Number of Samples • Univariate • Bivariate • Multivariate • Relationship of Samples • Independent • Dependent • Level of Measurement • Nominal • Ordinal • Interval/Ratio • Descriptive vs. Inferential
Flow Diagram for Choosing a Univariate Statistical Test Level of Measurement Nominal Interval Ordinal Ratio Number of Samples Number of Samples 1 2 or More 1 2 or More Relationship of Samples Relationship of Samples Independent Dependent Independent Dependent Chi-Square Kolmogorov Smirnov Chi-Square McNemar* Wilcoxon* Z-Test T-Test ANOVA Paired Difference T-Test Z-Test T-Test *Tests which are not discussed in the book but which are useful for some problems in marketing research.
CHI-SQUARE GOODNESS-OF-FIT TEST A statistical test to determine whether some observed pattern of frequencies corresponds to an expected pattern. Source:
Univariate Chi Square Goodness of Fit Test A manufacturer of facial soaps is considering changing the color of his best selling soap from pink to green. The marketing research department conducted personal intervals with 130 individuals to determine their package preference. The findings of the study are that 63 individuals prefer the current pink package and 67 individuals prefer the green package. Should the soap manufacturer change the package color?
Univariate Chi Square Goodness of Fit Test The Fly High Airlines has developed an advertising campaign aimed at business travelers. The marketing department is now trying to determine which business publication is most frequently read by business executives. A study of 165 business executives reports that 35 read Forbes, 30 read Industry Week, 45 read Fortune and 55 read Business Week. Which publication should Fly High select?
Bivariate Chi Square Test of Independence The marketing manager at Fly High Airlines has learned from reading the most recent trade publication that executives who have an advanced college degree are more likely to travel by air than other executives. How would this new information affect Fly High’s selection of a publication. The education level information is as follows:Advanced No Advanced Degree DegreeForbes 20 15Industry Week 15 15Fortune 25 20Business Week 30 25
KOLMOGOROV-SMIRNOV TEST A statistical test employed with ordinal data to determine whether some observed pattern of frequencies corresponds to some expected pattern; also used to determine whether two independent samples have been drawn from the same population or from populations with the same distribution. Source:
Kolmogorov-Smirnov Test A manufacturer of cosmetics is testing four different shades of makeup foundation compound: very light, light, medium and dark. The company has hired a marketing research firm to determine whether any distinct preference exists toward either extreme. If so, the company will manufacture only the preferred shades. Otherwise, it will market all shades. Suppose that in a sample of 100, 50 persons prefer the very light shade, 30 the light shade, 15 the medium shade, and 5 the dark shade. Do these results indicate some kind of preference?
Summary Table on Inferences About a Single Mean Known Unknown Small n: Use z= x-~N(0,1) Small n: Use t=x- where s_ =s/ n and x _ x s_ x and refer to t table for n-1 degrees of freedom (Xi-x )2 s= n-1 Distribution of Variable in Parent Population is Normal or Symmetrical Since t distribution approaches the normal as n increases, use Large n: Use z=x-~N(0,1) Large n: _ x x - t= s_ x for n>30.
x- Summary Table on Inferences About a Single Mean Known Unknown Small n: There is no theory to support the parametric test. One must either transform the variate so that it is normally distributed and then use the z test or one must use a distribution free statistical test. Small n: There is no theory to support the parametric test. One must either transform the variate so that it is normally distributed and then use the t-test or one must use a distribution free statistical test. Distribution of Variable in Parent Population is Asymmetrical Large n: If the sample is large enough so that the Central Limited Theorem is operative, use Large n: If sample is large enough so that: 1. The Central Limit Theorem is operative. 2. s is a close estimate of , use ^ x - ~N(0,1). z= ~N(0,1) z= s_ x _ x
Hypotheses about One Mean The marketing research department of a large beer brewer is trying to determine which college campuses to target for a new advertising campaign. The students must be over 21 to drink beer. However, younger “legal” students are believed to be heavier beer drinkers than older students. The marketing research department has decided to target campuses with an average age of 23. A study of 100 students at UNC Charlotte found that the average age for the sample was 24 with a standard deviation of 5. Should UNC Charlotte be targeted?
Hypotheses about One Mean A major bottler of a cola product is trying to estimate the average level of cola consumption on college campuses. If college students on average report drinking 100 gallons of cola products per year, the bottler plans to enlarge his operations and promote his product extensively to college students. A study of 15 college students is performed. The study reports that the average annual consumption of cola in the sample was 120 gallons with a standard deviation of 7. Should the bottler expand production and target the college market with advertising?
where 12 + 22 _ _ x1-x2 N2 N1 (1 + 1) n2 n1 (x1-x2)-(1-2) s_ _ x1-x2 Summary Table on Inferences About the Difference in Two Means Known Unknown Small n: Small n: Can you assume 1 = 2? 1. Yes: Use pooled variance t-test where (x1-x2)-(1-2) ~N(0.1) Use z = (x1-x2)-(1-2) x1-x2 t= = s_ _ x1-x2 and s_ _ x1-x2 = (Xil-x1)2 + (Xi2-x2)2 Distribution of Variable in Parent Population is Normal or Symmetrical n1 + n2-2 with n1+n2-2 degrees of freedom. 2. No: Approach is shrouded in controversy. Might use Aspin-Welch test. Large n: Use z = Large n: Use z= (x1-x2)-(1-2) _ _ x1-x2 and use pooled variance if variances can be assumed equal and unpooled variance if equality assumption is not warranted.
Summary Table on Inferences About the Difference in Two Means Known Unknown Small n: There is no theory to support the parametric test. One must either transform the variates so that they are normally distributed and then use the z-test or one must use a distribution free statistical test. Small n: There is no theory to support the parametric test. One must either transform the variates so that they are normally distributed and then use the t-test or one must use a distribution free statistical test. Large n: If the individual samples are large enough so that the Central Limit Theorem is operative for them separately, it will also apply for their sum or difference. Use Large n: One must assume that n1 and n2 are large enough so that the Central Limit Theorem applies to the individual sample means. Then it can also be assumed to apply to their sum or difference. Use Distribution of Variables in Parent Population are Asymmetrical ( x1-x2)-( z= s_ _ x1-x2 ( x1-x2)-( ~N(0,1) z= employing a pooled variance if the unknown parent population variances can be assumed equal and unpooled variance if the equality assumption is not warranted. _ _ x1-x2
Hypotheses about Two Independent (Unrelated) Means A health service agency has designed a public service campaign to promote physical fitness and the importance of regular exercise. Since the campaign is a major one, the agency wants to make sure of its potential effectiveness before running it on a national scale. To conduct a controlled test of the campaign’s effectiveness, the agency has identified two similar cities: City 1 will serve as the test city, and City 2 will serve as a control city. A preliminary random survey of 300 adults in City 1 and 200 adults in City 2 was conducted to measure the average time per day spent on some form of exercise by a typical adult in each city. The survey showed that the average was 30 minutes per day (standard deviation of 22 minutes) in City 1 and 35 minutes per day (standard deviation of 25minutes) in city 2. From the results, can the agency conclude confidently that the two cities were well matched for the controlled test? The agency does not want to allow more than a 5% chance of inferring that the cities are not matched when they truly are matched.
Hypotheses about Two Dependent (Related) Means Sales Per Store Before and After a Promotional Campaign Store Number Before After Change in Promotion Promotion Sales 1 250 260 10 2 235 240 5 3 150 151 1 4 145 140 -5 5 120 124 4 6 98 100 2 7 75 70 -5 8 85 95 10 9 180 200 20 10 212 220 8
Hypotheses about One Proportion Excell Microcomputer Company has been selling a new model of microcomputers in the Charlotte market for the past three months. Management of Excell will continue to sell the computer in charlotte only if it is able to maintain a 25% share of the computer market in the Charlotte metro area. A study of 30 computer retailers in Charlotte indicates that 30% of their computer sales is of the Excell computer. Should Excell continue to sell in Charlotte?
Hypotheses about Two Proportions A cosmetics manufacturer is interested in comparing male college students and male nonstudents in terms of their use of hair spray. Suppose random samples of 100 male students and 100 male nonstudents in Austin, Texas are selected and their use of hairspray in the last three months is determined. Suppose further that 30 of these students and 20 of those nonstudents have used hair spray within this period. Does the evidence indicate that a significantly higher percentage of male college students than male nonstudents use hair spray?