Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics of the population? Are two or more groups the same or different? Are two or more variables related in a systematic way? Can we predict one variable if we know one or more other variables? 1. Formulate H 0 (null hypothesis) and H 1 (alternative hypothesis) 2. Select appropriate test 3. Choose level of significance 4. Calculate the test statistic (SPSS) 5. Determine the probability associated with the statistic. Determine the critical value of the test statistic. General Procedure for Hypothesis Test 6 a) Compare with the level of significance, α b) Determine if the critical value falls in the rejection region. (check tables) 7 Reject or do not reject H 0 8 Draw a conclusion 1. Formulate H 1 and H 0 The hypothesis the researcher wants to test is called the alternative hypothesis H 1. The opposite of the alternative hypothesis is the null hypothesis H 0 (the status quo)(no difference between the sample and the population, or between samples). The objective is to DISPROVE the null hypothesis. The Significance Level is the Critical probability of choosing between the null hypothesis and the alternative hypothesis The selection of a proper Test depends on: Scale of the data nominal interval the statistic you seek to compare Proportions (percentages) means the sampling distribution of such statistic Normal Distribution T Distribution χ 2 Distribution Number of variables Univariate Bivariate Multivariate Type of question to be answered Testing for Differences Between Mean of the Sample and Mean of the Population The manager of Pepperoni Pizza Restaurant has recently begun experimenting with a new method of baking its pepperoni pizzas. He believes that the new method produces a better-tasting pizza, but he would like to base a decision on whether to switch from the old method to the new method on customer reactions. Therefore he performs an experiment. 1
The Experiment For 40 randomly selected customers who order a pepperoni pizza for home delivery, he includes both an old style and a free new style pizza in the order. All he asks is that these customers rate the difference between pizzas on a -10 to +10 scale, where -10 means they strongly favor the old style, +10 means they strongly favor the new style, and 0 means they are indifferent between the two styles. Old pizza New pizza -10 0 +10 1. Formulate H 1 and H 0 One-Tailed Versus Two-Tailed Tests The form of the alternative hypothesis can be either a one-tailed or two-tailed, depending on what you are trying to prove. A one-tailed hypothesis is one where the only sample results which can lead to rejection of the null hypothesis are those in a particular direction, namely, those where the sample mean rating is positive. A two-tailed test is one where results in either of two directions can lead to rejection of the null hypothesis. 1. Formulate H 1 and H 0 1. Formulate H 1 and H 0 One-Tailed Versus Two-Tailed Tests -- continued Once the hypotheses are set up, it is easy to detect whether the test is one-tailed or two-tailed. One tailed alternatives are phrased in terms of > or < whereas two tailed alternatives are phrased in terms of The real question is whether to set up hypotheses for a particular problem as one-tailed or two-tailed. There is no statistical answer to this question. It depends entirely on what we are trying to prove. As the manager you would like to observe a difference between both pizzas If the new baking method is cheaper, you would like the preference to be for it. Null Hypothesis Alternative µ= mu=population mean H 0 µ=0 (there is no difference between the old style and the new style pizzas) (The difference between the mean of the sample and the mean of the population is zero) H 1 µ 0 or H 1 µ >0 Two tail test One tail test What we want to test is whether consumers prefer the new style pizza to the old style. We assume that there is no difference (i.e. the mean of the population is zero) and want to know whether our observed result is significantly (I.e. statistically) different. The one-sample t test is used to test whether the mean of the sample is equal to a hypothesized value of the population from which the sample is drawn. Type I Error Rejecting the null hypothesis that the pizzas are equal, when they really are perceived equal by the customers of the entire population. Type II error Accepting the null hypothesis that the pizzas are equal, when they are really perceived to be different by the customers of the entire population. 2
3. Choose Level of Significance The ratings of 40 randomly selected customers produces the following table and statistics Significance Level selected is typically.05 or.01 I.e 5% or 1% From the summary statistics, we see that the sample mean is 2.10 and the sample standard deviation is 4.717 The positive sample mean suggests a slight preference for the new pizza, (alternative hypothesis) but there is a fair degree of variation. What we don t know is whether this preference is significant 4. Calculate the Test Statistic t = X- 0 s/ n T(n-1) 2.10 0 t value = = 2.816 4.717 / 40 5. Determine the Probability-value (Critical Value) We use the right tail because the alternative is one-tailed of the greater than variety The probability beyond this value in the right tail of the t distribution with n-1 = 39 degrees of freedom is approximately 0.004 The probability, 0.004, is the p-value for the test. It indicates that these sample results would be very unlikely if the null hypothesis is true. 6. Compare with the level of significance, α (.05)and determine if the critical value falls in the rejection region Do not Reject H 0 1-α Reject H 0 Reject H 0-2.023 7. Reject or do not reject H 0 0 2.023 2.816 8 Conclusion the sample evidence is fairly convincing that customers, on average, prefer the new-style pizza. Should the manager switch to the new-style pizza on the basis of these sample results? Depends. There is no indication that the new-style pizza costs any more to make than the old-style pizza. Therefore, unless there are reasons for not switching (for example, costs) then we recommend the switch. Since the statistic falls in the rejection area we reject H o and conclude that the perceived difference between the pizzas is significantly different from zero. 3
Comparing Means Suppose you are the brand manager for Tylenol, and a recent TV ad tells the consumers that Advil is more effective (quicker) at treating headaches than Tylenol. An independent random sample of 400 people with a headache is given Advil, and 260 people report they feel better within an hour. Another independent sample of 400 people is taken and 252 people that took Tylenol reported feeling better. Is the TV ad correct? Or, in other words, is there a difference between the means of the two samples Hypothesis Test for Two Independent Samples Test for mean difference: Null Hypothesis Alternative H 0 µ 1 = µ 2 H 1 µ 1 µ 2 Under H 0 µ 1 - µ 2 = 0. So, the test concludes whether there is a difference between the means or not. Comparison of means: Graphically Are the means equal? Or are the differences simply due to chance? In this example we have two independent samples Other examples populations of users and non-users of a brand differ in perceptions of the brand high income consumers spend more on the product than low income consumers The proportion of brand-loyal users in Segment 1 (eg males) is more than the proportion in segment II (e.g. females) The proportion of households with Internet in Canada exceeds that in USA Can be used for examining differences between means and proportions The two populations are sampled and the means and variances computed based on the samples of sizes n 1 and n 2 If both populations are found to have the same variance then a t-statistic is calculated. The comparison of means of independent samples assumes that the variances are equal. If the variances are not known an F-test is conducted to test the equality of the variances of the two populations. F Unequal variances: The problem α 0 f 4
Tylenol vs Advil We would need to test if the difference is zero or not. H 0 : π A - π T = 0; H 1 : π A - π T 0 t = mean 1 mean 2 p A = 260/400= 0.65 p T = 252/400= 0.63 Variability of random means.65 -.63 z = (.65)(.35)/400+ (.63)(.37)/400 = 0.66 Differences Between Groups when Comparing Means Ratio scaled dependent variables t-test When groups are small When population standard deviation is unknown z-test When groups are large For large samples the t-distribution approaches the normal distribution and so the t-test and the z-test are equivalent. Degrees of Freedom d.f. = n - k where: n = n 1 + n 2 k = number of groups The degrees of freedom is (n 1 + n 2 2) Tylenol vs Advil α = 0.10 Critical value = 1.64 α -1 - -1.64 0 0.66 1.64 Since 0.66 is less than the critical value of 1.64 we accept the null hypothesis: there is no difference between Advil and Tylenol users Test for Means Difference on Paired Samples What is a paired sample? When two sets of observations relate to the same respondents When you want to measure brand recall before and after an ad campaign. Shoppers consider brand name to be more important than price Households spend more money on pizza than on hamburgers The proportion of a bank s customers who have a checking account exceeds the proportion who have a savings account Since it is the same population that is being sampled the observations are not independent. The appropriate test is a paired-t-test Example Q1. When purchasing golf clubs rate the importance 1-5 of price Q2. When purchasing golf clubs rate the importance 1-5 of brand H 0 There is no difference in importance between brand and price H 1 One tailed H 1 Two Tailed Price is more important than brand There is a difference in importance between brand and price 5
What is an ANOVA? One-way ANOVA stands for Analysis of Variance Purpose: Extends the test for mean difference between two independent samples to multiple samples. Employed to analyze the effects of manipulations (independent variables) on a random variable (dependent). What does ANOVA tests? The null hypothesis tests whether the mean of all the independent samples is equal H 0 µ 1 = µ 2 = µ 3..= µ n H 1 µ 1 µ 2 µ 3.. µ n The alternative hypothesis specifies that all the means are not equal Definitions Dependent variable: the variable we are trying to explain, also known as response variable (Y). Independent variable: also known as explanatory variables or Factors (X). Research normally involves determining whether the independent variable has an effect on the variability of the dependent variable Comparing Antacids The maker of Acid-off, an antacid stomach remedy wants to know which type of ad results in the most positive brand attitude among consumers. Non comparative ad: Acid-off provides fast relief Explicit Comparative ad: Acid-off provides faster relief than Tums Non explicit comparative ad Acid-off provides the fastest relief Three groups of people are exposed to one type of ad and asked to rate their attitude towards the ad. Brand Attitude Type of Ad Comparing Antacids Non Comparative Explicit Comparative Means Non Explicit Comparative The dependent variable (denoted by Y) is called the response variable and in this case it is brand attitude (I.e. we want to know what effect ad type has on attitude toward the brand) The independent variables are called factors, in this case type of ad: non-comparative, explicit comparative, non-explicit comparative The different levels of the factor are called treatments. In this case the treatments are the different ratings for each of the three types of ads. There will be two sources of variation. Variation within the treatment (e.g. within the noncomparative ad etc.) Variation between the treatments (I.e. between the three types of ads) 6
The whole idea behind the analysis of variance is to compare the ratio of between group variance to within group variance. If the variance caused by the interaction between the samples is much larger when compared to the variance that appears within each group, then it is because the means are different. F Variance = Variance Degrees of Freedom between within groups groups The F statistics has DF for both numerator (between group) and denominator (within group) DF between group = (c-1) where c=number of groups DF within group = (N-c) where N is sample size Within Category Variation SS within Decomposition of the Total Variation Category Mean Independent Variable X Categories Total Sample X 1 X 2 X 3. X c Y 1 Y 1 Y 1. Y 1 Y 1 Y 2 Y 2 Y 2. Y 2 Y 2 Total Variation SS y Y n Y n Y n. Y n Y n Y 1 Y 2 Y 3 Y c Y Between Category Variation SS between Grand Mean ANOVA Test The null hypothesis would be tested with the F distribution F distribution Reject H 0 df (c-1)/(n-c) α One way ANOVA investigates: Main effects factor has an across-the-board effect e.g., type of ad Or age or involvement A TWO-WAY ANOVA investigates: INTERACTIONS effect of one factor depends on another factor e.g., larger advertising effects for those with no experience importance of price depends on income level and involvement with the product 7