Five types of statistical analysis

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?

General Procedure for Hypothesis Test

1. Formulate H0 (null hypothesis) and H1

(alternative hypothesis)2. Select appropriate test3. Choose level of significance4. 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,

b) Determine if the critical value falls in the

rejection region. (check tables)

7 Reject or do not reject H0

8 Draw a conclusion

• The hypothesis the researcher wants to test is called the alternative hypothesis H1.

• The opposite of the alternative hypothesis is the null hypothesis H0 (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

1. Formulate H1and H0

• 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 Distribution2 Distribution

– Number of variables• Univariate• Bivariate• Multivariate

– Type of question to be answered

2. Select Appropriate Test

Example

A tire manufacturer believes that men are more aware of their brand. To find out, a survey is conducted of 100 customers, 65 of whom are men and 35 of whom are women.

The question they are asked is:

Are you aware of our brand: Yes or No. 50 of the men were aware and 15 were not whereas 10 of the women were aware and 25 were not.

Are these differences significant?

Men WomenAware 50 10

Unaware 15 25 65 35

We want to know whether brand awareness is associated with gender. What are the Hypotheses

1. Formulate H1and H0

H0:

H1:

There is no difference in brand awareness based on gender

There is a difference in brand awareness based on gender

• Used to discover whether 2 or more groups of one variable (dependent variable) vary significantly from each other with respect to some other variable (independent variable).

• Are the two variables of interest associated:

– Do men and women differ with respect to product usage (heavy, medium, or light)

– Is the preference for a certain flavor (cherry or lemon) related to the geographic region (north, south, east, west)?

H0: Two variables are independent (not associated)

H1: Two variables are not independent (associated)

• Must be nominal level, or, if interval or ratio must be divided into categories

X2 (Chi Square)2. Select Appropriate Test

Aware 50/39 10/21 60

Unaware 15/26 25/14 40 65 35 100

Men Women Total

Awareness of Tire Manufacturer’s Brand

Estimated cell Frequency n

CRE

jiij

Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample sizeEij = estimated cell frequency

3. Choose Level of Significance• Whenever we draw inferences about a population, there is a

risk that an incorrect conclusion will be reached

• The real question is how strong the evidence in favor of the alternative hypothesis must be to reject the null hypothesis.

• The significance level states the probability of incorrectly rejecting H0. This error is commonly known as Type I error,

The value of is called the significance level of the test

• In the example a Type I error would be committed if we said that

There is a difference between men and women with respect to brand awareness when in fact there was no difference

• Significance Level selected is typically .05 or .01

• i.e 5% or 1%

•In other words we are willing to accept the risk that 5% (or 1%) of the time the results we get indicate that there is a difference between men and women with respect to brand awareness when in fact there is no difference

3. Choose Level of Significance

• We commit Type error II when we incorrectly accept a null hypothesis when it is false. The probability of committing Type error II is denoted by .

• In our example we commit a type II error when we say that.

there is NO difference between men and women with respect to brand awareness (we accept the null hypothesis) when in fact there is

Accept null Reject null

Null is true

Null is false

Correct-Correct-no errorno error

Type IType Ierrorerror

Type IIType IIerrorerror

Correct-Correct-no errorno error

Type I and Type II Errors

Which is worse?

• Both are serious, but traditionally Type I error has been considered more serious, that’s why the objective of hypothesis testing is to reject H0 only when there is enough evidence that supports it.

• Therefore, we choose to be as small as possible without compromising .

• Increasing the sample size for a given α will decrease β (I.e. accepting the null hypothesis when it is in fact false)

Aware 50/39 10/21 60

Unaware 15/26 25/14 40 65 35 100

Men Women Total

Awareness of Tire Manufacturer’s Brand

Estimated cell Frequency n

CRE

jiij

Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample sizeEij = estimated cell frequency

21

)2110(

39

)3950( 222

X

14

)1425(

26

)2615( 22

161.22

643.8654.4762.5102.32

2

1)12)(12(..

)1)(1(..

fd

CRfd

4. Calculate the Test StatisticChi-Square Test: Differences Among Groups

Chi-square test results are unstable if cell count is lower than 5

While there will be n such squared deviations only (n - 1) of them are free to assume any value whatsoever.

This is because the final squared deviation from the mean must include the one value of X such that the sum of all the Xs divided by n will equal the obtained mean of the sample.

All of the other (n - 1) squared deviations from the mean can, theoretically, have any values whatsoever..

Degrees of Freedom

the number of values in the final calculation of a statistic that are free to vary

For example To calculate the standard deviation of a random sample, we must first calculate the mean of that sample and then compute the sum of the squared deviations from that mean

5. Determine the Probability-value (Critical Value)

•The p-value is the probability of seeing a random sample at least as extreme as the sample observed given that the null hypothesis is true. • given the value of alpha, we use statistical theory to determine the rejection region.• If the sample falls into this region we reject the null hypothesis; otherwise, we accept it• Sample evidence that falls into the rejection region is called statistically significant at the alpha level.

Significance from p-values -- continued

• How small is a “small” p-value? This is largely a matter of semantics but if the – p-value is less than 0.01, it provides “convincing”

evidence that the alternative hypothesis is true;– p-value is between 0.01 and 0.05, there is “strong”

evidence in favor of the alternative hypothesis;– p-value is between 0.05 and 0.10, it is in a “gray

area”;– p-values greater than 0.10 are interpreted as weak

or no evidence in support of the alternative.

Chi-square Test for Independence

Under H0, the probability distribution is approximately distributed by the Chi-square distribution (2).

Chi-square

2

Reject H0 3.84

22.16

X2 with 1 d.f. at .05 critical value = 3.84

5. Determine the Probability-value (Critical Value)

6 a) Compare with the level of significance,

b) Determine if the critical value falls in the rejection region. (check tables)

22.16 is greater than 3.84 and falls in the rejection area

In fact it is significant at the .001 level, which means that the chance that our variables are independent, and we just happened to pick an outlying sample, is less than 1/1000

7 Reject or do not reject H0

Since 22.16 is greater than 3.84 we reject the null hypothesis

8 Draw a conclusionMen and women differ with respect to brand awareness, specifically, men are more brand aware then women

Example 2:• 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.

• 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.

The Experiment

-10 +100

Old pizza New pizza

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 H1and H0

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.

1. Formulate H1and H0

1. Formulate H1and H0

• 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 Two tail

testOne tail

test= mu=population mean

–H0 =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)

–H1 0 or H1 >0

2. Select Appropriate 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 data sample is equal to a hypothesized value of the population from which the sample is is drawn.

Not rejecting the null hypothesis that the pizzas are

equal, when they are perceived to be different by the

customers of the entire population.

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

3. Choose Level of Significance

Significance Level selected is typically .05 or .01

•I.e 5% or 1%

The ratings of 40 randomly selected customers and produce the following table and statistics

From the summary statistics, we see that the sample mean is 2.10 and the sample standard deviation is 4.717

Summary Statistics

The positive sample mean provides some evidence in favor of the alternative hypothesis, but given the rather large standard deviation does it provide enough evidence to reject H0?

816.240/717.4

010.2

valuet

4. Calculate the Test Statistic

t =X- 0s/n T(n-1)

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.

1-

Reject H0 Reject H0

Do not Reject H0

0 2.8.162.074-2.074

/2/2

6. Compare with the level of significance, (.05)and determine if the critical value falls in the rejection region

Since the statistic falls in the rejection area we reject Ho

and conclude that the perceived difference between the pizzas is significantly different from zero.

7. Reject or do not reject H0

8 Conclusion

• Should the manager switch to the new-style pizza on the basis of these sample results?

• We would probably recommend “yes”. There is no indication that the new-style pizza costs any more to make than the old-style pizza, and the sample evidence is fairly convincing that customers, on average, will prefer the new-style pizza.

• Therefore, unless there are reasons for not switching (for example, costs) then we recommend the switch.

Example 3• 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

–H0 1= 2

–H1 1 2

–Under H0 1- 2 = 0. So, the test concludes whether there is

a difference between the parameters or not.

Comparison of means: Graphically

Are the means equal?

2. Select Appropriate TestIn 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 is more than the proportion in segment II

•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 n1 and n2

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.

2. Select Appropriate Test

0

F

f

Unequal variances: The problem

Tylenol vs Advil

• We would need to test if the difference is zero or not.

H0: A - T = 0;

H1: A - T 0

z =

.65 - .63 -0

(.65)(.35)/400+ (.63)(.37)/400= 0.66

pA = 260/400= 0.65

pT = 252/400= 0.63

For large samples the t-distribution approaches the normal distribution and so the t-test and the z-test are equivalent.

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

Degrees of Freedom

• d.f. = n - k• where:

– n = n1 + n2

– k = number of groups

The degrees of freedom is (n1 + n2 –2)

Tylenol vs Advil

- 0 0.66

= 0.10 N(0,1) = 1.64

1.64-1.64

/2/2

-1

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.

• Since both samples are not independent we employ the differences as a random sample

di=x1i-x2i i=1,2,…,n

• The appropriate test is a paired-t-test

Test for Means Difference on Paired Samples

Q1. When purchasing golf clubs rate the importance 1-5 of price

Q2. When purchasing golf clubs rate the importance 1-5 of brand

Example

H0

H1 One tailed

H1 Two Tailed

There is no difference in importance between brand and price

There is a difference in importance between brand and price

Price is more important than brand

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).

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

What does ANOVA tests?• The null hypothesis tests whether the mean of all

the independent samples is equal

H0 1= 2 = 3 …..= n

H1 1 2 3 ….. n

• The alternative hypothesis specifies that all the means are not equal

Comparing Antacids

• 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

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.

Three groups of people are exposed to one type of ad and asked to rate their attitude towards the ad.

Comparing Antacids

BrandAttitude

Type of Ad NonComparative

ExplicitComparative

Non ExplicitComparative

Means

The dependent variable is called the response variable and in this case it is brand attitude

The independent variables are called factors, in this case type of ad

And the different levels of the factor are called treatments. In this case the treatments are each of the three types of ads: non-comparative, explicit comparative, non-explicit comparative.

There will be two sources of variation.

Variation within the treatment (e.g. within the non-comparative ad)

Variation between the treatments (I.e. between the three types of ads)

Independent Variable X

Categories Total Sample

X1 X2 X3 …. Xc

Y1 Y1 Y1 …. Y1 Y1

Y2 Y2 Y2 …. Y2 Y2

Yn Yn Yn …. Yn Yn

Y1 Y2 Y3 Yc Y

Decomposition of the Total Variation

TotalVariatio

nSSy

Between Category Variation SSbetween

CategoryMean

WithinCategoryVariation

SSwithin

GrandMean

ANOVA Test• The null hypothesis would be tested with

the F distribution

f(c-1)(N-c)

Reject H0 F distribution

•Degrees of Freedom

–cn-1 where c=number of groups, n= number of observations in a group

– 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