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Week08_CoreStatAnalysisII.pdf

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Week 08 Agenda

• M: core statistics II

• W & F: – Excel lab sessions using National

Insurance Case (course pack); A survey to determine customers’ perceptions about National’s service; Excel dataset on UBLearns

– F: Release assignment 2 (1-2 students)

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Core Statistical Analysis II

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Practical Steps of Hypothesis Testing

Formulate

Hypotheses

Choose statistical

test

Compute P- value using sample data

Compare to alpha and Conclude

• In hypothesis testing, we can never accept the null but we can accept the alternative hypothesis. So we formulate outcome that is of managerial interest as the alternative hypothesis • there is some effect, some difference

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Select Appropriate Statistical Tests

Nominal/ordinal data:

• Chi-squared goodness-of- fit test

• Chi-squared test of association

• Rank correlation (later)

Interval/ratio data:

• T-test – One sample t-test

– Independent samples t-test

– Paired sample t-test

• Correlation (later)

• Depends on type of data and hypothesis • Test statistic (e.g., t-stat) extracts relevant information from

sample data to measure how close the sample has come to the null hypothesis

• Technical detail: test statistic often follows a well-know distribution, e.g., normal, t, or chi-square distribution

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T-Tests (Review)

• T-test is for testing differences in population means – Suitable for interval/ratio variable (rating, market share,

expenditure, etc.)

– One sample t-test

– Independent samples t-test

– Paired sample t-test

• Test statistics reflect how different sample mean(s) is(are) from population mean assumed in null hypothesis – larger difference, more rare to observe such sample given null is

true, smaller p-value, more likely to reject null

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One Sample t-tests: an Example • We want to know whether the average market share of

Fage is different from the target market share 6% – Null hypothesis:

• Scenario 1: sample average market share is 6.2%, sample size is 1000, standard error is 1%. Use alpha=0.05.

• Scenario 2: sample average market share is now 8%; all else the same

• Question: which sample is more likely to lead to rejection of null hypothesis?

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One Sample t-tests: an Example • Scenario 1: sample average market share is 6.2%, sample size is 1000,

standard error is 1%. Use alpha=0.05.

; ; . % %

p-value = TDIST(t-stat, n-1, 2) = TDIST(0.2, 999, 2) = 0.842 > 0.05

• Conclusion: fail to reject null; we do not find sufficient evidence that average market share is (statistically) significantly different from 6%

• Scenario 2: sample average market share is 8%

; ; % %

• p-value = TDIST(t-stat, n-1, 2) = TDIST(2, 999, 2) = 0.048 < 0.05

• Conclusion: reject null; average market share is (statistically) significantly different from 6%, in particular, it’s significantly higher than 6%

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Independent Samples t-Test • Compare mean of same variable for two different groups

– the response of first group has no effect on the second group

: mean of group 1 = mean of group 2

• T-stat (complicated formula) measures difference between the sample means of group 1 and group 2 – Larger the difference, smaller the p-value, hence more inclined to

reject the null

• E.g., average expenditure on Greek yogurt for high-income households versus low-income households

Scenario Sample mean expenditure High Income

Sample mean expenditure Low Income

P-value

1 $35 $25 Will be smaller

2 $30 $28 Will be larger

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Paired Sample T-Test • Compare mean of different variables for same respondents

: mean of variable X = mean of variable Y

• T-stat (complicated formula) measures difference between the sample means of variable X and variable Y – Larger the difference, smaller the p-value, hence more inclined to

reject the null

• E.g., consumers brand awareness before and after seeing an ad

Scenario Sample mean awareness before

Sample mean awareness after

P-value

1 3 4 Will be smaller

2 3 3.3 Will be larger

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Statistical Test for Categorical Data

• Categorical: nominal / ordinal

• Tests related to frequency distribution:

• Chi-squared goodness-of-fit test – Test whether the frequency distribution of a single

categorical variable is consistent with a hypothesized distribution

– E.g., is gender distribution consistent with 1:1?

• Chi-squared test of association – Test whether there is any association between two

categorical variables

– E.g., are gender and tablet usage associated?

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Chi-square Goodness of Fit • Test whether the frequency distribution of a single

categorical variable is consistent with a hypothesized distribution

• E.g., we often are interested in testing customer profile

– Is the gender distribution of customers different from 1:1?

• Sample has 53 female and 47 male

• Is 53:47 significantly different from 1:1? – could be due to chance, we did not obtain 1:1 in the sample

– could be that population distribution is systematically different from 1:1

– When difference is large enough, we think that it is less likely due to chance

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Chi-square Goodness of Fit

: population distribution = a hypothesized distribution

: … not the same

• The test statistic captures the difference between the observed frequency counts in the sample data and the expected frequency counts given by the hypothesized distribution – When difference is large enough, we think that it is less likely

due to chance

• Larger the difference, smaller the p-value. When p<alpha, reject null; population distribution is significantly different from the hypothesized distribution

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• : gender distribution is 1:1 • : gender distribution is different from 1:1 • Construct observed frequency table for the sample data • Construct expected frequency table using hypothesized distribution

• Test statistic

• Excel: P-value = CHITEST(Observed Range, Expected Range)=0.55; fail to reject null

Chi-square Goodness of Fit

Category Observed frequency

female 53

male 47

Category Expected frequency

female 50= ½*(53+47)

male 50= ½*(53+47)

- O: observed counts - E: expected counts - n is number of categories

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Chi-squared Test of Association

• Test whether there is any systematic association between two categorical variables

• Association means that knowing the level of variable A can help predict the level of Variable B. Note however, the relationship is not necessarily causal, in the sense that one variable "causes" the other

• E.g., are gender and tablet usage (light/heavy) associated? Are gender and voting preference associated?

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Chi-squared Test of Association • H0 : variable A and variable B are independent

• H1 : variable A and variable B are not independent (or are associated)

• Use cross-tab frequency table to construct test statistic

• The idea is to think about what the cross-tab should look like if the two variables are independent; then compare this expected cross-tab with the one we actually observe – Greater the discrepancies between observed and expected

cross-tabs, less likely the two variables are independent

– Greater the discrepancies, smaller the p-value; when p< alpha, reject null; the two variables are associated

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Male (M) Female (F) Total

Light (L) 20 30 50

Heavy (H) 35 35 70

Total 55 65 120

Observed Cross-tab

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Male (M) Female (F) Total

Light (L) 50

Heavy (H)

Total 55 65 120

Expected Cross-tab

• Er,c = (nr * nc) / n • where Er,c is the expected frequency count for cell in row r and column c • nr is the total number of sample observations at row r • nc is the total number of sample observations at column c • n is the total sample size

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Male (M) Female (F) Total

Light (L) 50

Heavy (H)

Total 55 65 120

Expected Cross-tab

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Male (M) Female (F) Total

Light (L) 50

Heavy (H)

Total 55 65 120

Expected Cross-tab

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Chi-squared Test of Association • Test statistic measures the difference between observed

frequencies and expected frequencies under independent assumption

– Where O represents observed counts, E expected counts, r for rows, c for columns

• Excel: P-value = CHITEST(Observed Range, Expected Range) = 0.28; fail to reject null; do not find evidence that they are associated – Do not include “total” in the range!

• Examine cells to determine nature of relationship

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National Insurance Case (Course Pack)

• A survey to determine customers perceptions about National’s service – Excel dataset on UBLearns

• W&F: Bring your laptop and course pack to class – Get ready for excel!