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Week07_CoreStatAnalysisI1.pdf

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Core Statistical Analysis I

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Inferential Statistics

• Unlike descriptive statistics which are used to describe characteristics of a sample, inferential statistics are used to make inferences about a population from a sample of that population – Based on a random sample of 1000

supermarkets, what is your estimate of market share of Fage Greek yogurt in all supermarkets in the U.S.? Did Faye reach its goal? Did it increase compared to last year?

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Why Making Inference?

• Why not study the entire population? – Time-consuming and expensive to do census

– Not all units in the population can be identified

– Not feasible to conduct test on the whole population

Population:

Total collection of objects or people to be studied

Sample:

Subset of a population

Make inference

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Sample Statistics and Population Parameters (Revisit)

• Population parameters are characteristics of the population

• Generally we do not know these population parameters so we use the corresponding sample statistics as estimates – Sample statistics are measures computed from

the sample data

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Uncertainty in Making Inference • However, these sample statistics are not exactly the

value of the population parameter – Sample market share is 5.2%, what is the true market

share? Remember confidence interval?

• So we need to take into account the inaccuracy of the sample statistics when we want to make a judgement about the population parameter – Sample market share is 5.2%, 95% confidence interval is

5.1% to 5.3%, did Faye reach its targeted market share of 5%?

• Hence the whole process of doing hypothesis testing

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What is Hypothesis Testing?

• Hypothesis is a belief about population parameters of interests – E.g., population mean, population proportion,

population standard deviation, etc.

– We don’t test sample statistics

• Hypothesis testing is the procedure of testing a hypothesis based on what we observe in the sample

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Formal Steps for Hypothesis Testing

1. Formulate hypotheses

2. Choose appropriate statistical test

3. Set a decision rule

4. Calculate test statistic and p-value

5. Make conclusion

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1. Formulate Hypotheses • Formulate null and alternative hypotheses about

population parameter of interest – We never form hypotheses about sample statistics

• Null hypothesis ( , pronounced as “H-naught”): a statement of the status quo, one of no difference or no effect – E.g., average market share has not changed from last year

• Alternative hypothesis ( ): a statement in which some difference or effect is expected – E.g., average market share is different from last year

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2. Select Appropriate Statistical Tests

Nominal/ordinal data:

• Chi-squared goodness-of- fit test

• Chi-squared test of association

• Rank correlation

Interval/ratio data:

• T-test – One sample t-test

– Independent samples t-test

– Paired sample t-test

• Correlation

• 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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3. Set Decision Rule

• Rule used to determine whether to reject or fail to reject ; stated in terms of significance level ( )

• Choosing a 0.05 significance level ( ) means that we limit the chance that we would make a mistake to reject a true to be 5%

• Technical detail: Probability of making Type I error is 5%. Namely, if the null hypothesis is in fact true, the probability of observing a sample that leads to rejection of is 5%.

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4. Calculate Value of Test Statistic and P-value

• Collect data and calculate test statistic using sample data

• Calculate the corresponding p-value. P-value is a probability (between 0 and 1) that measures how extreme/rare the test statistic is, assuming that is true – Lower the p-value, more rare the test stat (data)

– If we observe very rare data, most likely ଴ is not true

଴: son has brushed teeth

• Test statistic: dryness of toothbrush

• If found toothbrush very dry, it is very unlikely that son has brushed his teeth

• Question: how rare the sample should be so that we reject the assumed to be true null hypothesis?

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4. Calculate Value of Test Statistic and P-value

• Technical detail: p-value is the probability of getting a value of test statistic (if draw many random samples from the same population and compute the same test statistic) as extreme as, or more extreme than, the one currently observed, assuming that the null hypothesis is true

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5. Make Conclusion • When p-value < alpha, reject null and accept

H1; otherwise fail to reject null; NEVER say we accept the null – Technical detail: In classical hypothesis testing,

there is no way to prove that the null is true

• Wet toothbrush doesn’t mean that son has brushed teeth. He could have run his toothbrush under the tap before heading to bed

• Make managerial conclusion

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T-Tests • T-test is for testing differences in population means.

• Suitable for interval/ratio variable (rating, market share, expenditure, etc.)

• One sample t-test – Compare mean of a single variable against a given standard

– E.g., is average market share different from 6%?

• Independent samples t-test – Compare mean of same variable for two different groups

– E.g., average expenditure on Greek yogurt among high-income households versus low-income households?

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

• Paired sample t-test – Compare means of different variables for same respondents

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

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One Sample t-tests • Compare mean of a single variable against a given

standard

• Null Hypothesis: Population Mean of X = Test Value μ0

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– Where is the sample mean

is standard error of X, , where is the standard deviation, is sample size

• Excel command: p-value = TDIST(tstat, df, tails=2) – If there are n observations, degrees of freedom df = n-1

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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%

• Sample average market share is 6.2%, sample size is 1000, standard error is 0.1%. Use alpha=0.05.

p-value = TDIST(2, 999, 2)=0.046<0.05

Conclusion: reject null; the average market share is statistically different from 6%

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One sample t-tests: an Example Cont.

• Is the average market share statistically higher than 6%?

• Yes because the sample mean 6.2% > 6% and the previous hypothesis testing shows that their difference is statistically significant

• Technical detail: here we use two-tailed test (one is statistically different from the other) to make directional inference; compared to one-tailed test (one is larger than the other), two-tailed test is a bit more conservative in reaching significance

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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

• E.g., average expenditure on Greek yogurt of high-income households is different from low-income households

: mean expenditure of high income HH = mean expenditure of low income HH

: … … • EXCEL Command to obtain p-value: Data -> Data

Analysis -> t-test for Equal/Unequal Variances • If p-value < alpha, reject null; the difference is statistically

significant; can also comment on the direction • Otherwise, fail to reject null; fail to find evidence that the

difference is statistically significant

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

: mean of variable X = mean of variable Y

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

: average consumer brand awareness before seeing ad = average … after seeing ad

: … …

• EXCEL Command to obtain p-value: Data -> Data Analysis -> Paired t-test

• If p-value < alpha, reject null; the difference is statistically significant; can also comment on the direction

• Otherwise, fail to reject null; fail to find evidence that the difference is statistically significant