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**Major Points An example Sampling distribution Hypothesis testing**

The null hypothesis Test statistics and their distributions The normal distribution and testing Important concepts

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**Distribution of M&M’s in the population**

Yellow 20% Brown 30% Orange 10% Blue 10% Red 20% Green 10% Original Distribution

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**Testing one sample against population**

There are normally 5.65 red M&M’s in bag (a population parameter) Mean number of red M&M’s in a halloween candy bag = 4.56 (a sample statistic) Are there sufficiently more red M&M’s to conclude significant differences / the two numbers come from different populations. I have provided a simplified version of the study because it is much easier to begin by comparing a sample mean with a population mean. You will probably need to explain why I did this. You can come back and work with the two-sample example later.

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**Understanding the theoretical and statistical question**

Theoretical Question Did M&M use a different proportion of red ones for Halloween? Statistical Question: Is the difference between 5.65 and 4.25 large enough to conclude that it is a real (significant) difference? Would we expect a similar kind of difference with a repeat of this experiment? Or... Is the difference due to “sampling error?”

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**Sampling Error Often differences are due to sampling error**

Sampling Error does not imply doing a mistake Sampling Error simply refers to the normal variability that we would expect to find from one sample to another, or one study to another

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**How could we assess Sampling Error?**

Take many bags of regular M&M candy. Record the number of red M&M’s. Plot the distribution and record its mean and standard deviation. This distribution is a “Sampling Distribution” of the Mean

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**Sampling Distribution**

Means of various samples of Brown M&M’s 7.25 7.00 6.75 6.50 6.25 6.00 5.75 5.50 5.25 5.00 4.75 4.50 4.25 4.00 3.75 Sampling Distribution Number of Red M&M’s in the population Frequency 1400 1200 1000 800 600 400 200 Std. Dev = .45 Mean = 5.65 Distribution ranges between 3.76 and 7.25 Mean is 5.65 SD is .45 Mean of 4.00 is not likely, mean of 5 is likely We can calculate these prob1abilities. Right now lets focus on the logic of the analysis. We can calculate how likely it is to get a score as low as 4.25 by chance. Convert score to z score (z distribution has a mean of 0 and standard deviation of 1) =score-mean/standard deviation Look up Table E10, smaller portion 4.25=-3.1=.0006 5.60 = -.11= .45 7.5=3.5=.0002

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**What is Sampling Distribution**

The distribution of a statistic over repeated sampling from a specified population. Can be computed for many different statistics

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**Distribution ranges between 3.76 and 7.25 Mean is 5.65, SD is .45 **

Mean Number Aggressive Associates 7.25 7.00 6.75 6.50 6.25 6.00 5.75 5.50 5.25 5.00 4.75 4.50 4.25 4.00 3.75 Number of Red M&M’s Frequency 1400 1200 1000 800 600 400 200 Std. Dev = .45 Mean = 5.65 N = Distribution ranges between 3.76 and 7.25 Mean is 5.65, SD is .45 Mean of 4.00 is not likely, mean of 5 is more likely

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**How likely is it to get score as low as 4.25 by chance.**

Convert score to z score (z distribution has a mean of 0 and standard deviation of 1) score - mean / standard deviation Look up Table E10, smaller portion Score Z Score Probability

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Hypothesis Testing A formal way of testing if we should accept results as being significantly different or not Start with hypothesis that halloween M&M’s are from normal distribution The null hypothesis Find parameters of normal distribution Compare halloween candy to normal distribution

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**The Null Hypothesis (H0)**

Is the hypothesis postulating that there is no difference, that two things are from the same distribution. The hypothesis that Halloween candy came from a population of normal M&M’s The hypothesis we usually want to reject. Alternative Hypothesis: Halloween and Regular M&M’s are from different distributions

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**It is easier to prove Alternative Hypothesis than Null Hypothesis**

Hypothesis: All crows are black Sample: 3000 crows Results: all are black Conclusion: Are all crows black?

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**Observation: One white crow**

Conclusion: Statement :Every crow is black” is false It is easier to prove alternative hypothesis (all crows are not black) than null hypothesis (all crows are black)

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**Another example of Null & Alternative Hypothesis**

Hypothesis: Shopping on Amazon.com is different than on BN.com Study: 100 usability tests comparing the two shopping process on both Results: Both sites performed similarly Conclusion: ?

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**Observation: On Test No:101 Amazon did better**

Conclusion: Amazon.com and BN are different.

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**Steps in Hypothesis Testing**

Define the null hypothesis. Decide what you would expect to find if the null hypothesis were true. Look at what you actually found. Reject the null if what you found is not what you expected.

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**Important Concepts Concepts critical to hypothesis testing Decision**

Type I error Type II error Critical values One- and two-tailed tests

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Decisions When we test a hypothesis we draw a conclusion; either correct or incorrect. Type I error Reject the null hypothesis when it is actually correct. Type II error Retain the null hypothesis when it is actually false.

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Possible Scenarios

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M&M candy example

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Type I Errors Assume Halloween and Regular candies are same (null hypothesis is true) Assume our results show that they are not same (we reject null hypothesis) This is a Type I error Probability set at alpha () usually at .05 Therefore, probability of Type I error = .05

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Type II Errors Assume Halloween and Regular Candies are different (alternative hypothesis is true) Assume that we conclude they are the same (we accept null hypothesis) This is also an error Probability denoted beta () We can’t set beta easily. We’ll talk about this issue later. Power = (1 - ) = probability of correctly rejecting false null hypothesis.

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Critical Values These represent the point at which we decide to reject null hypothesis. e.g. We might decide to reject null when (p|null) < .05. Our test statistic has some value with p = .05 We reject when we exceed that value. That value is the critical value.

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**One- and Two-Tailed Tests**

Two-tailed test rejects null when obtained value too extreme in either direction Decide on this before collecting data. One-tailed test rejects null if obtained value is too low (or too high) We only set aside one direction for rejection.

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**One- & Two-Tailed Example**

One-tailed test Reject null if number of red in Halloween candies is higher Two-tailed test Reject null if number of red in Halloween candies is different (whether higher or lower)

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**Designing an Experiment: Feature Based Product Advisors**

Identify some good online product advisors Feature Based Filtering: CNET Digital Camera Advisor Basic Sony Decision Guide Dealtime Feature Based Choice Multi-Dimension (including features) based filtration Sony Advanced Decision Guide Review Based Choosing Epinions CNET Computers / Cameras / both; Sample Size; Experimental Questions

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