1. 11E The Chi-Square Test of Independence
Unit 3: Statistical Applications
11E
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2. Chi-Square Test of Independence
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used to determine if two factors (or variables) from the same sample are independent (the occurrence of one does not affect the occurrence of the other) inferential statistical tests are used to test a claim Does “not independent” indicate dependence? No. The test is designed to examine whether two variables are independent or not independent.
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3. Testing Legal Claims
prosecution's claim: “The defendant is guilty.” null hypothesis H0: The defendant is not guilty. alternative hypothesis H1: The defendant is guilty.
 CLAIM
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4. Testing Legal Claims
defense’s claim: “The defendant is not guilty.” null hypothesis H0: The defendant is not guilty. alternative hypothesis H1: The defendant is guilty. If at the end of the trial you have: rejected the null hypothesis, then there was sufficient evidence to demonstrate guilt. failed to reject the null hypothesis, then was not sufficient evidence to demonstrate guilt. Insufficient evidence to demonstrate guilt DOES NOT mean that the defendant was innocent. It means there was, based on the evidence presented at trial, reasonable doubt that the defendant did not commit the crime.
 CLAIM
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5. Testing Statistical Claims
At the end of a statistical test you are we use evidence from the data to either reject the null hypothesis or fail to reject the null hypothesis. claim: “A is independent of B.” null hypothesis H0: A is independent of B. alternative hypothesis H1: A is not independent of B. If at the end of the statistical test you have: rejected the null hypothesis, then there was sufficient evidence to reject the claim. failed to reject the null hypothesis, then was not sufficient evidence to reject the claim.
 CLAIM
11E
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6. Testing Statistical Claims
At the end of a statistical test you are we use evidence from the data to either reject the null hypothesis or fail to reject the null hypothesis. claim: “A is not independent of B.” null hypothesis H0: A is independent of B. alternative hypothesis H1: A is not independent of B. If at the end of the statistical test you have: rejected the null hypothesis, then there was sufficient evidence to support the claim. failed to reject the null hypothesis, then was not sufficient evidence to support the claim.
 CLAIM
11E
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7. Chi-Square Statistic
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determined by examining the difference between the observed (fo) and expected (fe) values small differences between observed and expected frequencies (which yields a small chi- square statistic) indicate independence between the two factors large differences between observed and expected frequencies (which yields a large chi- square statistic) suggest that the factors are not independent
11E
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8. Example contingency table (observed frequency table)
expected frequency table
For each cell, multiply the row sum by the column sum and then divide by the overall total.
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9. Example
The chi-square statistic is small. This is a general indicator that the characteristics are independent since what you observed is close in value to what you would expect to see (with no assumptions made). How can technology help to quickly calculate the chi-square statistic “by hand”? And how can we use technology to find it quickly “not by hand”?
11E
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10. Certainty (or Lack Thereof)
“Sufficient evidence” does not indicate absolute certainty regarding which of the hypotheses is true and which is false. Statistical tests always begin by assuming the null is true until there is enough evidence to reject it. But is every individual found guilty actually guilty? Is every individual pronounced not guilty actually innocent? Why can’t we ever be certain about the results of a statistical hypothesis test?
11E
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11. Significance Level
the greatest acceptable probability of incorrectly rejecting the null hypothesis (also known as making a Type I error) used to find chi-square critical values from the chi-square distribution model will be given to you on the IB exam will need to choose, with a reason, for the project
11E
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12. Significance Level
common: 1%, 5% (scientific norm), 10% 5%: at most a 5% chance that you will state your data is significant when actually it is not (your allowance for a type I error to occur) The lower the significance level, the more stringent you are in allowing a “significant discrepancy” in your data and the more confident you can be that the conclusion is correct.
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13. Degrees of Freedom
use the original contingency table (without the total row or total column) to determine the degrees of freedom for the example:
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14. Limitations of the Chi-Square Test
often need to fix for the project will never encounter on the IB exam this statistical test is assumed to be unreliable if: any of the expected frequencies are less than 5 resolution: combining data; collecting more data; alter investigation the degrees of freedom is 1 resolution: cannot use p-value method; must use Yates’ continuity correction (must work by hand and cannot use technology)
11E
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15. Practice
p. 336: 1(b)(d) (Must show work by hand) p.336: 3(a)(b) (use GDC and steps from your project booklet) Read and follow all instructions. List the page and problem numbers alongside your work and answers in your notes. Use the back of the book to check your answers.
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11E
4/9/ :18 PM