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Elementary Statistics

Elementary Statistics. Chapter 1. The Scientific Method. Step 1: Formulate a Theory. Step 2: Collect data to test the theory. Step 3: Analyze the results. Step 4: Interpret the results and make a decision. Definitions.

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Elementary Statistics

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  1. Elementary Statistics Chapter 1

  2. The Scientific Method • Step 1: Formulate a Theory. • Step 2: Collect data to test the theory. • Step 3: Analyze the results. • Step 4: Interpret the results and make a decision.

  3. Definitions • Population - the entire group of objects or individuals under study. • Sample - a part of the population that is actually used to get information. • Statistical inference - the process of drawing conclusions about the population based on information from a sample of that population.

  4. Statistical Hypothesis A statistical hypothesis consists of two parts, the null hypothesis and the alternate hypothesis.

  5. Statistical Hypothesis • Null Hypothesis - denoted by H0 - the conventional belief - the status quo, or prevailing viewpoint, about a population. Alternate Hypothesis - denoted by H1 - an alternate to the null hypothesis - the change in the population that the researcher hopes to find evidence for.

  6. Statistically Significant • The data collected are said to be statistically significant if they are very unlikely to be observed under the assumption that H0 is true. If the data are statistically significant, then our decision would be to reject H0.

  7. What if we make a mistake? • Type I Error: Rejecting the null hypothesis when in fact it is true. Type II Error: Accepting the null hypothesis when in fact it is not true.

  8. Decisions, Decisions

  9. Definition • The Significance Level number  is the chance of committing a Type I error -- that is the chance of rejecting the null hypothesis when it is in fact true.

  10. Frequency (or Dot) Plot • A frequency plot displays a set of observations by representing each observation value with a symbol positioned along a horizontal scale. If there are two or more observations with the same value, the symbols are stacked vertically.

  11. Definition • The number n of observations in a sample is called the sample size.

  12. Decision Rule • A decision rule is a formal rule that states, based on the data obtained, when to reject the null hypothesis H0. Generally, it specifies a set of values based on the data to be collected, which are contradictory to the null hypothesis H0 and which favor the alternative hypothesis H1.

  13. Direction of Extreme • The Direction of extreme corresponds to the position of the values that are more likely under the alternative hypothesis H1 than under the null hypothesis H0. If the larger values are more likely under H1 than under H0, then the direction of extreme is said to be to the right.

  14. Most Extreme Value • The value under the null hypothesis H0 that is least likely, but at the same time is very likely under the alternative hypothesis H1, is called the most extreme value.

  15. Rejection Region • A rejection region is the set of values for which you would reject the null hypothesis H0. Such values are contradictory to the null hypothesis and favor the alternative hypothesis.

  16. Rejection Region • An acceptance region is the set of values for which you would accept the null hypothesis H0. A cutoff value, or critical value, is a value that marks the starting point of a set of values that comprise the rejection region.

  17. One or Two - Sided? • A rejection region is called one-sided if its set of extreme values are all in one direction, either all to the right or all to the left. A rejection region is called two-sided if its set of extreme values are in two directions, both to the right and to the left.

  18. P-value • The p-value is the chance, computed under the assumption that H0 is true, of getting the observed value plus the chance of getting all of the more extreme values.

  19. Two Approaches for Decision Making • Classical Approach: Choose a level of significance, , and use it to create a decision rule. P-value Approach: Collect the data and calculate the likelihood of the data based on the assumption that the null hypothesis is true. If the p-value is less than , reject H0.

  20. Think About It • The significance level is  = 0.1, the chance of committing a Type I error. The corresponding decision rule is: “Reject H0 if the selected voucher is $50 or more.” A voucher is selected and it turns out to be $60. Your decision is to reject the null hypothesis and conclude that the data are statistically significant at the 10% level. You reject H0. Could you have made a mistake? What type of mistake could you have made? What is the chance that you have made a mistake?

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