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Simple Hypothesis Testing

Simple Hypothesis Testing. Detecting Statistical Differences In The Simplest Case:  and  are both known I The Logic of Hypothesis Testing: The Null Hypothesis II The Tail Region, Critical Values: α III Type I and Type II Error. The Fundamental Idea.

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Simple Hypothesis Testing

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  1. Simple Hypothesis Testing Detecting Statistical Differences In The Simplest Case:  and  are both known I The Logic of Hypothesis Testing: The Null Hypothesis II The Tail Region, Critical Values: α III Type I and Type II Error Anthony Greene

  2. The Fundamental Idea • Apply a treatment to a sample • Measure the sample mean (this means using a sampling distribution) after the treatment and compare it to the original mean • Remembering differences always exist due to chance, figure out the odds that your experimental difference is due to chance. • If its too unlikely that chance was the reason for the difference, conclude that you have an effect Anthony Greene

  3. Null and Alternative Hypotheses Null hypothesis: A hypothesis to be tested. We use the symbol H0 to represent the null hypothesis. Alternative hypothesis: A hypothesis to be considered as an alternate to the null hypothesis. We use the symbol Ha to represent the alternative hypothesis. Anthony Greene

  4. The Distribution of Sample Means As The Basis for Hypothesis TestingThe set of potential samples is divided into those that are likely to be obtained and those that are very unlikely if the null hypothesis is true. Anthony Greene

  5. The Logic of the Hypothesis Test • We start with knowledge about the distribution given no effect (e.g., known parameters or a control group) and the data for a particular experimental treatment • Begin with the assumption that there is no experimental effect: this is the null hypothesis • Compute the probability of the observed data given the null hypothesis • If this probability is less than  (usually 0.05) then reject the null hypothesis and accept the alternative hypothesis Anthony Greene

  6. The Logic of the Hypothesis TestThe critical region (unlikely outcomes) for  = .05. Anthony Greene

  7. zobs zcrit Avoiding Confusion About zzcrit vs. zobs Anthony Greene

  8. Air Puff to Eyeblink Latency (ms) Anthony Greene

  9. 95% of all samples of 25 eyeblinks have mean within 1.96 standard deviations of  Anthony Greene

  10. Probability that the sample mean of 450 ms is a chance difference from the null-hypothesis mean of 454 ms z = -2.56 Anthony Greene

  11. Using More Extreme Critical Values The locations of the critical region boundaries for three different levels of significance:  = .05,  = .01, and  = .001. Anthony Greene

  12. Test Statistic, Rejection Region, Nonrejection Region, Critical Values Anthony Greene

  13. Rejection regions for two-tailed, left-tailed, and right-tailed tests While one-tailed tests are mathematically justified, they are rarely used in the experimental literature Anthony Greene

  14. Graphical display of rejection regions for two-tailed, left-tailed, and right-tailed tests Anthony Greene

  15. α α α/2 α for 1 and 2-tailed tests Anthony Greene

  16. α for 1 and 2-tailed tests for α = 0.05 Anthony Greene

  17. Correct and incorrect decisions for a hypothesis test Anthony Greene

  18. + + = = Correct and incorrect decisions for a hypothesis test 1.00 1.00 Anthony Greene

  19. Type I and Type II Errors Type I error: Rejecting the null hypothesis when it is in fact true. Type II error: Not rejecting the null hypothesis when it is in fact false. Anthony Greene

  20. Significance Level The probability of making a Type I error, that is, of rejecting a true null hypothesis, is called the significance level, a, of a hypothesis test. That is, given the null hypothesis, if the liklihood of the observed data is small, (less thana) we reject the null hypothesis. However, by rejecting it, there is still an a (e.g., 0.05) probability that rejecting the null hypothesis was the incorrect decision. Anthony Greene

  21. Relation Between Type I and Type II Error Probabilities For a fixed sample size, the smaller we specify the significance level, a, (i.e., lower probability of type I error) the larger will be the probability, b, of not rejecting a false null hypothesis. Another way to say this is that the lower we set the significance, the harder it is to detect a true experimental effect. Anthony Greene

  22. Possible Conclusions for a Hypothesis Test • If the null hypothesis is rejected, we conclude that the alternative hypothesis is true. • If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis. Anthony Greene

  23. Critical Values, α = P(type I error) Suppose a hypothesis test is to be performed at a specified significance level, a. Then the critical value(s) must be chosen so that if the null hypothesis is true, the probability is equal to a that the test statistic will fall in the rejection region. Anthony Greene

  24. Some important values of z Anthony Greene

  25. Power The power of a hypothesis test is the probability of not making a Type II error, that is, the probability of rejecting a false null hypothesis. We have Power = 1 – P(Type II error) = 1 –  The power of a hypothesis test is between 0 and 1 and measures the ability of the hypothesis test to detect a false null hypothesis. If the power is near 0, the hypothesis test is not very good at detecting a false null hypothesis; if the power is near 1, the hypothesis test is extremely good at detecting a false null hypothesis. For a fixed significance level, increasing the sample size increases the power.

  26. M = 42 Fail to reject M = 48 Reject null hypothesis Conclude Effect Basic Idea Zcrit = 1.64 μ0 = 40 H0: Parent distribution for your sample if there IS NO effect Anthony Greene

  27. Basic Idea Zcrit = 1.64 μa = ? μ0 = 40 Ha:Parent distribution for your sample if there IS an effect H0: Parent distribution for your sample if there IS NO effect Anthony Greene

  28. Basic Idea Zcrit = 1.64 1 - α μ0 = 40 α H0: Parent distribution for your sample if there IS NO effect Anthony Greene

  29. Basic Idea Zcrit = 1.64 1 - β β μa = ? Ha:Parent distribution for your sample if there IS an effect Anthony Greene

  30. Zcrit = 1.28 Zcrit = 2.58 Basic Idea We can move zcrit μa = ? μ0 = 40 Ha:Parent distribution for your sample if there IS an effect H0: Parent distribution for your sample if there IS NO effect Anthony Greene

  31. Basic Idea Zcrit = 1.64 We can increase n μa = ? μ0 = 40 Ha:Parent distribution for your sample if there IS an effect H0: Parent distribution for your sample if there IS NO effect Anthony Greene

  32. The one-sample z-test for a population mean (Slide 1 of 3) Step 1 The null hypothesis is H0:  = 0and the alternative hypothesis is one of the following: Ha:   0Ha:  < 0Ha:  > 0 (Two Tailed) (Left Tailed) (Right Tailed) Step 2 Decide on the significance level,  Step 3 The critical values are ±z/2 -z+z (Two Tailed) (Left Tailed) (Right Tailed) Anthony Greene

  33. α α α/2 The one-sample z-test for a population mean (Slide 2 of 3) Anthony Greene

  34. The one-sample z-test for a population mean (Slide 3 of 3) Step 4 Compute the value of the test statistic Step 5 If the value of the test statistic falls in the rejection region, reject H0, otherwise do not reject H0. Anthony Greene

  35. Synopsis Anthony Greene

  36. P-Value To obtain the P-value of a hypothesis test, we compute, assuming the null hypothesis is true, the probability of observing a value of the test statistic as extreme or more extreme than that observed. By “extreme” we mean “far from what we would expect to observe if the null hypothesis were true.” We use the letter P to denote the P-value. The P-value is also referred to as the observed significance level or the probability value. Anthony Greene

  37. P-value for a z-test • Two-tailed test: The P-value is the probability of observing a value of the test statistic z at least as large in magnitude as the value actually observed, which is the area under the standard normal curve that lies outside the interval from –|z0| to |z0|, • Left-tailed test: The P-value is the probability of observing a value of the test statistic z as small as or smaller than the value actually observed, which is the area under the standard normal curve that lies to the left of z0, • Right-tailed test: The P-value is the probability of observing a value of the test statistic z as large as or larger than the value actually observed, which is the area under the standard normal curve that lies to the right of z0, Anthony Greene

  38. Guidelines for using the P-value to assess the evidence against the null hypothesis Anthony Greene

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