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1. 1 Testing Statistical HypothesisThe One Sample t-Test Heibatollah Baghi, and
2. 2 Parametric and Nonparametric Tests Parametric tests estimate at least one parameter (in t-test it is population mean)
Usually for normal distributions and when the dependent variable is interval/ratio
Nonparametric tests do not test hypothesis about specific population parameters
Although appropriate for all levels of measurement most frequently applied for nominal or ordinal measures
3. 3 Parametric and Nonparametric Tests Nonparametric tests are easier to compute and have less restrictive assumptions
Parametric tests are much more powerful (less likely to have type II error)
4. 4 Two Types of Error Alpha: a
Probability of Type I Error
P (Rejecting Ho when Ho is true)
Predetermined Level of significance
Probability of Type II Error
P (Failing to reject Ho when Ho is false)
5. 5 Types of Error in Hypothesis Testing Ho: Hand-washing has no effect on bacteria counts
6. 6 Types of Error in Hypothesis Testing Ho: Hand-washing has no effect on bacteria counts
7. 7 Power & Confidence Level Power
Probability of rejecting Ho when Ho is false
Probability of failing to reject Ho when Ho is true
8. 8 Level of Significance a is a predetermined value by convention usually 0.05
a = 0.05 corresponds to the 95% confidence level
We are accepting the risk that out of 100 samples, we would reject a true null hypothesis five times
9. 9 Sampling Distribution Of Means
10. 10 Sampling Distribution Of Means A sampling distribution of means is the relative frequency distribution of the means of all possible samples of size n that could be selected from the population.
11. 11 One Sample Test Compares mean of a sample to known population mean
12. 12 The One Sample t Test Testing statistical hypothesis about when s is not known OR sample size is small
13. 13 An Example Problem Suppose that Dr. Tate learns from a national survey that the average undergraduate student in the United States spends 6.75 hours each week on the Internet composing and reading e-mail, exploring the Web and constructing home pages. Dr. Tate is interested in knowing how Internet use among students at George Mason University compares with this national average.
Dr. Tate randomly selects a sample of only 10 students. Each student is asked to report the number of hours he or she spends on the Internet in a typical week during the academic year.
14. 14 Steps in Test of Hypothesis Determine the appropriate test
Establish the level of significance:a
Determine whether to use a one tail or two tail test
Calculate the test statistic
Determine the degree of freedom
Compare computed test statistic against a tabled value
15. 15 1. Determine the appropriate test If sample size is more than 30 use z-test
If sample size is less than 30 use t-test
Sample size of 10
16. 16 2. Establish Level of Significance a is a predetermined value
a = .05
a = .01
a = .001
In this example, assume a = 0.05
17. 17 3. Determine Whether to Use a One or Two Tailed Test H0 : = 6.75
Ha : ? 6.75
18. 18 4. Calculating Test Statistics 1818
19. 19 4. Calculating Test Statistics 1919
20. 20 4. Calculating Test Statistics 2020
21. 21 4. Calculating Test Statistics
22. 22 4. Calculating Test Statistics
23. 23 4. Calculating Test Statistics
24. 24 4. Calculating Test Statistics
25. 25 5. Determine Degrees of Freedom Degrees of freedom, df, is value indicating the number of independent pieces of information a sample can provide for purposes of statistical inference.
Df = Sample size Number of parameters estimated
Df is n-1 for one sample test of mean because the population variance is estimated from the sample
26. 26 Degrees of Freedom Suppose you have a sample of three observations:
27. 27 Degrees of Freedom
Why n-1 and not n?
Are these three deviations independent of one another?
No, if you know that two of the deviation scores are -1 and -1, the third deviation score gives you no new independent information ---it has to be +2 for all three to sum to 0.
28. 28 Degrees of Freedom Continued For your sample scores, you have only two independent pieces of information, or degrees of freedom, on which to base your estimates of S and
29. 29 6. Compare the Computed Test Statistic Against a Tabled Value a = .05
Df = n-1 = 9
Therefore, reject H0
30. 30 Decision Rule for t-Scores If |tc| > |ta| Reject H0
31. 31 Decision Rule for P-values If p value < a Reject H0
32. 32 Example of Decision Rules In terms of t score:
|tc = 2.449| > |ta= 2.262| Reject H0
In terms of p-value:
If p value = .037 < a = .05 Reject H0
33. 33 Constructing a Confidence Interval for
34. 34 Constructing a Confidence Interval for for the Example Sample mean is 9.90
Critical t value is 2.262
Standard deviation of sample means is 1.29
9.90 + 2.262 * 1.29
The estimated interval goes from 6.98 to 12.84
35. 35 Distribution of Mean of Samples In drawing samples at random, the probability is .95 that an interval constructed with the rule
will include m
36. 36 Sample Report of One Sample t-test in Literature An example of a report in a paper. The null hypothesis that the sample mean is the same as the mean of the population can be rejected when t-values exceed the critical t-value. For 99 degrees of freedom, the critical t-value at 0.05 is 1.99. Therefore the t-value of 2.5 is higher than the critical value and therefore we can reject this hypothesis at this level. This is statistically significant (not due to chance)
The critical value for 99 degrees of freedom for alpha of 0.01 is 2.64 and therefore we can reject the hypothesis that the in vitro fertilization mean is same as the population. An example of a report in a paper. The null hypothesis that the sample mean is the same as the mean of the population can be rejected when t-values exceed the critical t-value. For 99 degrees of freedom, the critical t-value at 0.05 is 1.99. Therefore the t-value of 2.5 is higher than the critical value and therefore we can reject this hypothesis at this level. This is statistically significant (not due to chance)
The critical value for 99 degrees of freedom for alpha of 0.01 is 2.64 and therefore we can reject the hypothesis that the in vitro fertilization mean is same as the population.
37. 37 Testing Statistical Hypothesis With SPSS The SPSS output includes the confidence interval (CI) on MEAN DIFFERENCE. Slide number 19 includes the CI is on the MEAN.
The SPSS output includes the confidence interval (CI) on MEAN DIFFERENCE. Slide number 19 includes the CI is on the MEAN.
38. 38 Take Home Lesson Procedures for Conducting & Interpreting One Sample Mean Test with Unknown Variance