1 Level of Significance α is a predetermined value by convention usually 0.05 α = 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

2 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.

3 One Sample Test Compares mean of a sample to known population mean –Z-test –T-test This lecture focuses on one sample t-test

4 The One Sample t – Test Testing statistical hypothesis about µ when σ is not known OR sample size is small

5 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 , 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. Populaon mean Small sample Population variance is unknown & estimated from sample

6 Steps in Test of Hypothesis 1.Determine the appropriate test 2.Establish the level of significance: α 3.Determine whether to use a one tail or two tail test 4.Calculate the test statistic 5.Determine the degree of freedom 6.Compare computed test statistic against a tabled value

7 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

8 2. Establish Level of Significance α is a predetermined value The convention α =.05 α =.01 α =.001 In this example, assume α = 0.05

9 3. Determine Whether to Use a One or Two Tailed Test H 0 :µ = 6.75 H a :µ ≠ 6.75 A two tailed test because it can be either larger or smaller

10 4. Calculating Test Statistics Sample mean

11 4. Calculating Test Statistics Deviation from sample mean

12 4. Calculating Test Statistics Squared deviation from sample mean

13 4. Calculating Test Statistics Standard deviation of observations

14 4. Calculating Test Statistics Calculated t value

15 4. Calculating Test Statistics Standard deviation of sample means

16 4. Calculating Test Statistics Calculated t

17 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

18 Degrees of Freedom Suppose you have a sample of three observations : Σ= X

19 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

20 6. Compare the Computed Test Statistic Against a Tabled Value α =.05 Df = n-1 = 9 Therefore, reject H 0

21 Decision Rule for t-Scores If |t c | > |t α | Reject H 0

22 Decision Rule for P-values If p value < α Reject H 0 Pvalue is one minus probability of observing the t-value calculated from our sample

23 Example of Decision Rules In terms of t score: |t c = | > |t α= | Reject H 0 In terms of p-value: If p value =.037 < α =.05 Reject H 0

24 Constructing a Confidence Interval for µ Sample mean Standard deviation of sample means Critical t value

25 Constructing a Confidence Interval for µ for the Example Sample mean is 9.90 Critical t value is Standard deviation of sample means is * 1.29 The estimated interval goes from 6.98 to 12.84