1 Testing Statistical Hypothesis The One Sample t-Test Heibatollah Baghi, and Mastee Badii

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 Distribution-free tests Although appropriate for all levels of measurement most frequently applied for nominal or ordinal measures

3 Parametric and Nonparametric Tests This lecture focuses on One sample t-test which is a parametric test What is type two error? 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 Two Types of Error Alpha: α –Probability of Type I Error –P (Rejecting H o when H o is true) –Predetermined Level of significance Beta: β –Probability of Type II Error –P (Failing to reject H o when H o is false)

5 Types of Error in Hypothesis Testing Ho: Hand-washing has no effect on bacteria counts True False True (Accept H o) Type II error Probability =  False (Rejects H o) Type I error Probability = 

6 Types of Error in Hypothesis Testing Ho: Hand-washing has no effect on bacteria counts True False True (Accept H o) Correct decision Probability = 1-  Type II error Probability =  False (Rejects H o) Type I error Probability =  Correct decision Probability = 1- 

7 Power & Confidence Level Power –1- β –Probability of rejecting H o when H o is false Confidence level –1- α –Probability of failing to reject H o when H o is true True False True (Accept H o) Correct decision Probability = 1-  Type II error Probability =  False (Rejects H o) Type I error Probability =  Correct decision Probability = 1- 

8 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

9 Population of IQ scores, 10-year olds Etc Sample 2 Sample 1 Sample 3 n = 64 µ=100 σ=16 Sampling Distribution Of Means

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

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

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

14 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

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 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

17 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

18 4. Calculating Test Statistics Sample mean

19 4. Calculating Test Statistics Deviation from sample mean

20 4. Calculating Test Statistics Squared deviation from sample mean

21 4. Calculating Test Statistics Standard deviation of observations

22 4. Calculating Test Statistics Calculated t value

23 4. Calculating Test Statistics Standard deviation of sample means

24 4. Calculating Test Statistics Calculated t

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

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

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

31 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

32 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

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

34 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

35 Distribution of Mean of Samples In drawing samples at random, the probability is.95 that an interval constructed with the rule will include 

36 Sample Report of One Sample t-test in Literature Attitude Toward Mean Standard deviationt-valueP value In Vitro Fertilization <.05 Artificial Insemination NS Adoption <.01 Remaining Childless <.001 df = 99 One Sample t-test Testing Neutrality of Attitudes Towards Infertility Alternatives

37 Testing Statistical Hypothesis With SPSS NMean Std. Deviation Std. Error Mean Number of Hours Test Value = 6.75 tc dfSig. (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper Number of Hours SPSS Output: One-Sample Statistics One-Sample Test

38 Take Home Lesson Procedures for Conducting & Interpreting One Sample Mean Test with Unknown Variance