Chapter 8 Testing Means OneSample t Test With

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Chapter 8 Testing Means One-Sample t Test With Confidence Intervals Privitera, Essential Statistics for

Chapter 8 Testing Means One-Sample t Test With Confidence Intervals Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 1

Chapter Outline • • Going from z to t Degrees of Freedom Reading the

Chapter Outline • • Going from z to t Degrees of Freedom Reading the t Table One-Sample t Test Effect Size for the One-Sample t Test Confidence Intervals for the One-Sample t Test APA in Focus: Reporting the t Statistic and Effect Size Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 2

Going From z to t • To compute a zobt score, population variance must

Going From z to t • To compute a zobt score, population variance must be known • In behavioral science it is rare that the variance in a population is known • An alternative was proposed by Gosset • We substitute the population variance with the sample variance in the formula for standard error • This substitution, called the estimated standard error, is the denominator of the test statistic for a t test • This is acceptable because sample variance is an unbiased estimator of the population variance Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 3

T-Test • Compares two means • Since a mean has t be calculates, only

T-Test • Compares two means • Since a mean has t be calculates, only interval and ratio data can be used. • Three types of t-test: • One Sample t-Test • Paired Sample t-Test • Independent Samples t-Test 4

One Sample t-Test • Compares two means of a variable in a sample with

One Sample t-Test • Compares two means of a variable in a sample with a known population mean. • Example: • Does the Salary of Clerical workers match that of the national average ($35, 000)? 5

Going From z to t (cont. ) • Estimated Standard Error – an estimate

Going From z to t (cont. ) • Estimated Standard Error – an estimate of the standard deviation of a sampling distribution of sample means selected from a population with an unknown variance. • It is an estimate of the standard error or standard distance that sample means deviate from the value of the population mean stated in the null hypothesis • Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 6

Going from z to t (cont. ) • Using the substitution of sample variance

Going from z to t (cont. ) • Using the substitution of sample variance for population variance, a new test statistic is introduced • The t statistic • Used to determine the number of standard deviations in a t distribution that a sample deviates from the mean value or difference stated in the null • tobt = M – μ , where s. M = SD s. M √n Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 7

Going from z to t (cont. ) • Using the estimated standard error in

Going from z to t (cont. ) • Using the estimated standard error in the denominator of the test statistic led to a new sampling distribution known at the t distribution • It is like a normal distribution, but with greater variability in the tails Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 8

Degrees of Freedom (df) • The df for a t distribution are equal to

Degrees of Freedom (df) • The df for a t distribution are equal to the df of the sample variance: n – 1. • As the sample size increases, sample variance more closely resembles population variance • Degrees of freedom of the sample variance increases, the shape of the t distribution changes • The probability of outcomes in the tails become less likely and the tails approach the x-axis faster as n increases Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 9

Reading the t Table • To locate probabilities and critical values in a t

Reading the t Table • To locate probabilities and critical values in a t distribution, a t table is used. • We will use the t distribution and critical values listed in this table to compute t tests • Need to know: n, α, and location of rejection region Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 10

One-Sample t Test • A statistical procedure used to test hypotheses concerning a single

One-Sample t Test • A statistical procedure used to test hypotheses concerning a single group mean in a population with an unknown variance • Three assumptions are made: • 1. Normality – assume data in the population being sampled is normally distributed • 2. Random Sampling – assume that the data were obtained using a random sampling procedure • 3. Independence – assume that probabilities of each measured outcome in a study are independent Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 11

One-Sample t Test (cont. ) Privitera, Essential Statistics for the Behavioral Sciences © 2016

One-Sample t Test (cont. ) Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 12

Example 8. 1: One-Sample t Test • Albert and colleagues (2007) studied relatives of

Example 8. 1: One-Sample t Test • Albert and colleagues (2007) studied relatives of patients with obsessive-compulsive disorder (OCD). Social functioning of relatives was recorded using a 36 -item short form health survey (SF-36). Scores ranged from 0 (worst possible health) to 100 (best possible health). • The mean score for social functioning in the population of interest was 77. 43. • Suppose that researchers selected a sample of 18 relatives and recorded a mean social functioning score equal to 62. 00± 20. 94 (M±SD). • Table 8. 3 • Test whether or not the mean score in this sample significantly differs from that in the general population at a. 05 level of significance. Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 13

Example 8. 1: One-Independent Sample t Test (cont. ) • Step 1: State the

Example 8. 1: One-Independent Sample t Test (cont. ) • Step 1: State the hypotheses • H 0: μ = 77. 43 • The mean social functioning score is 77. 43, the same as that of the population • H 1: μ 77. 43 • The mean social functioning score is not equal to 77. 43 Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 14

Example 8. 1: One-Sample t Test (cont. ) • Step 2: Set the criteria

Example 8. 1: One-Sample t Test (cont. ) • Step 2: Set the criteria for making a decision • Level of significance for this test is. 05 • Since n = 18, the df = 17 (18 – 1 = 17) • To locate the critical values that make the cutoffs under a t distribution equal to. 05 in two tails combined, find 17 listed in the rows of Table B. 2 in Appendix B and go across the column for. 05 in two-tails combined. The critical values are 2. 110 Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 15

Example 8. 1: One-Sample t Test (cont. ) • Step 3: Compute the test

Example 8. 1: One-Sample t Test (cont. ) • Step 3: Compute the test statistic • Compute sample mean and standard deviation from information provided in Table 8. 3 (or in previous slide) • M = 62. 00; SD = 20. 94 • To find the t statistic, compute the estimated standard error and substitute values into the formula for t • To compute standard error, divide the sample SD by the square root of the sample size (n) 20. 94 4. 94 √ 18 • Find the t statistic by substituting values for the sample mean, the population mean stated in the null hypothesis, and the estimated standard error 62. 00 – 77. 43 -3. 126 4. 94 Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 16

Example 8. 1: One-Sample t Test (cont. ) • Step 4: Make a decision

Example 8. 1: One-Sample t Test (cont. ) • Step 4: Make a decision • To decide whether to reject or retain the null hypothesis, compare the obtained value to the critical value • Figure 8. 3 shows that the obtained value ( = -3. 126) falls beyond the critical value in the lower tail • The decision is to reject the null hypothesis Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 17

Example 9. 1: One-Sample t Test (cont. ) • Conclusion (in APA format): •

Example 9. 1: One-Sample t Test (cont. ) • Conclusion (in APA format): • Social functioning scores among relatives who care for patients with OCD (M = 62. 00) were significantly lower than scores in the general healthy population t(17) = -3. 126, p <. 05 Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 18

Effect Size for the One-Sample t Test • To estimate the size of the

Effect Size for the One-Sample t Test • To estimate the size of the effect in the population, effect size is computed • Typically only calculated after finding a significant effect (reject null) • Three measures of effect size: • Estimated Cohen’s d • Eta-Squared (proportion of variance) • Omega-Squared (proportion of variance) Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 19

Effect Size for the One-Sample t Test (cont. ) • Estimated Cohen’s d –

Effect Size for the One-Sample t Test (cont. ) • Estimated Cohen’s d – measure of effect size in terms of the number of SDs that mean scores shift above or below the population mean stated by the null hypothesis • The larger the value of d, the larger the effect • • Example 9. 1: • M = 62. 00; μ = 77. 43; SD = 20. 94 62. 00 – 77. 43 = -0. 74 , relatives caring for OCD patients • d = 20. 94 reduces mean social functioning scores by 0. 74 SD below the population mean Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 20

Effect Size for the One-Sample t Test (cont. ) • Proportion of Variance –

Effect Size for the One-Sample t Test (cont. ) • Proportion of Variance – measure of effect size in terms of the proportion or percent of variability in a dependent variable that can be explained or accounted for by a treatment • Treatment – any unique characteristic of a sample or any unique way that a researcher treats a sample • Can change value of a dependent variable • Associated with variability in a study Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 21

Effect Size for the One-Sample t Test (cont. ) • Eta-Squared ( ) •

Effect Size for the One-Sample t Test (cont. ) • Eta-Squared ( ) • • Eta-squared tends to overestimate proportion of variance explained by treatment • Omega-Squared ( ) • • Omega-squared gives a more conservative estimate Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 22

Effect Size for One-Sample t Test (cont. ) Privitera, Essential Statistics for the Behavioral

Effect Size for One-Sample t Test (cont. ) Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 23

Estimation for the One-Sample t Test • In example 9. 1, we computed a

Estimation for the One-Sample t Test • In example 9. 1, we computed a one-sample t test examining the social functioning of relatives of individuals with OCD compared to the general healthy population. • Scores ranged from 0 to 100 • In 18 participants, M = 62. 00; SD = 20. 94 • Find the 95% CI for these data Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 24

Estimation for the One-Sample t Test (cont. ) • Step 1: Compute the sample

Estimation for the One-Sample t Test (cont. ) • Step 1: Compute the sample mean and standard error • M = 62. 00 SD • SM = 4. 94 ( ) √n • Step 2: Choose the level of confidence and find critical values at that level of confidence • You want to find 95% confidence interval, so choose a 95% level of confidence • Step 2 (cont. ) • A 95% CI corresponds to a two-tailed test at a. 05 level of significance • Look in the t table. The degrees of freedom are 17 (df = n – 1 for onesample t test) • The upper critical value for the interval estimate is t = 2. 110 Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 25

Estimation for the One-Sample t Test (cont. ) • Step 3: Compute the estimation

Estimation for the One-Sample t Test (cont. ) • Step 3: Compute the estimation formula to find confidence limits for a 95% confidence interval • Multiply t times the SE: • = 2. 110(4. 94) = 10. 42 • = 62 + 10. 42 = 72. 42 • = 62 – 10. 42 = 51. 58 • The 95% confidence interval in this population is a score between 51. 58 and 72. 42 • Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 95% confident that the population mean falls within this range 26

APA in Focus: Reporting the t Statistic and Effect Size • When reporting results

APA in Focus: Reporting the t Statistic and Effect Size • When reporting results of a t test, you must include • The value for the test statistic • Degrees of freedom • p value • In addition, a figure or table is often used to summarize the means and standard error or standard deviations • Cohen’s d is most often reported with t tests Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 27

Question 1 Does the salary of clerical workers match that of the national average

Question 1 Does the salary of clerical workers match that of the national average ($35, 000)? a. Determine which type of t-test should be used for the research question. A Population Mean = $35, 000 One Sample t-test 28

Question 1 • b. Indicate the major null hypothesis (Ho) to be used for

Question 1 • b. Indicate the major null hypothesis (Ho) to be used for the research question. Ho: There is no difference between the salary of clerical workers and the national average ($35, 000). 29

Question 1 c. Use SPSS to run the test which we decided in the

Question 1 c. Use SPSS to run the test which we decided in the first step. One-Sample Test Value = 35000 t df Sig. (2 -tailed) Mean 95% Confidence Interval of the Difference Lower Current salary -13. 505 226 . 000 -7162. 952 -8208. 13 Upper -6117. 77 30

Question 1 • d. If an independent samples t-test is used, testing the equality

Question 1 • d. If an independent samples t-test is used, testing the equality of variances is needed to decide which pair of probability and t value should be used. § Indicate Ho to be used for testing the equality of variances. § Make a decision to accept or reject Ho. The independent samples t-test is not used in this research question. 31

Question 1 e. Make a decision to accept or reject the major null hypothesis

Question 1 e. Make a decision to accept or reject the major null hypothesis (Ho). The probability (0. 000) calculated with the test statistic (t = -13. 505) is less than alpha (0. 05), so we reject the null hypothesis (Ho). 32

Question 1 • f. Present finding in a graphic form and words. There was

Question 1 • f. Present finding in a graphic form and words. There was a statistically significant difference between the salary of clerical workers and the national average ($35, 000). On average, clerical workers made $7162. 952 less than the national average ($35, 000). 33

Question 2 2. Does the salary of security officers match that of the national

Question 2 2. Does the salary of security officers match that of the national average ($25, 000)? a. Determine which type of t-test should be used for the research question. A Population Mean = $25, 000 One Sample t-test 34

Question 2 b. Indicate the major null hypothesis (Ho) to be used for the

Question 2 b. Indicate the major null hypothesis (Ho) to be used for the research question. Ho: There is no difference between the salary of security officers and the national average ($25, 000). 35

Question 2 • c. Use SPSS to run the test which we decided in

Question 2 • c. Use SPSS to run the test which we decided in the first step. 36

Question 2 d. If an independent samples t-test is used, testing the equality of

Question 2 d. If an independent samples t-test is used, testing the equality of variances is needed to decide which pair of probability and t value should be used. § Indicate Ho to be used for testing the equality of variances. § Make a decision to accept or reject Ho. The independent samples t-test is not used in this research question. Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 37

Question 2 e. Make a decision to accept or reject the major null hypothesis

Question 2 e. Make a decision to accept or reject the major null hypothesis (Ho). The probability (0. 000) calculated with the test statistic (t = 14. 593) is less than alpha (0. 05), so we reject the null hypothesis (Ho). Privitera, Essential Statistics for the Behavioral Sciences © 2016 SAGE Publications, Inc. 38

Question 2 f. Present finding in a graphic form and words. There was a

Question 2 f. Present finding in a graphic form and words. There was a statistically significant difference between the salary of security officers and the national average ($25, 000). The security officers made on average $5938. 889 more than the national average ($25, 000). 39