Inferential Statistics z test and t test Chapter 8:

Inferential Statistics z test and t test Chapter 8 1

Objectives:

Objectives Brief review of symbols from Chapter 5 z- test symbols z- test calculations t- test symbols t- test calculation 2

It’s All Greek to Me… :

It’s All Greek to Me… µ = population mean (can also be for sample mean) ∑ = “the sum of” X = represents the individual scores N = represents the number of scores = mean for research sample = sum of X (adding the total number of scores) 3

It’s All Greek to Me… :

It’s All Greek to Me… µ = population mean (can also be for sample mean) X = represents the individual scores N = represents the number of scores = mean for research sample 4

It’s All Greek to z… :

It’s All Greek to z … z = z- score formula (used for individual score or data point) ơ = formula for population standard deviation (SD) 5

It’s All Greek to Me…(back to last week) :

It’s All Greek to Me…(back to last week) ơ = formula for sample standard deviation (SD ) (INCORRECT) S = formula for population SD (INCORRECT) s = formula for standard deviation to estimate population SD from sample data 6

It’s All Greek to Me…(back to last week) :

It’s All Greek to Me…(back to last week) ơ = formula for population standard deviation (SD ) S = formula for sample SD s = formula for standard deviation to estimate population SD from sample data 7

It’s All Greek to z… :

It’s All Greek to z … z- test formula Standard error of the mean (SEM) (Central limit theorem) 8

Sample Research Problem (p. 197) :

Sample Research Problem (p. 197) A researcher is wanting to find out whether a sample of children in an academic after-school program have a higher IQ mean than children in the general population. µ = 100 ơ = 15 N = 75 = 103.5 ( z -test is used when N = 31 or more) 9

Null and Alternative Hypotheses :

Null and Alternative Hypotheses For a one-tailed z -test (shows direction of difference)… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For a two-tailed z -test (difference but no indication of which direction)… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For either a one- or two-tailed t -test the objective is to reject the null hypothesis ( H 0 ) so the research can be statistically significant. In other words, you want the alternative hypothesis ( H a ) to be true. 10

z-test Calculation Demonstration :

z -test Calculation Demonstration µ = 100 ơ = 15 N = 75 = 103.5 We now use = 1.73 in the z -test formula 11

Null and Alternative Hypotheses :

Null and Alternative Hypotheses For a one-tailed z -test… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For a two-tailed z -test… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For either a one- or two-tailed t -test the objective is to reject the null hypothesis ( H 0 ) so the research can be statistically significant. In other words, you want the alternative hypothesis ( H a ) to be true. 12

Null and Alternative Hypotheses :

Null and Alternative Hypotheses For a one-tailed z -test… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For a one-tailed t -test the objective is to reject the null hypothesis ( H 0 ) so the research can be statistically significant. The number of significance is 1.645. Any z -test score that falls outside this range is considered significant on the .05 probability level. In the case of the research problem, z = +2.02 and this falls outside of 1.645. Therefore, the null hypothesis is rejected and the alternative hypothesis is accepted. 13

Null and Alternative Hypotheses :

Null and Alternative Hypotheses For a two-tailed z -test… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For a two-tailed t -test the objective is to reject the null hypothesis ( H 0 ) so the research can be statistically significant. The number of significance is 1.96. Any z -test score that falls outside this range is considered significant on the .05 probability level. In the case of the research problem, z = +2.02 and this falls outside of 1.96. Therefore, the null hypothesis is rejected and the alternative hypothesis is accepted. 14

Null and Alternative Hypotheses :

Null and Alternative Hypotheses Let’s assume z = - 1.52… For a one-tailed z -test… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For a two-tailed z -test… H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For a one- or two-tailed z -test, the null hypothesis ( H 0 ) would have to be accepted (failed to reject) as the score falls within 1.645 (one-tail) and 1.96 (two-tail). In this case, the alternative hypothesis could not be accepted. 15

Confidence Intervals Based on the z Distribution:

Confidence Intervals Based on the z Distribution Confidence interval: an interval of a certain width that we feel confident will contain μ Formula for the confidence interval: Statisticians recommend a 95% or a 99% confidence interval 16

Confidence Intervals :

Confidence Intervals 95% Confidence Interval formula: CI = ơ ) CI = ơ ) 99% Confidence Interval formula: CI = ơ ) CI = ơ ) 17

It’s All Greek to t… :

It’s All Greek to t … t = = (estimated standard error of the mean) s = formula for standard deviation to estimate population SD from sample data ( t -test is used when N = 30 or less) 18

It’s All Greek to t… :

It’s All Greek to t … Calculation and hypotheses processes very similar to z-test For a one-tailed t -test … H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population For a two-tailed t -test … H 0 : µ 0 µ 1 , µ academic program µ general population H a : µ 0 µ 1 , µ academic program µ general population 19

It’s All Greek to t… :

It’s All Greek to t … The differences are in application of critical values… One tailed z- test = 1.645 Two-tailed z- test = 1.96 For a t-test, critical values are determined with the help of using degrees of freedom ( df ) df = N – 1 Table A.3 (p. 409) 20

It’s All Greek to t… :

It’s All Greek to t … Using numbers from book N = 10 df = N – 1 (9) Critical values (from Table A.3) One tailed t- test = Two-tailed t - test = Using the t -value of +2.06, the null hypothesis is rejected for a one-tailed test; however, the hypothesis is accepted for a two-tail test. 21

Confidence Intervals Based on the t Distribution:

Confidence Intervals Based on the t Distribution For a one-sample t test, the confidence interval is determined by: Typically, statisticians recommend using either the 95% or 99% confidence interval Use the two-tailed level to obtain the critical value for t in the CI formula 22

Sample Research Problem (p. 197) :

Sample Research Problem (p. 197) A researcher is wanting to find out whether a sample of children in an academic after-school program have a higher IQ mean than children in the general population. µ = 100 ơ = 15 N = 75 = 103.5 ( z -test is used when N = 31 or more) 23

Key Take Aways :

Key Take Aways z- test and t -test are similar in terms of calculation processes z- test is used when N = 31 or greater t-test is used when N = 30 or less One-tailed hypothesis testing show direction of difference Two-tailed hypothesis testing shows on difference Objective in research is to reject the null hypothesis and accept alternative hypothesis 24

Key Take Aways :

Key Take Aways Critical values One-tailed z -test is 1.645 Two-tailed z -test is 1.96 One- or two-tailed t -tests is determined by degrees of freedom ( df ; N -1) using Table A.3 in Jackson text Confidence Intervals for z- test Use Use Confidence Intervals for t - test Use two-tailed critical values as based on df 25

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