Chapter 9 Notes

©2008 by S. Gramlich

updated 12/3/2020

Hypothesis Testing (for 2 samples)

! = Important Note

! These Notes are not meant to replace Reading. Read Chapter first.

Hypothesis (Significance) Test (HT) = procedure for testing a claim about 2 population parameters

For 2 samples

9-2 z-test for 2 population Proportions P₁ & P₂

9-3 t-test for 2 population Means μ₁ & μ₂ (independent samples)

!Part 1 only: Assume Unknown Population Variances/Standard Deviations

9-4 t-test for 2 populations Means μ₁ & μ₂ (matched pairs)

! we will only focus on 2 tailed tests in this chapter

1) Identify Hypotheses:

H₀ = Null Hypothesis, always contains equals, nothing happening

for z-test: H₀: P₁ = P₂ --> P₁ - P₂ = 0

for t-test (independent): H₀: μ₁ = μ₂ --> μ₁ - μ₂ = 0

for t-test (matched): H₀: μ₁ = μ₂ --> μ₁ - μ₂ = 0 --> μ_d = 0

H₁ = Alternative Hypothesis, does not contain equals, something happening

for z-test: H₁: P₁ ≠ P₂ --> P₁ - P₂ ≠ 0

for t-test (independent): H₁: μ₁ ≠ μ₂ --> μ₁ - μ₂ ≠ 0

for t-test (matched): H₁: μ₁ ≠ μ₂ --> μ₁ - μ₂ ≠ 0 --> μ_d ≠ 0

2) Write given (alpha, n, sample stats) and check Assumptions

alpha α = significance level = area of critical region shaded in 2 tails

! since Alternative Hypothesis is not equal to (≠) then 2 tailed test (α/2)

sample stats:

for z-test: 2 sample proportions of successes (phat₁ & phat₂)

for t-test (independent): 2 sample means (xbar₁ & xbar₂) and 2 sample standard deviation (s₁ & s₂)

for t-test (matched): 1 sample difference mean (dbar) and 1 sample difference standard deviation (s_d)

Assumptions (for each sample):

for z-test: Binomial (2 outcomes: "Success" and "Failure"), Normality [x & (n-x) >=5], SRS

for t-tests: observations unrelated (Independent) or related (Dependent or Matched), Normal or n >30, SRS, σ₁ & σ₂ unknown

3) find critical value (cv) = cutoff value for the critical region

for z-test: find z_α/2 from Table A-2 NEGATIVE z Scores

for t-test (indep): find t_α/2 in Table A-3 (use smaller df)

for t-test (matched): find t_α/2 in Table A-3 (df= #pairs - 1)

4) calculate test statistic (ts) = convert sample stat to z or t score

for z-test: z = [(phat₁ -phat₂) - (P₁ - P₂)] / [√(pbar*qbar/n₁ + pbar*qbar/n₂)]

where pbar = (x₁+x₂) / (n₁+n₂), qbar = 1 - pbar

! for P₁ - P₂ use 0

for t-test(indep): t = [(xbar₁ - xbar₂) - (μ₁ - μ₂)] / √[ s₁²/n₁ + s₂²/n₂]

! for μ₁ - μ₂ use 0

for t-test, t = (dbar - μ_d) / (s_d/√n)

! for μ_d use 0

5) Draw bell curve & shade critical region.

for z-test: label hypothesized P₁ - P₂ & mean z = 0 in middle, cv & ts

for t-test (indep): label hypothesized μ₁ - μ₂ & mean t = 0 in middle, cv & ts

for t-test (matched): label hypothesized μ_d & mean t = 0 in middle, cv & ts

6) Traditional Method Decision Rule (DR)

if ts is visually inside critical region, reject null.

if ts is visually outside critical region, fail to reject null.

7) find P-value = area under curve corresponding to test statistic

for z-test: find ts z score and corresponding P-val from body of table; double this value for a 2 tailed test

for t-tests: find closest range of values for ts t score in body of table, P-val will be range of values in top row (2-tail)

8) P-Value Method DR

if p-val <= alpha, reject null.

if p-val > alpha, fail to reject null.

9) Calculate Confidence Interval (CI)

for z-test: (phat₁ -phat₂) - E < P₁ - P₂ < (phat₁ -phat₂) + E

where E = z_α/2 * √(phat₁*qhat₁/n₁ + phat₂*qhat₂/n₂)]

for t-test (indep): (xbar₁ - xbar₂) - E < μ₁ - μ₂ < (xbar₁ - xbar₂) + E

where E = t_α/2 * √[ s₁²/n₁ + s₂²/n₂]

for t-test (matched): dbar - E < μ_d < dbar+ E

E = t_α/2 * s_d /√n

10) CI DR

if CI does NOT contain zero, reject.

if CI contains zero, fail to reject.

11) State conclusion in words relative to the original claim.

TECHNOLOGY

using StatCrunch calculators:

! Don't need to enter any info in StatCrunch spreadsheet for critical value procedures

to find cv z_c: Stat - Calculators - Normal - (enter α or α/2 in last box) - Compute

to find cv t_c: Stat - Calculators - T - (df=n-1 in 1st box, enter α or α/2 in last box) - Compute

! 3rd box will give critical value in both above procedures

to find z P-value: Stat - Calculators - Normal - (enter ts in 3rd box) - Compute

to find t P-value: Stat - Calculators - T - (df=n-1 in 1st box, enter ts in 2nd box) - Compute

! in both procedures above, double the value to get P-val for a 2 tailed test

! StatCrunch HT procedures below don't give the cv

using StatCrunch procedures:

! StatCrunch HT procedures below don't give the cv

for 2 sample P z-test HT:

! Don't need to enter any info in StatCrunch spreadsheet for this procedure

Stat - Proportions - Two sample - with summary - (enter # successes, # observations for each sample) - Next -

-Hypothesis Test - (enter null & alternative) - Calculate

for 2 sample P z-CI:

! Don't need to enter any info in StatCrunch spreadsheet for this procedure

Stat - Proportions - Two sample - with summary - (enter # successes, # observations for each sample) - Next -

- Confidence Interval - (enter CL for Level) - Calculate

for μ t-test (indep) HT:

! This procedure can be used with summary or original data set

Stat - T statistics - Two Sample - (select column) - Next -

- Hypothesis Test - (enter null & alternative) - Calculate

for μ t-test (indep) CI:

! This procedure can be used with summary or original data set

Stat - T statistics - Two Sample - (select column) - Next -

- Confidence Interval - (enter CL for Level) - Calculate

for μ t-test (matched pair) HT:

! this procedure can only be done if you have the original data set (not the summary),

entered into a column of the StatCrunch spreadsheet first

Stat - T statistics - Paired - (select column) - Next -

- Hypothesis Test - (enter null & alternative) - Calculate

for μ t-test (matched pair) CI:

! this procedure can only be done if you have the original data set (not the summary),

entered into a column of the StatCrunch spreadsheet first

Stat - T statistics - Paired - (select column) - Next -

- Confidence Interval - (enter CL for Level) - Calculate

EXCEL commands:

! enter the given value inside the parentheses

! Excel "Dist" commands will only give area to the Left of z or x

Find Excel Command

cv z_c =NormSinv(α/2) {! will only give the negative critical z value}

cv t_c =tinv(α,df) {! will only give the positive critical t value; enter α and not α/2}

z P-value =NormSdist(ts) {! enter ts as negative and double for 2tail}

t P-value =tDist(ts, df, tails) {! can't enter ts as a negative value}

The "Using Technology" section in the text gives Excel procedures using DDXL.

The DDXL add-in can be found in CourseCompass under Chapter Contents (in left margin) - Tools for Success -DDXL.

! DDXL must be downloaded from CourseCompass and then added into the Excel software. If you are using a college computer, the add-in will be removed when the computer is reset.

! In Excel 2007, you have to highlight the data before launching into a DDXL procedure.

EXCEL Data Analysis Procedures for t-tests {! there is no procedure the z-test}

! This procedure can only be done if you have the ORIGINAL data set

! The Data Analysis add-in must be added in first

(indep) Tools - Data Analysis - t-test:Two-Sample Assuming Unequal Variances - ok - (highlight & enter data) - ok

(match) Tools - Data Analysis - t-test:Paired Two Sample for Means - ok - (highlight & enter data) - ok

! Excel 2007 the Data Analysis procedures are found the Data menu not the Tools menu