Chapter 8 Notes

©2008 by S. Gramlich

Hypothesis Testing (for 1 sample)

! = Important Note

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

 

8-2, 8-3, 8-5

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

For 1 sample

8-3 population Proportion P use z-test, 8-5 population Mean μ use t-test

! its unreasonable to assume that population standard deviation is known so we skip section 8-4

1)  Identify Hypotheses:

            H0 =  Null Hypothesis, always contains equals, nothing happening

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

 

2)  Write given (alpha, n, sample stats)

alpha α = significance level = area of critical region shaded in either 1 or 2 tails

! if Alternative Hypothesis is > or < then 1 tail (α), if not equal to (≠) then 2 tail (α/2)

! if alpha is not given then assume it be 5%

            sample stats:

            for z-test, sample proportion of successes (phat)

for t-test, sample mean (xbar) and sample standard deviation (s)

 

Assumptions:

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

for t-test:  observations Independent, Normal or n >30, SRS, σ unknown

 

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

            for z-test, find zc from Table A-2 NEGATIVE z Scores

            for t-test, find tc in Table A-3

            ! I use the subscript c instead of α/2 because α/2 is only for a 2 tailed test,

            you can alternatively adopt the notation α/2 for 2 tail and α for 1 tail as the subscript

 

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

            for z-test, z = (phat - P) / (√(P*Q/n)

            ! for P use Hypothesized P from H0

            for t-test, t = (xbar - μ) / (s/√n)

            ! for μ use Hypothesized μ from H0

 

5)  Draw bell curve & shade critical region.

            for z-test, label hypothesized P & mean z = 0 in middle, cv & ts

            for t-test, label hypothesized μ & 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

 

8)  P-Value Method DR

            if p-val <= alpha, Reject null.

            if p-val > alpha, Fail to Reject null.

 

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

 

Type I error (alpha represents %) = when reject null (based on HT) but in reality shouldn't have

Type II error = when fail to reject null (based on HT) but in reality shouldn't have

Power = reject null (based on HT) and this was, in reality, correct

 

TECHNOLOGY

using StatCrunch:

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

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

to find cv tc:   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 3rd 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

for P z-test:

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

Stat - Proportions - One sample - with summary - (enter # successes, # observations) - Next -

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

 

! If you have the original data set then use this procedure

Enter the data into a column of the StatCrunch spreadsheet first.

Stat - Proportions - One sample - with data - (enter # successes, # observations) - Next -

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

 

for μ t-test:

! 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 - One Sample -(select column) - Next -

-Hypothesis Test - (enter null & alternative) - 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 zc                =NormSinv(α or α/2 ) {! will only give the negative critical z value}

cv tc                =tinv(α,df)   {! will only give the positive critical t value, have to enter 2α for a 1 tailed test}

z P-value        =NormSdist(ts) or =NormDist(xbar, μ, s/√n, true)

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.