Chapter 7 Notes

©2008 by S. Gramlich

updated 5/15/2014

 

Confidence Intervals

! = Important Note

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

 

7-2

Steps to estimate Population Proportion of Successes (P) Confidence Interval (CI):

            1)  Check assumptions:  Binomial (2 outcomes: "Success" and "Failure"), Normality [x & (n-x) >=5], SRS

2)  given or find sample proportion of successes: phat = x/n (that's "p-hat" not "fat")

calculate sample proportion of failures: qhat = 1 - phat

3)  given Confidence Level (CL), find critical value zα/2 from Table A-2 NEGATIVE z Scores,

in the body of the table lookup closest value to (1-CL)/2

            ! 1-CL = α (alpha) which represents the area of the critical region

            ! critical value is the cutoff value (along horizontal axis) for the critical region

4)  calculate margin of error E = zα/2 * √(phat*qhat/n)

5)  calculate the CI:  phat - E < P < phat + E

 

Estimating Sample Size (N)

if phat known:  n = [zα/2]2 * phat*qhat/ E2

if phat unknown, let phat =50%:  n = [zα/2]2 * .5*.5/ E2 = [zα/2]2 * .25/ E2

*always round N up (not roundoff)

 

7-3

Student t (Gosset) Distribution Properties:

            1) Bell-Shaped & Symmetric (Frequency Polygon with zero skewness)

            2) Total Area Under the curve is 100%  or 1 (so each half is 50% or .5)

            3) Continuous

            4) Asymptotic to the horizontal axis (runs infinitesimally close to but never touches)

            5) Different for different sample sizes (approaches Normal as sample size gets large)

            6) mean (μ) = 0, standard deviation (σ) > 1

 

Determining whether to use t, z, or neither:

            1)  sigma (population standard deviation) known? yes-> z, no-> t

            2)  normal?

            3)  n > 30?

if answer is no to 2) & 3) then neither.

 

Steps to estimate Population Mean (μ) Confidence Interval (CI):

            1)  Check assumptions:  SRS, approximately Normal or n > 30, σ unknown

2)  given or find mean and standard deviation

3)  given Confidence Level (CL), find critical value tα/2 in body of Table A-3 with df = n-1,

! top row gives alpha α (alpha) = 1 - CL, always use 2 tail for CI"s

4)  calculate margin of error E = tα/2 * s/√n

5)  calculate the CI:  xbar - E < μ < xbar + E

 

TECHNOLOGY

using StatCrunch:

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

to find critical zα/2:   Stat - Calculators - Normal - (enter α/2 in last box) - Compute

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

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

 

! Both CI Procedures below will only give the CI and not the Critical Value or Margin of Error

for P CI:

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

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

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

 

for μ CI:

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

Stat - T statistics - One sample - with summary

-(select column) - Next - Confidence Interval - (enter CL for Level) - Calculate

 

! this procedure is for using the original data set (not the summary),

entered into a column of the StatCrunch spreadsheet first

Stat - T statistics - One sample - with data

-(select column) - Next - Confidence Interval - (enter CL for Level) - Calculate

 

EXCEL commands:

Find                             Excel Command

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

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

 

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.

 

to find Margin of Error E for μ CI:

Excel 2003:  Tools - Data Analysis - Descriptive Statistics - (check Confidence Level for Mean and enter CL) - OK

Excel 2007:  Data - Data Analysis - Descriptive Statistics - (check Confidence Level for Mean and enter CL) - OK