Review of Statistical Inference from STA 2023

 

Confidence Intervals and Significance Tests for:

·        one mean  

·        matched paired differences

·        difference of two independent means

·        one proportion  

·        difference of two independent proportions

 

 

 

Chapter 7:  CI and Significance Tests for means using the t distribution

 

Why do we use the t table?

 

 

 

What assumptions do we need to make?  How do we check them?

 

1.      Random Samples

 

 

2.      Population is Normal

 

 

 

Find Formulas on tables.

 

§         CI:

 

 

§         Sig. Test:   Ho:                           Ha:

 

                  TS:

 

                         p-value:

 

 

 

 

Read Minitab Output

 

 

Interpret Results

CI does NOT include # from Ho  Supports Ha

p-value small →  Supports Ha


Chapter 8 :  CI and Sig. Tests for p and p1-p2 using the z

 

Why do we use the z table?

 

What assumptions do we need to make? 

1.      Random Samples

 

2.      Check specific assumption for that test

§         One Sample Proportion (Confidence Interval)

Sample size must be at least equal to 5.

 

§         One Sample Proportion (Significance Test)

Must have at least ten successes and ten failures.

 

§         Two Sample Proportion (Confidence Interval)

Both sample sizes must be at least equal to 10.

 

§         Two Sample Proportions (Significance Test)

The number of successes and the number of failures must be at least 5 in each of the samples.

§                                             EASIEST – Use the most restrictive one for all cases and you will be ok.

 

Which estimator of p do we use?

·        One Sample Proportion (Confidence Interval)   

 

·        One Sample Proportion (Significance Test)     

 

·        Two Sample Proportion (Confidence Interval)           

 

·        Two Sample Proportions (Significance Test)     

 

 

 

 

 

 

 

Examples:  For each of the following examples, we will first determine which kind of problem it is.  The Minitab results for each problem are given in the last page. 

 

1.  Suppose that in a random sample of 50 babies conceived by an in vitro fertilization process (test tube babies), 35 are girls.  Does this sample evidence show convincingly that the in vitro process favors the female gender?  Explain.

  
 2.  Four sets of identical twins were selected at random.  One child was selected at random from each pair to form an "experimental group"- these four children were sent to school.  The other four children were kept at home as a control group.  At the end of the school year the IQ scores presented below were obtained.  Does this evidence justify the conclusion that lack of school experience has a depressing effect on IQ scores?

                                                Experimental          Control
                                Pair                Group                    Group

                 A                       110                      111
                 B                       125                      120
                 C                       139                      128
                 D                       142                      135
 

3.  A restaurant has recently started using a dessert cart that is pushed around the eating area for diners to view and, it is hoped, be unable to resist.  On the first ten evenings the cart was     used, the average expenditure per night for desserts was $130 compared to an average of $110 prior to the use of the cart.  The standard deviation for the ten evenings was $65.  Does it look like the cart will improve sales significantly?
 

4.  A social scientist believes that the fraction of Republicans in favor of the death penalty is greater than the fraction of Democrats in favor of the death penalty.  She asked random samples of 200 Republicans and 200 Democrats and found 46 Republicans and 34 Democrats in favor of it.  Do these data support the social scientist's belief?

 
 5.  The effect of alcohol consumption on the body appears to be much greater at high altitudes than at sea level.  To test this theory, a scientist randomly selects 12 subjects and  randomly divides them into two groups of six each.  One group is transported to an altitude of 12,000 feet, where each subject ingests a drink containing 100 cc of alcohol.  The second group receives the same drink at sea level.  After two hours, the amount of alcohol in the blood (grams per 100 cc) for each subject is measured.  The data are shown in the table.  Do the data provide sufficient evidence to support the theory that retention of alcohol in the blood is greater at high altitudes?

Sea Level

12,000 Feet

0.07

0.13

0.10

0.17

0.09

0.15

0.12

0.14

0.09

0.10

0.13

0.14

Test and Confidence Interval for One Proportion
Test of p = 0.5 vs p > 0.5

                                             Exact

 X  N  Sample p       95.0 % CI        P-Value
35 50  0.700000  (0.553918, 0.821382)    0.003

 

Test and CI for One Proportion with Wilson’s Estimate                                                     

Sample      X      N  Sample p         95.0% CI      

1          37     54  0.685185  (0.544476, 0.804761)

   

 

Paired T-Test and Confidence Interval
Paired T for exp - control

             N      Mean    StDev   SE Mean
exp          4    129.00    14.67      7.34
control      4    123.50    10.34      5.17
Difference   4      5.50     5.00      2.50

95% CI for mean difference: (-2.46, 13.46)

T-Test of mean difference = 0 (vs >0): T-Value = 2.20 P-Value = 0.058


 
T-Test of the Mean
Test of mu = 110.0 vs mu > 110.0

Variable  N   Mean  StDev  SE Mean     T     P
dessert  10  130.0   65.3     20.6  0.97  0.18

T Confidence Intervals

Variable N   Mean  StDev SE Mean   95.0 % CI
dessert 10  130.0   65.3    20.6  (83.3,176.7)
 
 
 

Test and Confidence Interval for Two Proportions

Sample    X    N   Sample p
     1   46  200   0.230000
     2   34  200   0.170000

Estimate for p(1) - p(2): 0.06
95% CI for p(1) - p(2): (-0.0181778, 0.138178)
Test for p(1) - p(2) = 0 (vs > 0):  Z = 1.50 P-Value = 0.066

 

Test and CI for Two Proportions with Wilson’s Estimate

Sample      X      N  Sample p

1          47    202  0.232673

2          35    202  0.173267

 

95% CI for p(1) - p(2):  (-0.0188205, 0.137632)



Two Sample T-Test and Confidence Interval
Two sample T for sea level vs 12000ft

         N     Mean   StDev   SE Mean
sea leve 6   0.1000  0.0219    0.0089
12000ft  6   0.1383  0.0232    0.0095

95% CI for mu sea leve - mu 12000ft:( -0.0678, -0.0089)

T-Test mu sea leve = mu 12000ft (vs <):  T = -2.94 P = 0.0082 DF= 9