Chapter 0 Comparig Two Proportios BPS - 5th Ed. Chapter 0
Case Study Machie Reliability A study is performed to test of the reliability of products produced by two machies. Machie A produced 8 defective parts i a ru of 40, while machie B produced 0 defective parts i a ru of 00. Defects Machie A 40 8 Machie B 00 BPS - 5th Ed. Chapter 0
Two-Sample Problems The goal of iferece is to compare the resposes to two treatmets or to compare the characteristics of two populatios. We have a separate sample from each treatmet or each populatio. The uits are ot matched, ad the samples ca be of differig sizes. BPS - 5th Ed. Chapter 0 3
Case Study Machie Reliability A study is performed to test of the reliability of products produced by two machies. Machie A produced 8 defective parts i a ru of 40, while machie B produced 0 defective parts i a ru of 00. This is a example of whe to use the two-proportio z procedures. Defects Machie A 40 8 Machie B 00 BPS - 5th Ed. Chapter 0 4
Iferece about the Differece p p Simple Coditios The differece i the populatio proportios is estimated by the differece i the sample proportios: ˆ p Whe both of the samples are large, the samplig distributio of this differece is approximately Normal with mea p p ad stadard deviatio p ( ) ( ) p p p + BPS - 5th Ed. Chapter 0 5
Iferece about the Differece p p Samplig Distributio BPS - 5th Ed. Chapter 0 6
Stadard Error Sice the populatio proportios p ad p are ukow, the stadard deviatio of the differece i sample proportios will eed to be estimated by substitutig ad for p ad p : SE = ( ) ( ) + BPS - 5th Ed. Chapter 0 7
BPS - 5th Ed. Chapter 0 8
( ) ± z Case Study: Reliability Compute a 90% cofidece iterval for the differece i reliabilities (as measured by proportio of defectives) for the two machies. Cofidece Iterval ( ) ( ) + 8 = ±.645 40 00 = 0.00± 0.048 = 0.0397 to 0.0439 8 40 40 8 40 00 00 00 We are 90% cofidet that the differece i proportio of defectives for the two machies is betwee -3.97% ad 4.39%. Sice 0 is i this iterval, it is ulikely that the two machies differ i reliability. + BPS - 5th Ed. Chapter 0 9
Adjustmet to Cofidece Iterval Plus Four Cofidece Iterval for p p The stadard cofidece iterval approach yields ustable or erratic ifereces. By addig four imagiary observatios (oe success ad oe failure to each sample), the ifereces ca be stabilized. This leads to more accurate iferece of the differece of the populatio proportios. BPS - 5th Ed. Chapter 0 0
Adjustmet to Cofidece Iterval Plus Four Cofidece Iterval for p p Add 4 imagiary observatios, oe success ad oe failure to each sample. Compute the plus four proportios. umber of successes + p ~ = ~ umber of successes p = + + Use the plus four proportios i the formula. ~ ( ) ( ) ( ) ~ ~ ~ ~ p p p p p ~ p ± z + + + + BPS - 5th Ed. Chapter 0
Case Study: Reliability Plus Four 90% Cofidece Iterval ~ p ( ~ p ~ p ) 8 + 40 + = = ± z ~ p 9 4 ( ~ p ) ~ p ( ~ p ) + + + ~ p 9 = ±. 645 4 0 = 0. 0040 ± 0. 0434 = 0. 0394 to 0. 0474 + 00 + = = 9 4 4 9 4 + 0 0 0 (This is more accurate.) 0 We are 90% cofidet that the differece i proportio of defectives for the two machies is betwee -3.94% ad 4.74%. Sice 0 is i this iterval, it is ulikely that the two machies differ i reliability. BPS - 5th Ed. Chapter 0
The Hypotheses for Testig Two Proportios Null: H 0 : p = p Oe sided alteratives H a : p > p H a : p < p Two sided alterative H a : p p BPS - 5th Ed. Chapter 0 3
Pooled Sample Proportio If H 0 is true (p =p ), the the two proportios are equal to some commo value p. Istead of estimatig p ad p separately, we will combie or pool the sample iformatio to estimate p. This combied or pooled estimate is called the pooled sample proportio, ad we will use it i place of each of the sample proportios i the expressio for the stadard error SE. = total umber of successes i both samples total umber of observatios i both samples pooled sample proportio BPS - 5th Ed. Chapter 0 4
Test Statistic for Two Proportios Use the pooled sample proportio i place of each of the idividual sample proportios i the expressio for the stadard error SE i the test statistic: z = ( ) ( ) + z z = = ( ) ( ) + ( ) + BPS - 5th Ed. Chapter 0 5
P-value for Testig Two Proportios H a : p > p P-value is the probability of gettig a value as large or larger tha the observed test statistic (z) value. H a : p < p P-value is the probability of gettig a value as small or smaller tha the observed test statistic (z) value. H a : p p P-value is two times the probability of gettig a value as large or larger tha the absolute value of the observed test statistic (z) value. BPS - 5th Ed. Chapter 0 6
BPS - 5th Ed. Chapter 0 7
Case Study Summer Jobs A uiversity fiacial aid office polled a simple radom sample of udergraduate studets to study their summer employmet. Not all studets were employed the previous summer. Here are the results: Summer Status Me Wome Employed 78 593 Not Employed 79 39 Total 797 73 Is there evidece that the proportio of male studets who had summer jobs differs from the proportio of female studets who had summer jobs. BPS - 5th Ed. Chapter 0 8
Case Study: Summer Jobs The Hypotheses Null: The proportio of male studets who had summer jobs is the same as the proportio of female studets who had summer jobs. [H 0 : p = p ] Alt: The proportio of male studets who had summer jobs differs from the proportio of female studets who had summer jobs. [H a : p p ] BPS - 5th Ed. Chapter 0 9
Case Study: Summer Jobs Test Statistic = 797 ad = 73 (both large, so test statistic follows a Normal distributio) Pooled sample proportio: 78 + 593 3 = = 797 + 73 59 stadardized score (test statistic): 78 593 z = 797 73 = 3 3 + 59 59 797 73 5. 07 BPS - 5th Ed. Chapter 0 0
Case Study: Summer Jobs. Hypotheses: H 0 : p = p H a : p p. Test Statistic: 3 3 + 59 59 797 73 3. P-value: P-value = P(Z > 5.07) = 0.000000396 (usig a computer) P-value = P(Z > 5.07) < ( 0.9998) = 0.0004 (Table A) [sice 5.07 > 3.49 (the largest z-value i the table)] 4. Coclusio: 78 593 z = 797 73 = 5. 07 Sice the P-value is smaller tha α = 0.00, there is very strog evidece that the proportio of male studets who had summer jobs differs from that of female studets. BPS - 5th Ed. Chapter 0