2uData ⮞Hypothesis Testing ⮞Comparing two proportions

Comparing two proportions

Consider two populations whose proportions of properties A are `pi_1` and `pi_2`. Drawing two sample with size `n_1` and `n_2` (`n_1,\ n_2>30`), and proportions of properties A are `p_1` and `p_2`.

In general, we would like to compare the proportions of properties A of these populations with confidence level (`1-alpha`) or significance level `alpha`. So the null hypothesis is:

    Ho : `pi_1=pi_2`(16)

Alternative hypothesis can be `pi_1!=pi_2` ; or `pi_1< pi_2` ; or `pi_1>pi_2` depend on cases.

To compare two proportions, put :

`p_c=(n_1p_1+n_2p_2)/(n_1+n_2)`(17)

`q_c=1-p_c`(18)

`sigma_c=sqrt(p_cq_c(1/n_1+1/n_2))`(19)

(`c` is the acronym of "common")

And test statistic is determined by :

`z=(p_1-p_2)/sigma_c`(20)

Because large samples are used, so test statistic conforms to standard normal distribution.

Example

To compare product quality, 30 products of company X and 40 products of company Y are controlled. The result shows that 12 products of company X and 14 products of company Y are classified as class A (the best class). Compare the proportions of class A products of these companies with confidence level of 95%.

From these data :

• sample size : `n_X=30` ; `n_Y=40`
• proportion of class A products : `p_X=12//30=0,40` ; `p_Y=14//40=0,35`

To compare the proportions of class A products of these companies `pi_X` and `pi_Y`, hypothesis testing is realized as follows:

• Construct pair of hypotheses :
  • Ho : `pi_X=pi_Y`
  • Ha : `pi_X!=pi_Y`
• Calculate :

    `p_c=(n_Xp_X+n_Yp_Y)/(n_X+n_Y)=(12+14)/(30+40)=0,3714`

    `q_c=1-p_c=1-0,3714=0,6286`

    `sigma_c=sqrt(p_cq_c(1/n_X+1/n_Y))=sqrt(0,3714xx0,6286xx(1/30+1/40))=0,1167`

• Test statistic for this hypothesis testing is :

    `z=(p_X-p_Y)/sigma_c`

• This is a two-sided test, test statistic conforms to standard normal distribution with `alpha=0,05`.
• So, critical value is : `z`*`=z_(0,025)=1,96` (percentage point table of standard normal distribution).
• From the data of samples :

    `z_o=(p_X-p_Y)/sigma_c=(0,40-0,35)/(0,1167)=0,428`

• Because `|z_o|< z`* ; we cannot reject Ho.
• The proportions of class A products of company X and company Y are equal.