2uData ⮞Hypothesis Testing ⮞Tests on the proportion

Tests on the proportion

Consider a population which the proportion of elements belonging to group A is `pi`. Take randomly samples of size `n` (`n>30`) from this population. The proportion of elements belong to group A in samples are `p`. We have to decide some statements about the relation of the proportion `pi` with a value `a`, with confidence level `1-alpha` (or significance level `alpha`).

Tests on proportion is realized similarly as that to the mean with `sigma` already known and equal to `sqrt(a(1-a))`, `a` is the compared value, sample proportion is considered as normally distributed.

• null hypothesis has the form : Ho : `pi=a`
• depend on context, alternative hypothesis can be `pi!=a` ; or `pi>a` ; or `pi< a`.
• test statistic is :
  

  `z=(p-a)/(sqrt((a(1-a))/n))`(6)

• test statistic is normally distributed
• test on the proportion is realized when `na` and `n(1–a)` are greater or equal than 5.

Example

In supermarket S, shirts are acceptable when proportion of small defects not greater than 5%. In a quality control on 120 shirts of company C, there are 7 shirts having small defects. Is the shirts of company C satisfied the requirement of supermarket S with confidence interval of 95%?

From the context of exercise : `a=0,05` ; `n=120` ; `p=7/120=0,0583`

Because : `na=120xx0,05=6`  `(>5)` ; `n(1-a)=120xx0,95=114`  `(>5)`

We can apply test on proportion in this case.

Using the symbol `SD` for the proportion of shirts having small defects, we have pair of hypotheses:

•   Ho : `SD=0,05`
•   Ha : `SD>0,05`

For hypothesis testing, we use test statistic in formula (6) and this statistic conforms to normal distribution, `alpha=0,05` and rejection region is in the right of `t"*"`

Using percentage point table of Student's distribution (the last row): `z"*"=z_(0,05)=1,6449`

From the collected data :

  `z_o=(0,0583-0,05)/(sqrt((0,05xx0,95)/120))=0,4172`

Because `z_o< z`*, we cannot reject Ho, the proportion of defected shirts is not greater than 5%, shirts of company C are acceptable in supermarket S.