Question

random variables and distributions
notation for the population proportion and sample proportion
zahra is a company specializing in high - end cosmetics. its signature nail polish color is soft lilac. one day, the inventory manager took a sample of 63 bottles of
nail polish at the companys factory. she found that 12 of those bottles were soft lilac. on that day, 7 out of every 26 bottles of nail polish in the factory were
soft lilac.
for the inventory managers sample, find and write with proper notation the population proportion and sample proportion of bottles that were soft lilac. write
the proportions as decimals (not percentages) rounded to two decimal places.
(a) population proportion select =
(b) sample proportion: select =

Explanation:

Step1: Calculate the population proportion

The population proportion \(p\) is given by the ratio of the number of Soft Lilac bottles in the population to the total number of bottles in the population. Here, \(p=\frac{7}{26}\).

$$p=\frac{7}{26}\approx0.27$$

Step2: Calculate the sample proportion

The sample proportion \(\hat{p}\) is given by the ratio of the number of Soft Lilac bottles in the sample to the total number of bottles in the sample. Here, \(\hat{p}=\frac{12}{63}\).

$$ \hat{p}=\frac{12}{63}=\frac{4}{21}\approx0.19 $$

Answer:

(a) Population proportion \(p = 0.27\)
(b) Sample proportion \(\hat{p}=0.19\)