Photo AI
Tiana is a quality controller in a clothes factory - AQA - A-Level Maths Pure - Question 18 - 2020 - Paper 3
Question 18
Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type... show full transcript
Worked Solution & Example Answer:Tiana is a quality controller in a clothes factory - AQA - A-Level Maths Pure - Question 18 - 2020 - Paper 3
Step 1
Find the probability that: a box contains exactly 5 shirts with a colour defect.
Answer
To calculate the probability of finding exactly 5 shirts with a colour defect in a box of 30 shirts, we use the binomial distribution:
Let ( X \sim B(30, 0.25) ).
The probability mass function is given by:
For our case:
Calculating this gives: .
Step 2
Step 3
Find the probability that: a box contains at least 20 shirts which do not have a fabric defect.
Answer
Let ( Y ) be the number of shirts with fabric defects. The probability of not having a fabric defect is ( 1 - 0.30 = 0.70 ).
Thus, we need ( P(Y \leq 10) ) for ( Y \sim B(30, 0.70) ):
[ P(Y \geq 20) = 1 - P(Y < 20) = 1 - P(Y \leq 19) \approx 0.7304. ]
Step 4
Using a 5% level of significance, find the critical region for x.
Answer
Given the null hypothesis ( H_0: p = 0.3 ) and the alternative hypothesis ( H_1: p < 0.3 ), we use:
( X \sim B(60, 0.3) ).
To find the critical region, we find the largest integer ( x ) such that ( P(X \leq x) \geq 0.05. )
After calculations, the critical region is found for ( x \leq 11. )
Step 5
Complete the test stating her conclusion in context.
Answer
In her sample, Tiana finds 13 shirts with a fabric defect.
Since 13 is greater than 11, we do not reject the null hypothesis ( H_0 ). Therefore, there is insufficient evidence to suggest that the proportion of shirts with a fabric defect has decreased.