Answer: (-0.620, 0.220)
Step-by-step explanation:
The formula to find the confidence interval for the difference in true proportion of the two groups. is given by :-
![\hat{p}_1-\hat{p}_2\pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}](https://tex.z-dn.net/?f=%5Chat%7Bp%7D_1-%5Chat%7Bp%7D_2%5Cpm%20z%5E%2A%20%5Csqrt%7B%5Cdfrac%7B%5Chat%7Bp%7D_1%281-%5Chat%7Bp%7D_1%29%7D%7Bn_1%7D%2B%5Cdfrac%7B%5Chat%7Bp%7D_2%281-%5Chat%7Bp%7D_2%29%7D%7Bn_2%7D%7D)
, where
= Sample size for first group.
= Sample size for second group.
= Sample proportion for first group.
= Sample proportion for second group.
z* = critical z-value.
Let first group be "group of prototypes by first process " and second group be "group of prototypes by second process".
= Proportion of defectives in the first batch.
= Proportion of defectives in the second batch.
From question , we have
,
,
,
By z-table , Critical value for 95% confidence interval is z* =1.96.
Substitute all values in formula , we get
![0.3-0.5\pm (1.96)\sqrt{\dfrac{0.3(1-0.3)}{10}+\dfrac{0.5(1-0.5)}{10}}](https://tex.z-dn.net/?f=0.3-0.5%5Cpm%20%281.96%29%5Csqrt%7B%5Cdfrac%7B0.3%281-0.3%29%7D%7B10%7D%2B%5Cdfrac%7B0.5%281-0.5%29%7D%7B10%7D%7D)
![-0.2\pm (1.96)\sqrt{0.021+0.025}](https://tex.z-dn.net/?f=-0.2%5Cpm%20%281.96%29%5Csqrt%7B0.021%2B0.025%7D)
![-0.2\pm (1.96)\sqrt{0.046}](https://tex.z-dn.net/?f=-0.2%5Cpm%20%281.96%29%5Csqrt%7B0.046%7D)
![-0.2\pm (1.96)(0.21448)](https://tex.z-dn.net/?f=-0.2%5Cpm%20%281.96%29%280.21448%29)
![-0.2\pm 0.4203808](https://tex.z-dn.net/?f=-0.2%5Cpm%200.4203808)
![=(-0.2-0.4203808,\ -0.2+0.4203808)\\\\=(-0.6203808,\ 0.2203808)\\\\\approx(-0.620,\ 0.220)](https://tex.z-dn.net/?f=%3D%28-0.2-0.4203808%2C%5C%20-0.2%2B0.4203808%29%5C%5C%5C%5C%3D%28-0.6203808%2C%5C%200.2203808%29%5C%5C%5C%5C%5Capprox%28-0.620%2C%5C%200.220%29)
Hence, a 95% confidence interval for the difference in the proportion of defectives is (-0.620, 0.220).