A candy manufacturer selects mints at random from the production line and weighs
ID: 3174458 • Letter: A
Question
A candy manufacturer selects mints at random from the production line and weighs them. For one week, the day shift weighed n_1 = 194 mints and the night shift weighed n_2 = 162 mints. The numbers of these mints that weighed at most 21 grams was y_1 = 28 for the day shift and y_2 = 11 for the night shift. Let p_1 and p_2 denote the proportions of mints that weigh at most 21 grams for the day and night shifts, respectively. Give a point estimate of p_1. Give the endpoints for a 95% confidence interval for p_1. Give a point estimate of p_1 - p_2. Find a one-sided 95% confidence interval that gives a lower bound for p_1 - p_2.Explanation / Answer
(a) p1=y1/n1=28/194=0.1443
(b) here p1=0.1443, n1=194 then SE(p1)=sqrt(p1(1-p1)/n1)=sqrt(0.1443*(1-0.1443)/194)=0.0252
(1-alpha)*100% confidence interval for p1=p1± z(alpha/2)*SE(p1)
95% confidence interval for sample p1=0.1443±z(0.05/2)*0.0252=
=0.1443±1.96*0.0252=0.1443±0.0494=(0.0949,0.1937)
(c)p2=y2/n2=11/162=0.0679
p1-p2=0.1443-0.0679=0.0764
(d) SE(p1-p2)=sqrt(p1(1-p1)/n1 + p2(1-p2)/n2)=
=sqrt(0.1443*(1-0.1443)/194 +0.0679*(1-0.0679)/162)=0.032
(1-alpha)*100% confidence interval for p1-p2=(p1-p2)± z(alpha/2)*SE(p1-p2)
95% confidence interval for sample p1-p2=0.0764±z(0.05/2)*0.032=
=0.0764±z(0.05/2)*0.032=0.0764±0.0627=(0.0137,0.1391)
lower bound is 0.0137
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