1. A substance used in biological and medical research is shipped by airfreight
ID: 2949580 • Letter: 1
Question
1. A substance used in biological and medical research is shipped by airfreight to users in cartons of 1000 ampules. The data (see Table 1), involving 10 shipments, were collected on the number of times the carton was transferred from one aircraft to another over the shipment route (X) and the number of ampules found to be broken upon arrival (Y). (note you can cut and paste the table below into R using the scan function) Table 1: Airfreight breakage 1 1 16 3 2 17 4 0 12 5 3 22 6 1 13 8 1 15 9 2 19 10 0 11 (a) Using a 0.1 carry out a test to see if the number of transfers has an impact on the number of broken ampules (b) Construct 95% confidence intervals for both the intercept and slope parametersExplanation / Answer
Solution
Given xi = number of transshipment transfers and yi = number of broken ampoules on arrival of the ith consignment, i = 1 to n (10)
Back-up theory in brief
The linear regression model Y = ?0 + ?1X + ?, ………………………………………..(1)
where ? is the error term, which is assumed to be Normally distributed with mean 0 and variance ?2.
Estimated Regression of Y on X is given by: Y = ?0cap + ?1capX, …………………….(2)
where ?1cap = Sxy/Sxx and ?0cap = Ybar – ?1cap.Xbar..…………………………….…..(3)
Mean X = Xbar = (1/n)sum of xi ………………………………………….……….….(4)
Mean Y = Ybar = (1/n)sum of yi ………………………………………….……….….(5)
Sxx = sum of (xi – Xbar)2 ………………………………………………..…………....(6)
Syy = sum of (yi – Ybar)2 ……………………………………………..………………(7)
Sxy = sum of {(xi – Xbar)(yi – Ybar)} ……………………………………………….(8)
All above sums are over i = 1, 2, …., n,n = sample size ……………………………..(9)
Estimate of ?2 is given by s2 = (Syy – ?1cap2Sxx)/(n - 2)……………………………..(10)
Standard Error of ?1cap is sb, where sb2 = s2/Sxx ……………………………………..(11)
Standard Error of ?0cap is sa, where sa2 = sb2{(sum of xi2 over i = 1, 2, …., n)/n}……(12)
100(1 - ?)% Confidence Interval (CI) for ?1 = ?1cap ± {SE(?1cap) x tn – 2, ?/2}…………(13)
100(1 - ?)% Confidence Interval (CI) for ?0 = ?0cap ± {SE(?0cap) x tn – 2, ?/2}…………(14)
?0cap represents the y-intercept and ?1cap represents the slope of the regression line …(15)
Preparatory Calculations
Data
i
xi
yi
1
1
16
2
0
9
3
2
17
4
0
12
5
3
22
6
1
13
7
0
8
8
1
15
9
2
19
10
0
11
Summary of Excel Calculations
Detail
Value
Back-up Theory
n
10
-
Xbar
1.00
4
ybar
14.2
5
Sxx
10
6
Syy
177.6
7
Sxy
40
8
?1cap
4
3
?0cap
10.2
3
s^2
2.2
10
sb^2
0.22
11
sa^2
0.44
12
s
1.4832
10
sb
0.4690
11
sa
0.6633
12
Now, to work out the solutions
Part (a) ??
Testing if number of transshipment transfers have any impact on the number broken ampules, is equivalent to testing ?1 = 0.
Null hypothesis H0: ?1 = 0 Vs Alternative H1: ?1 ? 0
Test statistic: t = ?1cap/SE(?1cap) = ?1cap/sb = 8.528
Under H0, t ~ tn - 2 and hence critical value = tn – 2,?/2 = t8, 0.05 = 1.860 [given ? = 0.1] and
p-value = P(|t8| > 8.528) = 2.7487E-05
Since tcal > tcrit, H0 is rejected [also confirmed by p-value < ?]
So, we conclude that
transshipment transfers have impact on ampule breakage. ANSWER 1
Part (b)
95% CI for ?1cap (slope parameter) = [3.128, 4.872] [vide (13)] ANSWER 2
95% CI for ?0cap (intercept) = [8.967, 11.433] [vide (14)] ANSWER 3
DONE
i
xi
yi
1
1
16
2
0
9
3
2
17
4
0
12
5
3
22
6
1
13
7
0
8
8
1
15
9
2
19
10
0
11
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