In: Statistics and Probability
The year was 1986 and the space shuttle Challenger exploded, and seven astronauts were killed. Prior to the launch, when predicted atmospheric temperature was for freezing weather at the launch site, engineers for Morton Thiokol (the manufacturer of the rocket motor) prepared charts to make the case that the launch should not take place due to the cold weather. These arguments were rejected, and the launch tragically took place. Upon investigation after the tragedy, experts agreed that the disaster occurred because of leaky rubber O-rings that did not seal properly due to the cold temperature. Data indicating the atmospheric temperature at the time of 23 previous launches and the O-ring damage index are given below.
Part 5. Interpret the meanings of ?1 in the context of the
problem.
a) Expected decrease in the o-ring damage when the temperature goes
down by one unit b) Expected value of the temperature when the
o-ring damage is zero
c) Expected decrease in the o-ring damage when the temperature goes
up by one unit
d) Expected decrease in the temperature when the o-ring damage goes
up by one unit
Do you think that the engineers’ claim at Morton Thiokol was
reasonable (use an alpha of 0.05)?
Part 6. Please state your hypothesis. a) H0: β1 = 0, Ha: β1 > 0
b) H0: β1 = 0, Ha: β1 < 0 c) H0: β1 = 0, Ha: β1 ≠ 0 d) H0: β0 =
0, Ha: β0 < 0
Part 7. What type of test you will be using to test the hypothesis? a) t-test b) independent samples t-test c) F-test d) z-test
Part 8. What is the value of the test statistic? a) 3.63 b) 3.93 c) -3.63 d) -3.93
Flight number |
Temperature |
Damage |
1 |
66 |
0 |
2 |
70 |
4 |
3 |
69 |
0 |
5 |
68 |
0 |
6 |
67 |
0 |
7 |
72 |
0 |
8 |
73 |
0 |
9 |
70 |
0 |
41-B |
57 |
4 |
41-C |
63 |
2 |
41-D |
70 |
4 |
41-G |
78 |
0 |
51-A |
67 |
0 |
51-B |
75 |
0 |
51-C |
53 |
11 |
51-D |
67 |
0 |
51-F |
81 |
0 |
51-G |
70 |
0 |
51-I |
67 |
0 |
51-J |
79 |
0 |
61-A |
75 |
4 |
61-B |
76 |
0 |
61-C |
58 |
4 |
ΣX | ΣY | Σ(x-x̅)² | Σ(y-ȳ)² | Σ(x-x̅)(y-ȳ) | |
total sum | 1591.00 | 33.00 | 1057.30 | 157.65 | -253.74 |
mean | 69.17 | 1.43 | SSxx | SSyy | SSxy |
Sample size, n = 23
here, x̅ = Σx / n= 69.174
ȳ = Σy/n = 1.435
SSxx = Σ(x-x̅)² = 1057.3043
SSxy= Σ(x-x̅)(y-ȳ) = -253.7
estimated slope , ß1 = SSxy/SSxx =
-253.7391/1057.3043= -0.2400
intercept,ß0 = y̅-ß1* x̄ = 1.4348- (-0.24
)*69.1739= 18.0356
Regression line is, Ŷ= 18.0356 +
( -0.2400 )*x
========
5) c) Expected decrease in the o-ring damage when the temperature goes up by one unit
6)
b) H0: β1 = 0, Ha: β1 < 0
7) ) t-test
8)
estimated std error of slope =Se(ß1) = Se/√Sxx =
2.1465/√1057.3043= 0.0660
t stat = estimated slope/std error =ß1 /Se(ß1) =
(-0.24-0)/0.066= -3.63 (answer)