Name ________________________ Stat 3411
Spring 2009
Exam 2
(1) (12 points) The gravitational constant, g, is estimated by measuring the length, L, and period,  of a
pendulum using the relationship
The measured values of L and  have the following means and standard deviations
L = 5 feet
L = 0.02 feet
 = 2.5 sec
 = 0.1 sec
Give an unsimplified numerical expression for the approximate standard deviation for the computed g
value based on measured values of L and 
g
g 39.5 39.5


L  2
2.52
39.5* L

2
g 2*39.5* L 2*39.5*5



3
2.53
 g 
 g 
 39.5 
 2*39.5*5 
2
St Dev( g )    Var  L     Var    
0.022  
 0.1
2 
3
2.5
 L 
  
 2.5 


2
2
2
2
(2) (12 points) Pistons produced from a grinding process have edge widths which are normally distributed
with population mean 0.5 inches and population standard deviation 0.02 inches. If 25 pistons are
randomly sampled, what is the probability that the sample mean of these 25 pistons, , is within ± 0.01
inches of 0.5 inches?
2
0.022
n
25
0.02
Var  X  =
=0.004
5
X-
X-0.5
Z=

Var  X  0.004
Var  X  

  0.5  0.01  0.5
 0.5  0.01  0.5 
P  0.5  0.01  X  0.5  0.01  P 
Z

0.004
0.004


 P  2.5  Z  2.5   1  2*0.0062  0.9876
(3) (10 points) We have data for lifetimes of motors at two different temperatures. To model the lifetimes
of such motors, others have used Weibull models with the same  value for both temperatures. You are
given the task of deciding if
1. The lifetimes in our motor data at these temperatures appear to follow Weibull distributions
2. The  values of the Weibull distributions appear similar for both temperatures.
(a) What would you do with the data to investigate these questions? Just give the basic idea. You don’t
need to give details.
Draw Weibull probability plots for both temperatures on the same plot.
(b) How would you decide whether the lifetimes of the motors appear to follow Weibull distributions?
Check if both plots are reasonably straight.
(c) How would you decide whether the  parameters of the Weibull distributions appear fairly similar for
both temperatures?
Since  corresponds to the slope of a Weibull probability, plot check if the two plots are reasonably
parallel.
(4) (23 points) In comparing fall times with and without a paper clip, three students recorded the
following rounded results when we flew helicopters in class. They were Xerox paper with rectangular
wings flown from the same height.
Student
Folded
Paper Clip
1
2
3
3.5
3.3
1.8
2.5
2.5
1.6
(a) Find the 95% confidence interval for the mean difference in fall times with and without a paper clip.
Show steps in finding your answer.
The first thing to ask yourself when comparing 2 groups is whether the data are paired or not. These data
are paired, since each student flew the same helicopter with and without a paper clip.
d  t0.025 SEd
SEd  Var  d  
S d2 S d

n
3
df  3  1  2
t0.025  4.303
d  0.67
S d  0.42
 0.42 
0.67  4.303 

 3 
Helicopter
Folded
Paper Clip
Difference
1
2.5
2.5
1.6
1.000
3
3.5
3.3
1.8
n
df
Confidence
tabled t
Mean
St Deviation
St Error
3
2
95%
4.303
0.667
0.416
0.240
2
0.800
0.200
Confidence Interval
t*SE
Lower Limit
Upper Limit
1.034
-0.368
1.701
(b) We are planning to investigate the effect of adding a lighter paper clip to the same type of helicopter.
We want to end up with a confidence interval that is more precise, not so wide as the confidence interval in
part (a). What are 2 things we could do to end up with a narrower, more precise confidence interval than in
part (a) for the difference between mean flight times of folded and paper clipped helicopters?
1. Increase sample size. Make and fly more paper helicopters.
2. Decrease variability
One of the 3 helicopters dropped quicker than the other 2.
Try to make the helicopter flights more consistent.
Make helicopters more consistently, and/or
Make the dropping of the helicopters more consistent
(5) (31 points) Students measured the stretch of two brands of 10 lb test fishing line under a load of 3.5 kg.
Brand A
0.95
0.97
0.99
Brand B
1.05
1.02
0.99
We are testing the null hypothesis that both brands have the same means versus the alternative that the
means are different. Perform the test assuming equal population variances.
(a) Based on the t-table, for what values of the calculated t would you reject the null hypothesis at the
 level?
The first thing to ask yourself when comparing 2 groups is whether the data are paired or not. These data
are not paired. These are line pieces of different brands. There is nothing in the problem to indicate that
these were paired in any way.
When assuming equal variances, df = df1 + df1 = 2 + 2 = 4
t Critical two-tail
2.776
Reject H0 for |t| ≥ 2.776
(Specify for what values of t you would reject H0 .
(b) Find the calculated value of t for these data. Show all steps in the calculation.
H 0 : 1  2  0
H a : 1   2  0
Estimate of 1  2 : X 1  X 2  0.97  1.02
SE X1  X 2  Var  X 1  X 2  
2
S p2
S p2


3
3
3
2
2
2
2
2
2
2
df1S1  df 2 S2 2S1  2S2 S1  S2 0.02  0.032
2
Sp 



 0.00065
df1  df 2
22
2
2
3

2
0.00065 0.00065

 0.0208
3
3
X  X 2  0  0.97  1.02   0
t 1

 2.40
SE X1  X 2
0.0208
SE X1  X 2 
Calculated t = -2.402 or 2.402 depending on which way you took differences.
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Brand A
0.97
0.0004
3
0.00065
0
4
-2.402
0.037
2.132
0.074
2.776
Brand B
1.02
0.0009
3
(c) Find the p-value for this test to within a range on the t-table.
2*0.025 < p < 2*0.0.5
0.05 < p < 0.10
(d) Based on your answers above, do you reject the null hypothesis? How did you decide?
We fail to reject H0 either since
 |t| = 2.402 < 2.776 or since
 p-value > 0.05
(e) Summarize your conclusions about the comparison of mean stretches for these two brands of fishing
line based on the hypothesis test results above.
We do not have sufficiently strong evidence to reject the possibility that We cannot conclude that
both brands have the same mean stretch; we only know that no difference is one of the possibilities we are
not rejecting. A confidence interval would be more useful in describing how much of a difference the
difference might be, at least with 95% confidence.
(6) (12 points) Another group of students plans to compare voltage measurements of D batteries at 20
degrees C and 0 degrees C. They plan to use the same sort of analysis as you used in problem 5.
(a) What 3 assumptions do they need to make in order to use the same analysis as in problem 5?
1. Both populations of voltages are normally distributed.
2. The populations have the same variances.
3. We have independent sample means from independent batteries.
(b) How would they assure themselves that those assumptions are being met for their data?
Draw normal probability plots for both sets of data on the same plot.
1. Straight trends  normal
2. Parallel  equal variances
3. We have two sets of independent batteries, one set measured at 20 degrees and then other set at 0
degrees. We should consider running the experiment as a paired test with the same batteries measured at
both temperatures, but to use the same analysis as in problem 5, we would need independent batteries.