Page 385 Exercise #3
Descriptive Statistics: Differences
Variable
Differen
N
50
Mean
0.00040
Median
0.00000
TrMean
0.00068
StDev
0.01160
SE Mean
0.00164
A. Make a 90% two-sided CI for the mean difference in digital and vernier readings for
this instrument.
x t*
sd
n
= .00040 ∓ (1.684) .01160/√50
= .00040 ∓ .0028
= (-.0024, .0032)
B. Assess the strength of the evidence provided by these differences to the effect that
there is a systematic difference in the readings produced by the two calipers.
1. Ho: µ = 0
Ha: µ ≠ 0
.0004  0
2. t 
 .244
.01160 / 50
3. df = 49, p >. 20
4. Accept Ho: There is not enough evidence to suggest that there is a
difference in the readings.
C. Compare your answers to (a) and (b). Discuss how the outcome of part b could easily
have been anticipated from the outcome of part a.
Problem #4. B Choi tested the stopping properties of various bike tires on various surfaces.
For one thing, he tested both treaded and smooth tires on dry concrete. The lengths of skid
marks produced in his study under the two conditions were recorded.
A. In order to make formal inferences about the difference of the means based on these
data, what must you be willing to use for model assumptions? Make a plot to
investigate the reasonableness of those assumptions.
Must be approximately normally distributed and variances must be approximately
equal.
Normal Probability Plot for Treaded
Normal Probability Plot for Smooth
ML Estimates - 95% CI
ML Estimates - 95% CI
99
99
ML Estimates
Mean
95
StDev
95
14.0406
90
Mean
35
StDev
17
90
80
Goodness of Fit
80
Goodness o
70
AD*
70
AD*
1.997
Percent
Percent
ML Estimat
384.833
60
50
40
30
60
50
40
30
20
20
10
10
5
5
1
1
340
390
440
300
350
Data
400
Data
Descriptive Statistics: Treaded, Smooth
Variable
Treaded
Smooth
N
6
6
Mean
384.83
359.83
Median
383.50
352.00
TrMean
384.83
359.83
StDev
15.38
19.17
SE Mean
6.28
7.82
B. Proceed under the necessary model assumptions to assess the strength of the
evidence of a difference in mean skid lengths.
1. Ho: µ1 = µ2
2. Ha: µ1 ≠ µ2
( x  x 2 )  0 (384.83  359.83)
25


 2.49
3. t  1
10.03
s P n11  n12
17.38 16  16
4. df = n1+n2-2 = 10
Q(.975) = 2.228
Q(.99) = 2.764
.02 < p <.05
C. Make a 95% two-sided CI for the difference assuming equal variability.
( x1  x2 )  t  s P
1
n1
 n12  25  (2.228)(17.38)
1
6
 16  25  22.36  (2.64,47.36)
D. Use the Satterthwaite method and make an approximate two-sided confidence
interval for the difference assuming only that skid mark lengths for both types are
normally distributed
ˆ 
 s12 s 22 
  
 n1 n2 
2
2
 1   s12   1   s 22 

    
  
 n1  1   n1   n2  1   n2 
2

10134.90
 9.55  9
310.85  750.27
Since df = 9 instead of 10, t = 2.262
Formulas are given in text on Pages 440-442
Problem 26 – Page 435
Kim did some crude tensile strength testing on pieces of some nominally .012 in diameter
wire of various lengths. The lengths are given in the data file WIRE in the class directory.
A. If one is to make a confidence interval for the mean measured strength of 25 pieces
of this wire, what model assumption must be employed? Make a probability plot
useful in assessing the reasonableness of this assumption.
B. Make a 95% two-sided confidence interval for the mean measured strength of the
25cm pieces of this wire
C. Give a 95% lower confidence bound for the mean measured strength for a 25cm piece
of wire.
D. In order to make formal inferences about the difference of means, what must you be
willing to use for model assumptions. Make a plot useful for investigating the
reasonableness of those assumptions.
E. Assess the strength of the evidence provided by the data to the effect that an increase
in specimen length produces a decrease in measured strength.
F. Give a 98% two-sided confidence interval for the difference.