Appendix 13
A13.1 Equal-variances t-test of μ1 − μ 2
H 0 : (μ1 − μ 2 ) = 0
H 1 : (μ1 − μ 2 ) < 0
( x 1 − x 2 ) − (μ 1 − μ 2 )
t=
⎛ 1 1 ⎞
s 2p ⎜⎜ + ⎟⎟
⎝ n1 n 2 ⎠
A B C
1 t-Test: Two-Sample Assuming Equal Variances
2
3 This Year 3 Years Ago
4 Mean 8.29 10.36
5 Variance 8.13 8.43
6 Observations 100 100
7 Pooled Variance 8.28
8 Hypothesized Mean Difference 0
9 df 198
10 t Stat -5.09
11 P(T<=t) one-tail 0.0000
12 t Critical one-tail 1.6526
13 P(T<=t) two-tail 0.0000
14 t Critical two-tail 1.9720
t = –5.09, p-value = 0. There is overwhelming evidence to conclude that there has been a decrease over the past
three years.
A13.2 a z-test of p1 − p 2 (case 1)
H 0 : (p1 − p 2 ) = 0
H1 : (p1 − p 2 ) > 0
(p̂ 1 − p̂ 2 )
z=
⎛ 1 1 ⎞
p̂(1 − p̂)⎜⎜ + ⎟⎟
n
⎝ 1 n 2 ⎠
A B C D E
1 z-Test of the Difference Between Two Proportions (Case 1)
2
3 Sample 1 Sample 2 z Stat 2.83
4 Sample proportion 0.4336 0.2414 P(Z<=z) one-tail 0.0024
5 Sample size 113 87 z Critical one-tail 1.6449
6 Alpha 0.05 P(Z<=z) two-tail 0.0047
7 z Critical two-tail 1.9600
331
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,z = 2.83, p-value = .0024. There is enough evidence to infer that customers who see the ad are more likely to make a
purchase than those who do not see the ad.
b Equal-variances t-test of μ 1 − μ 2
H 0 : (μ 1 − μ 2 ) = 0
H 1 : (μ 1 − μ 2 ) > 0
( x 1 − x 2 ) − (μ 1 − μ 2 )
t=
⎛ 1 1 ⎞
s 2p ⎜⎜ + ⎟⎟
⎝ n1 n 2 ⎠
A B C
1 t-Test: Two-Sample Assuming Equal Variances
2
3 Ad No Ad
4 Mean 97.38 92.01
5 Variance 621.97 283.26
6 Observations 49 21
7 Pooled Variance 522.35
8 Hypothesized Mean Difference 0
9 df 68
10 t Stat 0.90
11 P(T<=t) one-tail 0.1853
12 t Critical one-tail 1.6676
13 P(T<=t) two-tail 0.3705
14 t Critical two-tail 1.9955
t = .90, p-value = .1853. There is not enough evidence to infer that customers who see the ad and make a purchase
spend more than those who do not see the ad and make a purchase.
c z-estimator of p
p̂(1 − p̂)
p̂ ± z α / 2
n
A B C D E
1 z-Estimate of a Proportion
2
3 Sample proportion 0.4336 Confidence Interval Estimate
4 Sample size 113 0.4336 ± 0.0914
5 Confidence level 0.95 Lower confidence limit 0.3423
6 Upper confidence limit 0.5250
We estimate that between 34.23% and 52.50% of all customers who see the ad will make a purchase.
d t-estimator of μ
s
x ± tα/2
n
332
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, A B C D
1 t-Estimate: Mean
2
3 Ad
4 Mean 97.38
5 Standard Deviation 24.94
6 LCL 90.22
7 UCL 104.55
We estimate that the mean amount spent by customers who see the ad and make a purchase lies between $90.22 and
$104.55.
A13.3 t-test of μ D
H0 :μD = 0
H1 : μ D > 0
xD −μD
t=
sD / n D
A B C
1 t-Test: Paired Two Sample for Means
2
3 Before After
4 Mean 381.00 373.12
5 Variance 39001 40663
6 Observations 25 25
7 Pearson Correlation 0.96
8 Hypothesized Mean Difference 0
9 df 24
10 t Stat 0.70
11 P(T<=t) one-tail 0.2438
12 t Critical one-tail 1.7109
13 P(T<=t) two-tail 0.4876
14 t Critical two-tail 2.0639
t = .70, p-value = .2438. There is not enough evidence to conclude that the equipment is effective.
A13.4 Frequency of accidents: z -test of p1 − p 2 (case 1)
H 0 : (p1 − p 2 ) = 0
H1 : (p1 − p 2 ) > 0
(p̂ 1 − p̂ 2 )
z=
⎛ 1 1 ⎞
p̂(1 − p̂)⎜⎜ + ⎟⎟
n
⎝ 1 n 2 ⎠
333
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, A B C D E
1 z-Test of the Difference Between Two Proportions (Case 1)
2
3 Sample 1 Sample 2 z Stat 0.47
4 Sample proportion 0.0840 0.0760 P(Z<=z) one-tail 0.3205
5 Sample size 500 500 z Critical one-tail 1.6449
6 Alpha 0.05 P(Z<=z) two-tail 0.6410
7 z Critical two-tail 1.9600
z = .47, p-value = .32053. There is not enough evidence to infer that ABS-equipped cars have fewer accidents than
cars without ABS.
Severity of accidents Equal-variances t-test of μ 1 − μ 2
H 0 : (μ 1 − μ 2 ) = 0
H 1 : (μ 1 − μ 2 ) > 0
( x 1 − x 2 ) − (μ 1 − μ 2 )
t=
⎛ 1 1 ⎞
s 2p ⎜⎜ + ⎟⎟
n
⎝ 1 n 2 ⎠
A B C
1 t-Test: Two-Sample Assuming Equal Variances
2
3 No ABS ABS
4 Mean 2075 1714
5 Variance 450,343 390,409
6 Observations 42 38
7 Pooled Variance 421,913
8 Hypothesized Mean Difference 0
9 df 78
10 t Stat 2
11 P(T<=t) one-tail 0.0077
12 t Critical one-tail 1.6646
13 P(T<=t) two-tail 0.0153
14 t Critical two-tail 1.9908
Estimate of the difference between two means (equal-variances)
A B C D E F
1 t-Estimate of the Difference Between Two Means (Equal-Variances)
2
3 Sample 1 Sample 2 Confidence Interval Estimate
4 Mean 2075 1714 360.48 ± 290
5 Variance 450,343 390,409 Lower confidence limit 71
6 Sample size 42 38 Upper confidence limit 650
7 Pooled Variance 421,913
8 Confidence level 0.95
334
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or distributed without the prior consent of the publisher.
