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HIH
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I
I
3,—’
0
j
cD
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3
ci
2
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—Th
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w
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2
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4
.
A
r
(2 -k
c&k s
V’N
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s
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i
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cfre
S4
9m
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ç
Ij
N
c;cS
& N vA
S
S
S
-
§
L
5 L4”—r
y’
C
ci
6
that customers often
Do good smells bring good business? Businesses know
s Guguen and
Nicola
odors?
to
d
respond to background music. Do they also respon
on Satur
France
in
ant
restaur
his colleagues studied this question in a small pizza
through
spread
was
odor
er
day evenings in May. On one evening, a relaxing lavend
evening
third
a
odor;
lemon
the restaurant; on another evening, a stimulating
many
in
rable
compa
were
gs
served as a control, with no odor. The three evenin
as
data
the
regard
to
willing
ways (weather, customer count, and so on), so we are
Table 24.3
independent SRSs from spring Saturday evenings at this restaurant.
on each
ant
restaur
the
in
stayed
ers
custom
s)
contains data on how long (in minute
evenings.
of the three 5
c o4
of the distri
(a) Make stemplots of the customer times for each evening. Do any
from Nor
ons
butions show outliers, strong skewness, or other clear deviati
mality?
(h)
(c)
do the
Figure 24.5 gives the Minitab ANOVA output for these data. What
odors?
two
the
of
effects
the
about
mean times in the restaurant say
What are the values of tl ie ANOVA P statistic and irs P-value? What hy
potheses does F test? Briefly describe the conclusions you draw from these
surprising
d t Did you find anythini 7
Checking standard deviations Verify that the sample standard deviations for
these sets of data do allow use of ANOVA to compare the population means.
—
FIGURE 245
—
One-way ANOVA: time versus odor
Source
odor
Error
Total
S
14.50
MS
2285
210
SS
4569
17879
22448
DF
2
85
87
F-Sq
N
Level
lavender 30
28
lemon
30
noodor
Pooled StDev
20.35%
Mean
105.70
89.79
91.27
F
10.86
P
0.000
F Sq(adj)
=
Minitab output
ble 24.3 on time
tomers spend in
ercise 24.4.
18.48%
Individual 95% CIs For Mean Based on
Pooled StDev
+
+-+
StDev +
*
C
13.10
*
15.44
(
*
14.93
(
+
÷
+
+
105.0
98.0
91.0
84.0
14.50
D9
Road rage. Exercise 24.2 describes a study of road rage. Here are the means and
standard deviations for a measure of “ang/threatening driving” for random sam
pies of drivers in three age groups:
Age group
Less than 30 years
30 to 55 years
Over 55 years
n
244
734
364
s
2.22
1,33
0.66
3.11
2.21
1.60
(a)
The distributions of responses are somewhat right-skewed. ANOVA is
nonetheless safe for these data. Why?
(h)
Check that the standard deviations satisfy the guideline for ANOVA infer
ence.
(c)
Calculate the overall mean response
and the ANOVA F statistic.
(ci)
Which F distribution would you use to find the P-value of the ANOVA F
rest? Software gives P < 0.001. Write a brief conclusion based on the sample
means and the ANOVA F test.
,
the mean squares MSG and MSE,
()
3
Attitudes toward math. Do high school students from different racial/ethnic
groups have different attitudes toward mathematics? Measure the level of in
terest in mathematics on a 5-point scale for a national random sample of stu
dents. Here are summaries for students who were taking math at the time of the
survey:
10
Raciallethnic group
African American
White
Asian/Pacific Islander
Hispanic
Native American
(a)
(b)
(c)
n
809
1860
654
883
207
s
2.57
2.32
2.63
2.51
2.51
1.40
1.36
1.32
1.31
1.28
The conditions for ANOVA are clearly satisfied. Explain why.
Calculate the ANOVA table and the F statistic.
Software gives P < 0.001. What explains the small P-value? Do you think
the differences are large enough to be important?
TABLE 7
1
2
3
4
9.00
19.0
39.0
99.0
199
1
558
225
900
5,620
22,500
2
8,53
18.5
38.5
98.5
199
1
536
216
864
5,400
21,600
3
5,45
9.55
160
30.8
498
49.5
200
800
6,000
20,000
5
6
234
937
5,860
23,400
57,2158,2
230
922
5,760
23,100
10
12
15
20
30
60
120
) dx
2
, v
1
g(x; v
9
Jo
4
y
7
63.3
254
1.020
6.370
25,500
9,49
19.5
39.5
99.5
199
63.1
253
1.010
6.340
25,400
9 48
19.5
39.5
99.5
199
62.8
252
1.010
6,310
26,200
9 47
19.5
39 5
99.5
199
62.3
250
1,000
6,260
25,000
9.46
195
39.5
99.5
199
5.13
8 53
13.9
26.1
41.8
61 7
248
993
6.210
24,600
9.44
19.5
39.4
99.4
199
514
855
13.9
26 2
42.0
61.2
246
925
6,160
24,600
9,42
19.4
39.4
99,4
199
5.15
8.57
14.0
25.3
42.1
60.7
244
977
6,110
24,400
941
19.4
39.4
99,4
199
617
8.62
14.1
26.5
42.5
602
242
969
6,060
24,200
9.39
19.4
39.4
99.4
139
5.18
8 66
14.2
26.7
42.8
59.9
241
963
6,020
24,100
938
19.4
394
99.4
199
5.20
8,70
14.3
26.9
43.1
3.76
5.63
8.26
13.5
193
594
239
957
6,980
23,900
9.37
19.4
39,4
99.4
199
5.22
874
14.3
27.1
43.4
3.72
5.66
8.31
13.6
19.5
58.9
237
948
6,930
23,700
935
19.4
39.4
99.4
199
5 23
8.79
14.4
27.2
43 7
3.79
5.69
8 36
13.7
19.6
9.33
19.3
39.3
99.3
199
6.24
881
14.6
27.3
43 9
3.82
5.75
8 48
13.8
19.9
9.29
19.3
39.3
99.3
199
5.26
8.85
145
27.5
44.1
3.84
5.80
8 56
140
20.2
9.24
19.2
39.2
992
199
9.16
19.2
39 2
99.2
199
5.27
889
146
27.7
44.4
3.87
5.88
8.66
14,2
20 4
5.28
894
14.7
27.9
44 8
390
5.91
875
144
20 7
5.31
901
14.9
28.2
45.4
3.92
596
894
145
21.0
5.34
9.12
151
28.7
45.2
393
600
890
147
21.1
5,39
928!
15.4
295
47.5
3.95
604
8.98
14.8
21 4
3.11
4.37
602
9.02
121
3.98
609
9.07
15,0
21 6
3.12
440
607
911
12.3
401
616
920
152
22.0
5,54
101
17.4
34,1
55.6
39.9
161
648
4,050
16,200
2 degrees of freedom
1 and v
v of Snedecor’s Distribution with v
)
(v 2
7
f
,
100 x yth Percentiles 1
y
0.90
0.95
0.975
0.92
0.995
0.90
0.95
0975
0.99
0.996
090
0.95
0.975
099
0995
4
314
4.43
612
9.20
124
4.05
6.26
936
155
22.5
450
5.23
9,38
12.7
411
639
660
160
23.2
3.21
4.56
6.33
9.55
12.9
4.19
659
9,98
167
24.3
432
6.94
10.6
18.0
26.3
4.54
771
12.2
21.2
31 .3
090
0 95
0975
0.99
0.995
324
4 62
6,43
072
13.1
3.34
4 82
6,76
10.3
140
327
4 68
6 52
9.89
134
3,37
488
6 85
10.5
14.2
3.30
4 74
6 62
101
13.6
3 40
4 95
6 88
107
14.5
3.32
4 77
6 68
10.2
13.6
345
605
‘715
11.0
14.9
362
5 41
‘7 76
121
16,5
5
2.84
3.87
5.17
7.40
9.59
2.72
367
4.85
6.88
8.88
2.87
3.94
527
7.56
274
370
4905
6 97j
9005
9.81
2.76.
3.74!
4.96’
706
9.12
290
400
5.37
772
2.80
3.81
507
7.23’
9.36
10.0
104
7.981
2.96
410
5.52
2.94
406
5.46
7.815
10.2
106
2.985
4,15
560
810
3.52
5 19
7 39
114
19.6
3.$5
406
6 61
10 0
16.3
22.8
120
115
3.01
4,21
5.705
8.26!
3.78’
5.79
8.43
133
18.3
0 90
0.95
0975
0 99
0 995
‘
3.153
453
6,23
915,
108
3.29
476
680
978
3.05!
4.28
582
847
12.9
111
5.141
7.26’
10.9
3.11
4,39
5.99
75
14,5
13.7
186
3.75
5.99j
881
0110
0,95
0.975 6
0.99
0.995
-U
rn
2
02
2<
C)
>
-4
Cl,
m
F
01
-4
Cl,
0
-n
02
Ci,
01
C
-4
0
2
0
5
75
9
0
5
75
9
95
0
35
75
39
30
15
375
19
195
TA BLE 7 (continu.d)
15
090
0.95
0.975
0.99
0995
12
090
0 95
0975
0 99
0 995
10
090
0 95
0915
0 99
0 995
9
090
0 95
0975
0.99
0 935
8
090
0.95
0975
0 99
0 995
7
0.90
0.95
0.975
099
0995
2.16
2.71
3.29
414
4.85
2 21
2.79
341
4.32
5,07
227
2.90
3.58
4.56
5,37
20
2.33
3.00
3,73
4 82
5 76
239
3.11
3.89
506
6.07
252
3,33
4.24
564
6 87
2.61
3.48
447
599
734
289
363
4.72
6.42
7.96
281
386
508
699
872
292
4.07
5.42
7.59
9.60
311
4.46
6.06
865
11.0
3.26
4.74
6 54
9,55
124
3,59
5 59
807
12.2
162
2
1
209
259
3.12
389
4.54
2,24
235
351
450
535
228
2.91
3.61
4,64
552
2.38
3.07
385
506
6.12
2.41
3.14
395
520
630
246
3 22
407
5,39
654
281
329
4.20
5.61
6.88
2.55
3 37
432
5.80
7.13
261
3.48
448
608
7.47
267
3,58
4.65
6.37
7.95
2.73
3.69
4.82
6.63
8.30
2.81
3.84
5.05
7.01
8.81
2.83
3.87
5,12
7.19
9.16
2.88
3,97
5.29
7.46
952
2.98
4.12
5.52
7.85
101
3.07
435
5.89
845
109
4
3
2.48
326
412
5 41
852
261
349
4.47
5,95
723
281
389
510
6.93
851
318
4.75
6 55
9 33
118
2.73
371
483
655
808
292
410
546
756
943
3 29
4 98
6 94
100
128
301
426
571
802
10.1
3.36
512
721
106
136
346
5.32
7.57
11.3
147
214
269
325
4.02
482
228
292
359
451
524
225
287
351
443’
517
582:
2333
310
386
494
259
349
446
685
2881
417,
5571
30
9181
30
35
1
212
2.64
320
4.00
4.67
236
3.06
3.80
4.89
5.80
249
329
4.15
5.42
6.48
270
366
477
6.36
7.70
3 07
454
6.20
8.68
10.8
2971
435
5331
810
249
332
418
539
635
7561
60
120
2.39
315
393
498
580
279
400
529
708
849
4131
4731
5
2.16
2.71
329
476
198
242
287
347
395
4231
6
7
9
10
12
15
20
30
80
120
1.90
230
2.72
3 36
390
1 93
2.34
2.79
3.45
4 01
1,96
2.38
285
3.54
412
2.01
2.47
2.96
3.70
433
2.06
2.54
307
3.86
453
2.10
2.62
3.18
4.01
472
2.15
2.69
328
4,16
491
2.19
275
337
4.30
509
221
280
344
439
520
2.06
2.54
3 08
3 91
4.64
208
2,58
314
400
4.75
2,11
2,62
320
408
4.86
2.15
2.70
331
425
5.07
2.20
2.77
342
4.41
5.27
2.24
2.84
3.62
4.58
5.47
2.28
2.91
382
471
556
2.32
2.98
372
485
585
2.35
302
378
494
597
216
271
3 33
4.31
5.19
2.18
275
3.39
4.40
5.30
221
279
345
4.48
5.41
225
2.86
3.56
4,65
5.82
2.30
294
367
4,81
5.83
234
301
337
4.96
6.03
2,38
307
3.87
5.11
8.23
242
3.14
3.96
5.26
6.42
2,4.4
3.18
403
5.35
6.54
247
323
4.10
5.47
6.69
2.29
2.93
3.67
4.86
5.95
2.31
2.97
3:73
4.95
6.06
2.34
3.01
3.78
503
6.18
2.38
3.08
3.89
5.20
840
2.42
3.18
4.00
5.36
6.61
2,46
3.22
4.10
5.52
6.81
2.50
3.28
4.20
5.67
7.01
2.54
335
4.30
5.81
7.21
2.56
3.39
4.36
5.91
734
2.59
3.4.4
4.43
6.03
7.50
262
3.50
4,53
6.18
7.69
247
3.23
4.14
5.65
708
249
3.27
4.20
5.74
7.13
2.51
330
4,25
5.82
7.31
2.56
3.38
4.38
5.99
743
2.59
344
4.47
6.16
775
263
351
4.57
6.31
7,97
2.67
3,57
4.67
6,47
8.18
2.30’
3.641
4.76
6.62
8.38
2,72
3,68
4.82
6,72
8.51
2.75
3.73
490
6.84
8.68
2.78
3.79
499
6.99
8.89
1,74
2 04
2 35
278
1 79
212
246
294
3 32
1 84
2 20
2 57
309
3 50
164’
190
216
252
251
168
1 95
222
281
292
3.12!
1 54
1 74
1 94
221
242
1 61
1 94
2 07
2 39
263
1.67
1 93
220
2.55
2 82
1 72
201
231
2 70
301
177
209’
241
284
318
182
216
251
298
334
185
221
257
307
345
1 48
1 65
1 82
2 03
219
1
1
1
1
1
1 48
1 86
1 82
2.03
219
1
5
1
1
1
1,61
184
209
242
289
1.48
162
179
201
218
150
168
1337
211
230
135
147
158
173
183
1 40
1 53
1 67
1 84
196
41
55
69
86
98
‘3
1
1
1
1
32
43
53
66
75
1 24
1 32
1 39
1,47
1,53
1 34
1 46
1 57
1 70
1.79
1 42
1,57
1 71
1.88
200
1 49
1 67
183
2,04
219
165
175
194
218
2.36
252!
154
1 75
1,94
220
239
1,60
1,84
206
253
2 57
186
1 92
217
250
274
171
1 99
227
263
290
160
183
205
232
1.63
188
2,11
241
262
154
1 75
1 34
219
2 37
160
1 83
205
234
254
185
191
2.16
247
271
168
1.98
222
2.56
281
172
202
230
266
2.93
189’
228
268
323
368
194
235
277
337
385
198
239
284
346
396
200
245
291
356
409
1 881
227
265
317
358
193
233
275
330
374
1.14
2.04
233
272
301
177
2.101
2411
282
313
152
2,17
251
295
329
280!
8
1,76
207
240
287
3.26
1,79
2.11
2 46
2 96
3.37
182
216
252
3.05
3.48,
1.87
2.25
2.64
321
3.69
1.92
233
2.76
3.37
3.88
1.97
2.40
286
3.52
407
2,02
2,48
296
3,87
4,25
206
254
306
3.80
442
204
251
301
370
426
209
260
313
387
447
4.101
1 971
2.25
263j
312
349
1,95
237
279
334
3761
177
209
239
279
309
1
2 181
282
296
328
167
194
213
251
2,74
172
201
229
264
290,
177
210
241
185
221
2571
302,
194
237
279
332
2081
260
312
378
428
5301
1 901
2 39
2.87
317
3551
199
245
2.89
348
392
213
268
323
395
450
230
300
3.89
481j
205
283
303
370
204
253
3011
3651
4,14’
218
278
334
235
301
380
479
554j
275
392
5 15
885
818
271
384
502
93
788
1
995
129
1.39
148
160
169
119
1 25
1 31
138
‘343
26
35
43
53
81
117
1 22
1 27
1 32
1 36
100
100
100
100
100
I