Statistics Q&A

ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
Income
Year
443.9998
2006
518.1919
2006
457.2028
2006
638.1267
2006
595.2664
2006
398.7556
2006
596.6668
2006
500.2712
2006
455.1641
2006
488.4575
2006
480.1819
2006
634.4962
2006
553.0838
2006
341.6128
2006
668.2744
2006
500.1915
2006
423.6913
2006
544.162
2006
546.3877
2006
488.9709
2006
322.9193
2006
526.2674
2006
791.9488
2006
264.858
2006
475.2804
2006
335.743
2006
179.6449
2006
356.1493
2006
250.9318
2006
260.0431
2006
322.874
2006
574.7699
2006
689.6498
2006
327.9227
2006
340.4411
2006
436.7163
2006
619.4394
2006
726.9027
2006
540.2598
2006
400.5074
2006
627.5209
2006
287.0434
2006
380.5353
2006
637.0776
2006
235.6536
2006
434.3423
2006
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
650.3799
411.4362
527.9233
396.5026
292.8838
242.1864
326.7692
93.94352
382.5481
502.975
634.3537
631.5617
635.8267
567.4035
671.9119
471.1237
673.5814
408.8532
622.3092
464.8562
453.7009
279.3119
301.6292
428.7107
447.5113
244.0563
423.7787
382.7806
537.2543
605.3241
445.9468
471.0897
465.7643
580.839
503.4363
573.4019
473.6067
388.188
436.2697
596.75
105.1232
320.1841
430.2686
112.2081
338.3914
221.7399
575.2429
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
461.3519
304.6257
344.3349
665.0009
389.6631
594.9336
318.9562
499.6397
680.8865
368.8821
388.9169
660.6584
555.4425
413.3594
488.492
589.3905
154.9053
246.6738
335.0321
924.8065
420.1826
628.9947
413.444
442.0139
430.647
480.5691
497.2451
723.0537
278.2449
483.3891
303.4444
409.246
475.5955
612.8828
349.7205
596.5201
554.0402
416.2569
170.6135
426.0547
364.8947
245.6123
519.0946
475.4573
430.3107
365.9635
535.0384
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
499.8566
428.3444
631.1044
497.7854
595.926
229.0307
702.7072
425.8881
396.9214
456.4555
523.5416
499.8992
445.0558
530.5731
279.8124
511.3975
499.7111
260.8755
679.2622
521.0661
117.3504
528.6993
365.5097
440.275
247.9442
356.7114
299.145
335.5928
464.2153
630.1119
540.2542
496.7816
616.0628
530.9084
406.7826
330.9378
397.9343
435.1965
740.3614
420.6436
339.376
275.3584
804.5754
461.2243
493.9741
346.8506
515.8251
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
305.6638
757.0079
348.8662
231.7034
589.5571
599.5791
313.9024
169.7331
528.2697
275.0262
393.4
410.5102
590.9191
291.3543
392.7068
349.4216
409.8872
625.5873
310.59
315.8115
507.6193
478.4824
398.4804
140.1917
375.301
598.0191
354.97
376.8866
281.9073
684.4139
466.3107
601.1053
538.9324
595.3012
470.939
483.0768
515.8607
534.3528
358.5739
258.6275
-25.6428
323.7246
397.9553
318.3575
639.7722
374.656
649.9814
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
375.881
559.6718
540.3652
540.0804
723.1374
529.1539
445.9748
559.816
225.3568
269.4082
524.5976
418.8244
453.7314
387.5343
473.8211
248.6512
245.4269
591.5787
585.088
277.7988
407.6261
459.355
510.4445
301.2724
463.6098
526.9305
665.3462
422.7253
697.133
662.8377
287.0641
583.6917
442.9577
646.6466
381.1101
326.9841
844.8865
405.9679
583.4582
376.3799
463.5662
758.8672
388.2526
750.0523
595.6972
606.8591
176.6606
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
282.315
340.405
519.4712
389.1421
514.4849
132.0301
392.5933
488.2374
520.2148
439.1823
498.8222
425.4068
491.6532
629.8942
452.2877
684.8105
469.1462
331.0477
404.0885
578.3718
369.9177
431.5394
467.5747
557.6071
694.4188
695.695
563.8683
491.012
489.3538
487.5152
458.5433
182.1826
224.5177
545.0458
629.675
445.3963
236.058
228.5005
576.4625
485.0258
320.1143
294.2322
345.8936
277.4855
446.1266
548.75
130.3047
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
625.5144
419.7234
178.6045
674.978
462.0277
376.8844
475.834
618.5387
772.9379
334.934
757.7795
409.9998
536.5971
351.8905
360.4556
331.3349
379.0525
491.9526
395.8902
789.4186
554.0782
246.5322
729.1306
660.5266
456.7933
316.0313
707.5869
647.5475
406.1437
345.4173
211.0121
675.4944
350.1028
293.8121
670.0801
346.2176
518.3208
424.4732
384.3088
305.5888
154.2415
462.5568
677.5429
426.3263
545.4508
533.9221
407.2672
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
387.5061
459.5047
521.1501
237.0158
607.3718
383.5507
285.8844
568.8188
605.3689
379.0145
632.3909
807.8974
371.0955
233.2663
362.5129
525.3804
630.257
490.286
367.3094
599.9912
441.5846
364.1198
347.218
249.8743
381.8137
581.3082
617.823
229.8794
628.7153
405.6327
684.1359
626.88
325.6782
685.4593
272.939
491.4873
484.9938
385.1017
498.4247
588.1238
445.4215
451.3784
639.9473
289.5164
479.2464
531.9476
388.3644
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
243.766
705.1455
632.5752
440.7841
676.41
446.3659
302.0513
319.4097
517.4066
241.2216
510.3853
455.8076
202.2073
469.9094
512.4485
535.7998
475.2066
445.917
692.183
390.2139
521.019
619.9962
500.1613
313.3008
569.334
536.5528
354.538
272.2855
619.9039
386.9464
619.1844
795.2452
634.4361
274.4249
546.4082
420.2038
394.3929
184.4818
573.1316
503.0806
558.3259
522.3214
613.8772
208.793
578.4111
620.1302
601.978
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
394.3929
276.6466
456.2916
377.779
398.9769
394.3929
284.5824
400.3708
633.6742
402.5808
394.3929
228.2377
404.3818
571.748
334.0423
162.5838
140.6159
396.0635
434.2363
536.3834
366.543
380.1071
478.078
155.6578
182.3494
449.9547
419.6478
481.9924
333.7279
338.0989
107.8575
192.7389
385.2777
166.1642
375.3342
358.4053
484.23
397.8013
518.8611
518.7658
483.5086
491.4569
396.1401
293.0208
329.5756
274.5737
190.472
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
668.0785
646.2351
371.2904
368.4148
341.4028
343.7767
410.9385
204.9952
222.0821
243.0141
480.6048
674.9704
281.3379
370.3206
359.7799
284.3368
419.824
731.5129
630.8861
497.9195
408.395
548.6128
502.8478
804.4977
260.8195
341.9739
482.6024
153.5494
465.1911
471.5795
362.5651
533.0402
556.4929
194.2691
387.8026
274.1925
268.9504
303.3637
362.9808
942.0502
434.0075
1028.885
392.0152
444.8042
486.8417
613.1555
424.4956
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
352.3216
252.3216
465.3075
88.64745
539.6353
160.0387
164.9706
144.1892
356.0651
774.4746
566.8479
639.711
373.17
361.5881
320.7487
384.6577
376.4855
497.4465
516.2317
615.703
812.0918
500.4809
425.2215
264.2462
816.7034
522.8325
534.842
716.4211
485.3459
160.5061
286.96
167.9443
314.9388
268.9764
372.0157
322.3464
409.6241
585.7583
564.0079
55.48604
356.1265
718.0055
403.691
500.1595
-15.6122
780.7007
557.8978
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
516.0506
155.5769
432.7104
393.1309
300.1403
420.9992
23.97073
378.9509
426.6201
422.0406
257.9847
441.079
177.45
801.8225
273.3702
432.4227
306.0942
431.6191
628.1728
265.7571
411.276
286.9692
692.7943
454.1197
314.2948
554.7201
450.9511
564.1749
322.6734
480.5075
708.1708
343.727
257.1354
697.0625
184.1931
618.6891
445.503
111.3099
583.4596
836.5615
428.5339
243.1033
502.8246
446.6838
441.7341
228.9042
272.4514
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
667.314
402.7069
303.7048
597.8464
521.611
392.9299
472.8503
529.4198
866.9837
58.89715
440.1233
665.482
78.12055
437.2641
426.005
636.2707
346.3511
310.4184
322.4595
458.026
427.9941
832.1754
636.8801
270.6053
374.9692
488.2611
716.0491
409.8929
435.7656
185.0769
532.6653
434.086
109.7836
504.6512
440.7415
513.4819
280.2106
390.6245
716.7416
439.7264
598.5548
644.3812
788.9627
298.081
298.4675
321.5066
150.5636
2005
2005
2005
2005
2005
2005
2005
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2005
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735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
523.0786
526.9566
336.8767
211.0711
463.0917
391.3175
626.4634
349.9096
277.0814
368.4471
304.9948
439.3228
422.9962
431.9135
540.5571
124.9714
548.0603
681.0264
213.9382
338.8296
382.8224
518.6187
251.9944
223.151
646.0967
309.0243
468.1338
441.8497
347.7785
77.98461
651.2112
592.0623
578.332
451.6989
682.4156
425.6152
568.5628
561.777
642.2416
310.7267
489.5683
625.6391
557.0228
279.3522
167.7258
549.2023
471.1211
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
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2005
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769
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780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
528.2116
396.4384
517.4791
281.5444
436.6358
217.5481
545.43
452.4734
710.1691
303.5022
276.3662
824.7407
551.7059
95.09621
476.8767
379.6577
559.7304
388.5704
204.2514
131.1521
613.4983
298.2712
398.4014
450.1046
187.594
130.9394
651.5744
456.1833
562.188
202.205
721.2196
240.4625
702.9871
453.8282
759.0544
521.368
351.6888
20.10107
291.3056
362.7406
459.299
483.6439
716.0168
761.065
436.6067
607.4884
466.636
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
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2005
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2005
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2005
799
800
801
802
803
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811
812
813
814
815
816
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819
820
821
822
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826
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828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
381.8143
709.7187
542.6747
503.5595
277.3489
853.5286
753.3429
643.3938
565.9478
669.5898
648.3312
521.8454
450.4279
589.9874
308.1376
734.3702
267.6368
238.2762
687.5651
484.8261
535.1425
399.3237
116.7282
685.5781
621.6237
510.537
348.9994
448.619
465.8983
133.7722
298.0984
335.7086
419.6845
691.0737
513.058
462.6102
543.0465
22.00598
410.0836
315.7153
464.8736
497.4421
476.1737
393.5004
402.3347
455.1087
449.3632
2005
2005
2005
2005
2005
2005
2005
2005
2005
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2005
846
847
848
849
850
851
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855
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859
860
861
862
863
864
865
866
867
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877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
572.0995
128.6609
297.5126
348.5247
515.6245
581.4273
307.0453
279.6102
363.3369
587.4644
583.7917
290.0693
660.7159
606.7722
392.0901
389.737
314.3544
393.5128
538.0102
323.289
523.0217
420.0169
409.6142
358.9482
451.3785
466.2034
302.2512
604.8918
917.9313
386.1577
608.972
250.7236
518.4915
258.2303
490.9212
482.6233
455.4125
111.3358
245.3245
464.165
485.5687
241.1032
791.7749
291.8317
647.402
541.4394
656.8338
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
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893
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895
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902
903
904
905
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908
909
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911
912
913
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923
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925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
14.82675
570.8338
413.769
418.3516
438.4955
540.419
541.1814
510.6181
689.6134
158.9407
464.6167
194.4865
527.3174
629.4589
441.9685
230.1381
293.9645
308.7358
262.3723
391.2455
457.2594
281.1586
398.2656
395.8108
575.4398
425.9112
409.0877
365.0048
498.5446
268.9881
369.4139
551.5998
534.4469
646.6809
374.2733
557.9501
763.7451
687.0426
385.4653
521.456
229.1673
348.3794
608.4532
399.8535
632.3007
629.1749
46.06201
2005
2005
2005
2005
2005
2005
2005
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940
941
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971
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974
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976
977
978
979
980
981
982
983
984
985
986
529.6999
252.0743
438.9882
832.138
303.4115
350.0562
705.5892
196.853
295.3126
332.6511
454.9193
491.7493
423.3708
339.7737
522.3513
488.609
462.9581
209.5858
539.9256
287.7965
416.3447
476.6563
276.218
254.1295
491.6987
696.8151
564.3686
424.5976
239.6667
653.35
706.0244
369.7728
430.9584
366.6031
502.0425
571.5092
704.2838
379.0758
521.2214
551.4425
558.8114
358.5161
103.428
290.8247
500.2896
193.1939
639.267
2005
2005
2005
2005
2005
2005
2005
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2005
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
161.1128
244.3365
463.6609
543.1239
256.299
609.2958
-20.6361
449.509
547.2803
483.5796
455.9569
249.2816
559.0652
295.6544
2005
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2005
QUESTION 1 – DEMAND FOR FLU SHOTS
Attention is already turning to vaccinations during flu season, when falling temperatures
and the seasonal flu may lead to a 2nd wave of COVID-19. It is often difficult however to
predict the necessary numbers of doses to produce.
Suppose it is October 2016, and you are working on a team at the U.S. Centers for Disease
Control and Prevention (“CDC”) on flu-related issues. Concerned about a vaccine shortage
– which happens from time to time1 – you and your team make plans based on your best
guess that 53.5 percent of American adults intend to get vaccinated this year.
Then, on November 16, 2016, Rasmussen, a polling company, publishes the results of a
national poll of 1,000 American adults:2
You focus specifically on the percentage of American adults that say they intend to get a
flu shot this year. Assume that 560 of the 1,000 adults in the sample gave this response.
(a) The Basics. Define the following terms in this specific context:
1. Population
2. Sample
1
See, e.g., http://health.usnews.com/health-news/news/articles/2013/01/15/spot-shortages-of-fluvaccine-tamiflu-reported-fda-head-says. For more, see
http://www.cdc.gov/flu/about/qa/vaxdistribution.htm.
2
http://www.rasmussenreports.com/public_content/lifestyle/general_lifestyle/november_2016/
most_intend_to_get_a_flu_shot_this_year
1
3. Estimate
(b) Using the Rasmussen survey results, test the hypothesis that 53.5 percent of American
adults intend to get vaccinated this year. Follow the steps described in class:
1.
State the null hypothesis (H0).
2.
Set a significance level (α).
3.
Calculate the estimate from the sample.
4.
Define the sampling distribution and use it to calculate the p-value.3
5.
Reject or fail to reject H0 based on whether p-value < α. (c) What is the interpretation of the sampling distribution and the p-value in this context? Explain the result to your project leader, a statistical neophyte who is extremely concerned that the Rasmussen survey result (56 percent) is greater than the vaccination rate (53.5 percent) that you anticipated and used as the basis for your flu-season plans. (d) Calculate a 95% Confidence Interval for your estimate in (b), and explain this to your Statistics professor. (e) Suppose that you conduct a new poll to test the same null hypothesis, that 53.5 percent of American adults intend to get vaccinated this year. Suppose that exactly 56 percent of adults in your new sample intend to get vaccinated this year. What is the minimum number of individuals you would need in your new sample to be able to reject the null hypothesis at a 5 percent significance level? 3 Note that, assuming that the Central Limit Theorem applies (which you can check), you can calculate the p value using Excel. Suppose that you have a z-score Z. To calculate the two-tailed p-value, use the following Excel command: =2*(1-NORM.S.DIST(ABS(Z), TRUE)). For example, you could calculate the p-value of Z = -1.2 using =2*(1-NORM.S.DIST(ABS(-1.2), TRUE)). 2 QUESTION 2 – WELFARE PROGRAMS IN INDIA The Mahatma Gandhi National Rural Employment Guarantee Act (NREGA, http://nrega.nic.in/netnrega/home.aspx), one of the world’s largest welfare programs enacted by the Government of India in 2005, guarantees 100 days of paid employment to adults in rural India willing to do unskilled manual work at the minimum statutory wage. The program was rolled out over a three-year period across districts in rural India. You are interested to know whether the average monthly per capita expenditure in the Indian state of Rajasthan has changed in the aftermath of the program’s implementation. In 2006, you randomly sample 500 adults in Dungarpur district in Rajasthan, which was one of the 200 districts where NREGA was first implemented. You also have access to a random sample of 500 adults from the same district in 2005, before the program was implemented there.4 You can download these data (NREGA.xlsx) from the course website. (a) Use these data to conduct a hypothesis test at the 5 percent significance level to assess whether the monthly per capita expenditure has changed in the aftermath of the NREGA implementation. Use the 5 steps outlined in class: 1. State the null hypothesis (H0). 2. Set a significance level (α). 3. Calculate the estimate from the sample. 4. Define the sampling distribution and use it to calculate the p-value.5 5. Reject or fail to reject H0 based on whether p-value < α. (b) Use the data to construct a figure that visualizes the key takeaway from your analysis in (a), as you would want to present it to a policy-maker. Your figure should stand alone, without the need for further explanation. (You may assume that the policymaker is familiar with the NREGA program.) 4 5 The numbers quoted in this example were created for educational purposes and do not reflect official estimates by the Government of India or other analytical work. Note that, assuming that the Central Limit Theorem applies (which you can check), you can calculate the p value using Excel. Suppose that you have a z-score Z. To calculate the two-tailed p-value, use the following Excel command: =2*(1-NORM.S.DIST(ABS(Z), TRUE)). For example, you could calculate the p-value of Z = -1.2 using =2*(1-NORM.S.DIST(ABS(-1.2), TRUE)). 3 QUESTION 3 – THE BOUNDARY BETWEEN REJECT AND FAIL TO REJECT Suppose you draw a random sample from the population and conduct a hypothesis test based on the following four quantities: • H0: q = 0.40 • 𝑞̂ = 0.36 • n = 1000 •  = 0.05 (c) Do you reject or fail to reject the hypothesis that q = 0.40? Remember the Alamo. While holding the other three quantities constant, change each blank quantity below in turn so that you are just on the boundary between rejecting and failing to reject the null hypothesis. For each case, report the range of values for which you would reject the hypothesis and the range of values for which you would fail to reject the hypothesis.6 In addition, in each case describe intuitively why increasing/decreasing the quantity in question changes the p-value. 6 (b) H0: q = 0.40 𝑞̂ = n = 1000  = 0.05 (c) H0: q = 0.40 𝑞̂ = 0.36 n=  = 0.05 (d) H0: q = 0.40 𝑞̂ = 0.36 n = 1000 = For example, state, “I would reject the hypothesis for any value of __ that is [greater than/less than/between] _______, and I would fail to reject the hypothesis for any value of __ that is [less than/greater than/between] _______.” 4 QUESTION 4: POLLUTION AND HEALTH IN LOS ANGELES Particulate matter (PM) pollution is generated by industrial activity and motor vehicles. Of different types of pollution, PM 2.5 poses the greatest health risks – the particles are sufficiently fine to penetrate deep into the lungs. You are advising the current Mayor of Los Angeles, Eric Garcetti, on public health issues related to pollution. As part of your analysis, you run two regressions examining daily data. You use the number of cardiorespiratory and non-cardiorespiratory emergency room visits in the Los Angeles metropolitan area as the dependent variable and the average level of PM2.5 pollution on that day as the independent variable. Consider the following table of results.7 Table 1: Pollution and Emergency Room Visits, by Cause Note: Standard errors are in parentheses. (a) State a reasonable null hypothesis (in non-technical language) for the slope of the regression where cardiorespiratory emergency room visits is the dependent variable. Calculate the approximate p-value. (b) Construct a 95% confidence interval for the slope of the regression where cardiorespiratory emergency room visits is the dependent variable. (c) Although Mayor Garcetti is a graduate of Columbia School of International and Public Affairs (MIA, 1993), it’s been a while since he has taken statistics. Explain the main takeaways from the table using non-technical language in a short (100 words or less) paragraph. 7 These results are a simplified version of actual research that examines the relationship between pollution and health outcomes in http://faculty.haas.berkeley.edu/rwalker/research/SchlenkerWalker_Airports_2012.pdf 5 QUESTION 5: FINAL EXERCISE Go online to the survey question associated with Problem Set #2 and enter your preferred choice between the two topics: 1. Economic Effects of Expanded Medicaid Coverage 2. Clean Water Policies in Africa (Note that you need only do this once between both problem sets this week.) 6