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Confidence Interval for Conusmer Purchase Amounts

Purchase Amounts for all Customers Shopping at a Local Supermarket on a Given Friday
4. Data of the amount that customers spend in a supermarket on a specific Friday. Assume this data represents the population. (20 points)
a. What sample size would be required to be 95% sure that the estimate is within + $10?
b. How would the sample size value change for a one sided interval (+$10)
c. Use the data from the first 30 customer to create a sample n = 30. What is the 95% confidence interval for this sample data?
d. What is a generic 95% confidence interval for any sample n=30?

Customer Amount
1 $35.97
2 $52.23
3 $93.08
4 $2.90
5 $91.13
6 $150.69
7 $75.76
8 $95.74
9 $79.33
10 $36.69
11 $278.50
12 $82.54
13 $58.93
14 $131.56
15 $106.52
16 $215.11
17 $3.42
18 $0.42
19 $71.01
20 $97.99
21 $110.10
22 $289.33
23 $48.02
24 $1.60
25 $45.05
26 $305.44
27 $157.87
28 $26.89
29 $16.44
30 $37.69
31 $146.98
32 $31.96
33 $101.57
34 $12.06
35 $50.92
36 $117.43
37 $184.54
38 $62.78
39 $154.01
40 $91.97
41 $139.55
42 $285.63
43 $12.73
44 $179.99
45 $34.72
46 $132.84
47 $38.56
48 $51.41
49 $85.59
50 $139.34
51 $239.40
52 $5.10
53 $172.15
54 $136.67
55 $222.74
56 $138.40
57 $140.38
58 $23.91
59 $17.88
60 $153.83
61 $146.55
62 $52.36
63 $150.43
64 $141.97
65 $29.67
66 $280.48
67 $54.16
68 $101.26
69 $26.56
70 $32.16
71 $151.86
72 $158.88
73 $111.91
74 $109.35
75 $61.69
76 $48.12
77 $247.15
78 $105.98
79 $119.42
80 $107.41
81 $163.28
82 $184.66
83 $147.16
84 $46.01
85 $60.68
86 $24.19
87 $187.94
88 $37.38
89 $70.05
90 $42.18
91 $68.57
92 $76.19
93 $50.72
94 $160.46
95 $13.94
96 $93.98
97 $80.04
98 $109.65
99 $21.57
100 $112.20
101 $110.89
102 $68.21
103 $87.25
104 $40.76
105 $61.76
106 $240.38
107 $29.35
108 $102.58
109 $222.16
110 $169.74
111 $11.32
112 $17.30
113 $28.99
114 $52.03
115 $203.83
116 $80.59
117 $295.70
118 $40.85
119 $86.88
120 $38.21
121 $65.35
122 $228.90
123 $23.50
124 $210.61
125 $65.90
126 $84.68
127 $66.71
128 $108.78
129 $86.24
130 $312.29
131 $47.99
132 $167.38
133 $79.78
134 $154.43
135 $40.79
136 $9.02
137 $6.80
138 $76.07
139 $308.86
140 $35.48
141 $6.35
142 $136.21
143 $43.08
144 $43.19
145 $33.57
146 $11.70
147 $13.61
148 $86.46
149 $13.28
150 $14.85
151 $57.43
152 $54.08
153 $51.88
154 $217.07
155 $25.68
156 $18.45
157 $24.95
158 $33.05
159 $58.46
160 $40.03
161 $37.67
162 $66.44
163 $215.24
164 $66.53
165 $77.99
166 $20.73
167 $44.95
168 $160.72
169 $31.63
170 $263.41
171 $169.65
172 $41.39
173 $22.64
174 $113.86
175 $62.13
176 $89.41
177 $9.07
178 $22.87
179 $23.31
180 $83.42
181 $16.48
182 $32.46
183 $113.46
184 $181.54
185 $16.16
186 $294.98
187 $100.41
188 $97.79
189 $20.55
190 $36.51
191 $72.58
192 $15.75
193 $205.93
194 $18.60
195 $52.18
196 $99.29
197 $6.41
198 $53.78
199 $226.38
200 $178.51
201 $8.18
202 $142.63
203 $127.50
204 $64.80
205 $143.14
206 $133.78
207 $59.85
208 $185.79
209 $111.94
210 $59.02
211 $236.52
212 $41.30
213 $66.62
214 $23.03
215 $174.76
216 $6.67
217 $38.85
218 $38.98
219 $174.38
220 $65.61
221 $129.06
222 $204.94
223 $138.72
224 $129.99
225 $11.83
226 $53.43
227 $131.80
228 $0.34
229 $179.28
230 $89.40
231 $146.70
232 $30.76
233 $31.93
234 $94.46
235 $219.50
236 $8.62
237 $116.98
238 $21.38
239 $242.18
240 $9.89
241 $76.22
242 $271.54
243 $211.69
244 $28.70
245 $55.21
246 $33.21
247 $47.70
248 $38.22
249 $33.61
250 $243.79
251 $148.22
252 $233.69
253 $163.64
254 $156.91
255 $130.28
256 $101.21
257 $20.36
258 $159.02
259 $47.07
260 $263.21
261 $50.36
262 $23.12
263 $28.60
264 $39.07
265 $258.73
266 $3.35
267 $13.12
268 $183.31
269 $111.27
270 $233.97
271 $72.97
272 $63.20
273 $155.24
274 $30.81
275 $28.90
276 $79.75
277 $109.19
278 $247.41
279 $58.26
280 $7.25
281 $270.10
282 $56.11
283 $15.70
284 $14.86
285 $163.97
286 $8.14
287 $186.56
288 $63.63
289 $203.61
290 $166.87
291 $82.82
292 $108.59
293 $33.19
294 $219.79
295 $29.53
296 $88.11
297 $149.30
298 $1.30
299 $16.70
300 $23.45
301 $63.32
302 $6.12
303 $70.13
304 $26.82
305 $144.19
306 $30.42
307 $63.91
308 $172.26
309 $119.11
310 $19.05
311 $37.87
312 $145.39
313 $16.52
314 $60.39
315 $12.91
316 $53.27
317 $296.90
318 $61.59
319 $117.45
320 $54.53
321 $211.74
322 $204.18
323 $7.41
324 $239.72
325 $43.83
326 $154.09
327 $15.49
328 $35.93
329 $11.29
330 $26.35
331 $64.30
332 $212.53
333 $21.49
334 $98.76
335 $96.21
336 $29.30
337 $185.18
338 $33.02
339 $87.37
340 $116.38
341 $102.39
342 $234.61
343 $75.76
344 $81.68
345 $42.83
346 $33.06
347 $76.39
348 $94.10
349 $26.09
350 $27.07
351 $74.20
352 $73.99
353 $89.97
354 $173.32
355 $28.53
356 $91.30
357 $64.77
358 $5.35
359 $14.89
360 $49.55
361 $7.87
362 $17.18
363 $207.41
364 $76.41
365 $86.16
366 $52.95
367 $33.13
368 $24.96
369 $1.51
370 $53.93
371 $60.02
372 $184.81
373 $107.41
374 $128.61
375 $138.17
376 $12.63
377 $301.14
378 $89.13
379 $226.55
380 $83.96
381 $16.39
382 $94.31
383 $67.88
384 $18.75
385 $21.36
386 $150.07
387 $39.23
388 $66.17
389 $21.85
390 $52.15
391 $82.25
392 $3.59
393 $44.45
394 $175.12
395 $13.96
396 $85.40
397 $142.22
398 $1.45
399 $23.84
400 $105.40
401 $107.17
402 $116.65
403 $187.45
404 $63.87
405 $4.39
406 $74.53
407 $54.06
408 $29.96
409 $46.88
410 $43.05
411 $28.07
412 $9.07
413 $57.44
414 $38.11
415 $154.94
416 $5.66
417 $51.03
418 $35.12
419 $63.24
420 $47.25
421 $78.62
422 $33.34
423 $34.53
424 $40.38
425 $46.41
426 $41.40
427 $19.01
428 $218.19
429 $15.42
430 $3.27

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4. Data of the amount that customers spend in a supermarket on a specific Friday. Assume this data represents the population. (20 points)

a. What sample size would be required to be 95% sure that the estimate is within + $10?

Sample size (means)

confidence interval= 95%
Standard deviation =σ= $73.77 (calculated for the population using EXCEL worksheet function STDEVP)
Number of tails= 2
Error= ±z*σx= $10
Where σx is the standard error of the mean=σ/√n
Thus we have to ensure that Z* &#963;x < 10
Z corresponding to a confidence level of 95.% and 2 tail = 1.96
(from the tables or using EXCEL function NORMSINV)
Thus, &#963;x < 10/Z
or &#963;x < 5.102 =10/1.96
But, &#963;x=standard error of mean=&#963;/&#8730;n
&#963;x=&#963;/&#8730;n

Or, n=(&#963;^2)/(&#963;x^2)= 209.0637 =73.77^2/5.102^2

Therefore sample size required for ± 10 accuracy = 210 (rounding up)

Answer: sample size = 210

b. How would the sample size value change for a one sided interval (+$10)

Sample size (means)

confidence interval= 95%
Standard deviation =&#963;= $73.77 (calculated for the population using EXCEL worksheet function STDEVP)
Number of tails= 1
Error= + z*&#963;x= $10
Where ...

Solution Summary

This solution answers the questions on confidence interval and sample size in the attached Excel file.

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