Question

In January 2003, the FAA ordered that passengers be weighed before boarding 10- to 19-seat passenger planes. The order was instituted in response to a crash that occurred on January 8, 2003, in Charlotte, North Carolina, in which all 21 passengers, including the pilot and co-pilot,of a 19-seat Beech 1900 turboprop died. One possible cause of the crash was that the plane may have been carrying too much weight. The airlines were asked to weigh adult passengers and carryon bags randomly over a one-month period to estimate the mean weight per passenger (including luggage). A total of 426 people and their luggage were weighed, and the sample data are contained in a data file called FAA.
Required Task: 1. Indicate what sample size would be required if the margin of error in the estimate for the mean of all passengers is to be reduced by half.

Weight	Gender	Age
197	Male	55
171	Male	47
212	Male	62
239	Male	49
237	Male	36
252	Male	47
146	Male	46
199	Male	47
235	Male	52
176	Male	62
186	Male	56
159	Male	51
155	Male	67
179	Male	51
184	Male	56
148	Male	44
190	Male	53
194	Male	52
209	Male	55
195	Male	50
196	Male	51
195	Male	54
241	Male	47
203	Male	59
200	Male	57
191	Male	58
258	Male	52
228	Male	57
269	Male	50
187	Male	54
250	Male	42
161	Male	53
220	Male	50
229	Male	61
257	Male	42
203	Male	53
191	Male	48
223	Male	50
195	Male	44
225	Male	53
166	Male	64
182	Male	52
164	Male	65
195	Male	47
204	Male	51
206	Male	49
196	Male	55
264	Male	47
158	Male	61
185	Male	52
135	Male	70
244	Male	52
170	Male	50
187	Male	53
225	Male	55
218	Male	58
229	Male	49
221	Male	48
168	Male	53
175	Male	68
224	Male	45
214	Male	51
180	Male	56
198	Male	56
209	Male	53
220	Male	56
209	Male	47
180	Male	56
256	Male	53
218	Male	59
207	Male	54
227	Male	58
228	Male	48
188	Male	65
180	Male	55
235	Male	44
173	Male	47
163	Male	48
224	Male	46
222	Male	47
265	Male	49
244	Male	65
240	Male	55
208	Male	43
205	Male	56
217	Male	46
204	Male	60
177	Male	49
157	Male	49
227	Male	55
217	Male	59
222	Male	40
211	Male	43
177	Male	51
238	Male	55
197	Male	73
182	Male	48
183	Male	45
193	Male	61
193	Male	49
191	Male	42
228	Male	46
219	Male	43
189	Male	61
240	Male	62
157	Male	55
220	Male	43
202	Male	66
206	Male	57
187	Male	47
190	Male	62
228	Male	50
227	Male	50
217	Male	51
224	Male	65
249	Male	46
213	Male	51
221	Male	54
255	Male	51
196	Male	49
233	Male	53
209	Male	41
236	Male	62
201	Male	53
184	Male	48
234	Male	51
189	Male	62
219	Male	38
220	Male	60
196	Male	44
193	Male	40
168	Male	55
259	Male	60
190	Male	45
207	Male	50
199	Male	55
282	Male	56
239	Male	50
229	Male	47
241	Male	52
210	Male	54
220	Male	62
198	Male	34
172	Male	49
239	Male	59
197	Male	52
170	Male	60
226	Male	53
226	Male	54
217	Male	56
216	Male	54
188	Male	47
225	Male	53
219	Male	54
234	Male	51
130	Male	56
218	Male	58
245	Male	56
158	Male	43
206	Male	55
242	Male	56
251	Male	56
211	Male	40
209	Male	38
255	Male	59
204	Male	72
187	Male	54
229	Male	44
205	Male	53
233	Male	57
217	Male	46
244	Male	42
202	Male	73
179	Male	58
163	Male	50
136	Male	59
208	Male	56
213	Male	57
205	Male	57
210	Male	54
212	Male	57
245	Male	54
207	Male	60
175	Male	48
167	Male	50
210	Male	65
231	Male	45
164	Male	44
189	Male	48
219	Male	46
199	Male	61
196	Male	54
201	Male	52
193	Male	48
183	Male	57
222	Male	61
207	Male	61
239	Male	49
212	Male	54
207	Male	59
211	Male	48
211	Male	47
205	Male	48
172	Male	53
211	Male	66
209	Male	49
225	Male	44
185	Male	51
222	Male	45
211	Male	46
208	Male	46
228	Male	51
254	Male	50
170	Male	50
172	Male	58
220	Male	60
237	Male	54
181	Male	58
147	Male	55
181	Male	43
210	Male	36
217	Male	48
178	Male	58
182	Male	56
217	Male	43
208	Male	55
190	Male	41
208	Male	60
180	Male	60
181	Male	53
233	Male	55
249	Male	59
187	Male	49
200	Male	47
208	Male	51
255	Male	61
221	Male	52
250	Male	56
210	Male	40
247	Male	58
188	Male	48
179	Male	67
237	Male	47
230	Male	49
158	Male	64
239	Male	57
200	Male	53
174	Male	49
154	Male	35
183	Male	45
218	Male	57
230	Male	65
203	Male	53
204	Male	46
211	Male	60
195	Male	50
230	Male	48
192	Male	49
268	Male	60
225	Male	37
221	Male	61
179	Male	47
229	Male	45
244	Male	48
185	Male	58
197	Male	65
169	Male	55
242	Male	38
233	Male	63
210	Male	39
146	Male	49
219	Male	65
219	Male	54
238	Male	44
179	Male	52
238	Male	68
190	Male	48
185	Male	60
158	Male	41
242	Male	48
179	Male	50
247	Male	46
173	Male	52
194	Male	62
170	Male	40
248	Male	34
206	Male	56
214	Male	44
231	Male	55
258	Male	46
203	Male	59
177	Male	47
190	Male	47
228	Male	44
229	Male	44
219	Male	59
165	Male	62
167	Male	51
201	Male	56
188	Male	53
188	Male	53
229	Male	43
182	Male	59
202	Male	57
199	Male	59
246	Male	56
169	Male	49
180	Male	50
200	Male	54
176	Male	43
166	Male	53
147	Male	35
221	Male	46
196	Male	59
216	Male	44
150	Male	48
237	Male	47
152	Male	59
191	Male	45
179	Male	49
202	Male	49
209	Male	46
252	Male	52
191	Male	53
184	Male	27
185	Male	33
221	Male	51
240	Male	35
166	Male	33
231	Male	46
248	Male	30
187	Male	38
207	Male	40
183	Male	45
224	Male	46
211	Male	42
206	Male	54
177	Male	52
197	Male	29
225	Male	35
205	Male	45
232	Male	30
178	Male	27
225	Male	41
239	Male	28
146	Female	47
131	Female	45
139	Female	37
152	Female	45
139	Female	42
117	Female	37
123	Female	31
138	Female	36
109	Female	36
128	Female	35
123	Female	47
121	Female	32
171	Female	37
127	Female	48
96	Female	36
139	Female	37
130	Female	45
127	Female	28
160	Female	45
152	Female	48
164	Female	29
90	Female	39
107	Female	32
134	Female	27
108	Female	36
110	Female	33
114	Female	48
143	Female	31
168	Female	36
140	Female	43
135	Female	40
148	Female	43
160	Female	44
102	Female	37
154	Female	37
128	Female	37
144	Female	44
145	Female	36
120	Female	24
159	Female	47
135	Female	43
127	Female	55
136	Female	46
147	Female	41
157	Female	58
112	Female	57
133	Female	47
103	Female	43
156	Female	45
122	Female	52
159	Female	47
108	Female	73
171	Female	50
122	Female	55
163	Female	47
148	Female	53
159	Female	48
127	Female	50
132	Female	53
136	Female	55
123	Female	62
135	Female	55
128	Female	50
142	Female	60
116	Female	48
113	Female	48
124	Female	55
122	Female	48
155	Female	57
149	Female	49
113	Female	52
139	Female	59
182	Female	44
137	Female	70
115	Female	61
115	Female	58
152	Female	48
116	Female	58
93	Female	51
140	Female	49
103	Female	54

Answer 1

To get required sample size for 50% less margin of error, we should first find margin of error with given sample.

Before starting let's assume that we are considering 5% significance level i.e   = 5% = 0.05

So, z for   = 0.05 is 1.96

From above Data, Mean of Weight, M = 192.129

Sample Size, n = 426

Calculating standard deviation for weight, = 38.869

Margin of Error, E = (z * ) /

E = 3.691

So, in our current sample, margin of error in estimating our mean is 3.691

Now Margin of Error is reduced by half, So new Margin of error is, E_n = E / 2 = 1.8455

Now calculating sample size as per new required error is given by below formula:

Here, E is equal to E_n

n = [(1.96 * 38.869) / 1.8455]²

n = (41.28)²

n = 1704.03

n = 1704 is th sample size required if we want to reduce the margin of error by 50%