Confidence interval for the proportion of all applicants

Question

Assignment:

Harrigan University is a liberal arts university in the Midwest that attempts to attract the highest quality students, especially from its region of the country. It has gathered data on 178 applicants who were accepted by Harrigan. The data are in the file named Harrigan which is posted on my website.

The variables are:

Accepted: whether the applicant accepts Harrigan's offer to enroll.

MainRival: whether the applicant enrolls at Harrigan's main rival university.

HSClubs: number of high school clubs applicant served as an officer.

HSSports: number of varsity letters applicant has earned.

HSGPA: applicant's high school GPA.

HSPctile: applicant's percentile (in terms of GPA) is his or her graduating class.

HSSize: number of students in applicant's graduating class

SAT: applicant's combined SAT score

CombinedScore: a combined score for the applicant used by Harrigan to rank applicants.

The derivation of the combined score is a closely kept secret by Harrigan, but it is basically a weighted average of the various components of high school performance and SAT. Harrigan is concerned that it is not getting enough of the best students, and worse yet, it is concerned that many of the best students are going to Harrigan's main rival. Solve the following problems and then write an executive summary on whether Harrigan appears to have a legitimate claim.

1. Find the 95% confidence interval for the proportion of all applicants who accept Harrigan's invitation to enroll. Do the same for all applicants with a combined score less than or equal to the median of combined score. And then for the applicants with a combined score greater than the combined score median. Perform a hypothesis test to determine if there is a significant difference between these two proportions?

2. Find the 95% confidence interval for the proportion of all students with a combined score less than or equal to the median who chose Harrigan's rival over Harrigan. Do the same for those with a combined score greater than the median. Perform a hypothesis test to determine if there is a significant difference between the two?

3. Find the 95% confidence intervals for the mean combined score, the mean high school GPA, and the mean SAT score of all acceptable students who accept Harrigans invitation to enroll. Do the same for all acceptable students who choose to enroll else where. Then find the 95% confidence intervals for the differences between these means, where each difference is a mean for students enrolling at Harrigan minus the similar mean for students enrolling elsewhere.

4. Harrigan is interested in recruiting students who are involved in extracurricular activities. Does it appear to be doing so? Perform a hypothesis test to determine if at least half of those students that come to Harrigan have been officers of at least two clubs. Perform a similar test to determine if at least half of the students that come to Harrigan have at least four varsity letters in sports.

5. The combined score Harrigan calculates for each student gives some advantage to students who attended large high schools relative to those who attended small high schools. Is Harrigan correct in this assumption? (Split the data As a result, Harrigan believes it is more successful in attracting students from large high schools than from small high schools. Are they correct?

6. If the GPA, SAT score, the number of clubs where the student serves as an officer, and the number of letters in sports is used to calculate the combined score, is there a difference in any of these parameters when comparing large schools to small schools? Can you draw any possible conclusions from your results that might cause a shift in the combined score for a specific group of students?

Applicant	Accepted	MainRival	HSClubs	HSSports	HSGPA	HSPctile	HSSize	SAT	CombinedScore
1	Yes	No	1	5	2.89	78.1	388	1262	309
2	Yes	No	5	5	3.61	94	121	1341	382
3	Yes	No	1	3	3.05	82.1	343	1330	305
4	No	No	2	6	3.13	88.8	192	1072	322
5	Yes	No	4	3	3.19	88.2	181	1114	323
6	No	No	5	4	3.11	80.1	259	1159	342
7	No	Yes	2	3	4	97	425	1322	381
8	No	Yes	3	11	3.56	87.4	467	1205	422
9	Yes	No	3	6	3.98	97.1	475	1300	416
10	No	Yes	6	9	3.34	81	270	1310	414
11	No	No	2	7	3.06	69.4	258	1248	335
12	No	Yes	4	4	3.31	87.4	374	1389	365
13	Yes	No	2	2	3.53	88.3	290	1294	332
14	No	Yes	3	6	3.36	85.3	467	1118	365
15	Yes	No	2	6	3.67	87.5	250	1215	365
16	No	No	2	5	3.14	81	227	1070	316
17	Yes	No	3	9	3.41	86.9	248	1039	375
18	Yes	No	6	8	3.34	88.9	147	1285	398
19	Yes	No	5	2	3.54	91.6	338	1303	367
20	No	No	1	8	3.31	85.7	471	1091	357
21	No	No	1	12	2.91	73.8	408	1180	362
22	No	Yes	5	6	3.55	88.3	165	1318	388
23	Yes	No	6	4	3.61	97.6	300	1036	382
24	No	No	2	12	3.63	92.4	446	1405	435
25	Yes	No	4	3	2.83	81.4	45	1138	293
26	Yes	No	2	9	3.14	77	148	1245	351
27	Yes	No	3	9	3.63	98.6	516	1157	416
28	No	No	4	6	3.5	86.7	409	1158	382
29	Yes	No	3	6	3.33	85.6	318	1355	366
30	Yes	No	3	7	2.79	72.1	478	902	324
31	Yes	No	6	4	3.02	82.2	338	1034	346
32	No	No	2	7	3.33	78.2	156	1324	351
33	Yes	No	1	10	2.6	67.1	93	1002	301
34	Yes	No	3	6	3.24	86.9	353	1213	356
35	Yes	No	4	4	3.61	86.6	254	1127	361
36	No	No	5	4	3.47	92.5	243	1240	369
37	No	No	2	6	2.94	73.6	378	955	313
38	Yes	No	2	15	2.97	74.4	174	1141	385
39	Yes	No	6	4	2.87	79.5	176	1227	336
40	No	No	6	9	3.2	77.8	139	1176	391
41	Yes	No	0	10	3.35	94.5	403	1025	361
42	Yes	No	3	1	3.83	100	484	1387	373
43	No	No	2	12	2.77	74.3	348	993	351
44	Yes	No	2	6	3.57	92.7	328	1348	372
45	Yes	No	2	6	3.46	81.1	236	1031	342
46	Yes	No	5	4	4	95	276	1296	406
47	Yes	No	6	4	3.66	93.5	446	1058	396
48	No	No	4	2	3.57	86.3	244	1152	344
49	Yes	No	2	5	3.48	94	240	1258	349
50	No	No	4	11	3.58	90.4	505	1423	448
51	No	No	6	5	3.57	84.3	288	1047	385
52	No	No	4	6	3.04	79.7	174	964	329
53	Yes	No	5	2	3.18	85.8	271	1041	326
54	Yes	No	3	7	2.82	72.5	171	1194	324
55	Yes	No	5	6	3.3	85.8	276	1257	377
56	Yes	No	3	6	3.15	84.4	378	1012	342
57	No	No	4	5	3.44	81.8	495	1302	382
58	Yes	No	3	5	3.42	96	275	1174	354
59	Yes	No	2	5	3.5	84.7	356	1212	353
60	No	Yes	5	4	3.79	89.5	617	1467	424
61	No	Yes	3	2	3.59	85	364	1011	335
62	Yes	No	1	11	3.22	78.5	392	1236	376
63	No	Yes	4	9	3.05	79.3	369	1091	371
64	Yes	No	3	4	3.64	96.6	274	1162	358
65	Yes	No	6	0	3.92	96.2	274	1265	378
66	Yes	No	3	2	3.26	82.6	255	1054	310
67	No	Yes	4	7	3.52	88.8	286	1186	386
68	No	No	2	3	3.68	92.1	392	1079	346
69	Yes	No	3	12	2.98	80	124	1033	364
70	Yes	No	4	2	3.12	80.5	196	1080	309
71	Yes	No	1	5	2.91	76.1	228	1186	297
72	Yes	No	2	7	3.18	87.5	288	1023	337
73	Yes	No	6	0	3.63	90.6	135	1117	343
74	No	No	4	13	3.01	84.9	291	1120	399
75	No	Yes	2	4	3.81	96.3	302	1258	365
76	Yes	No	3	4	2.95	77.4	235	1075	306
77	Yes	No	1	9	3.04	77	151	1020	324
78	Yes	No	1	4	2.99	70.9	237	1181	293
79	No	No	4	10	3.29	76.9	374	1035	391
80	Yes	No	1	4	2.95	75.2	393	1255	304
81	No	Yes	2	7	3.39	86.4	236	1330	361
82	Yes	No	6	5	3.16	82.9	71	1101	349
83	Yes	No	5	8	3.17	80.8	280	1028	373
84	Yes	No	4	0	3.22	78.6	163	1158	301
85	No	No	3	5	3.52	91.3	213	1198	356
86	Yes	No	4	3	3.47	92.2	301	1140	350
87	Yes	No	5	1	3.33	90.2	97	1226	326
88	No	Yes	0	10	3.16	80.3	372	1197	352
89	No	No	3	10	3.38	86.3	461	1184	402
90	Yes	No	6	5	3.13	76.8	250	950	350
91	Yes	No	2	7	3.13	80.6	229	1241	340
92	Yes	No	1	5	3.19	85.5	238	1242	319
93	No	No	3	5	3.23	82.1	257	1067	333
94	Yes	No	3	7	2.58	67.9	379	1119	316
95	Yes	No	5	4	3.3	90.1	379	1306	371
96	No	Yes	6	3	3.55	93	329	1149	377
97	No	No	3	4	3.33	86.8	275	1273	343
98	Yes	No	0	6	2.77	74.1	163	1042	275
99	Yes	No	3	3	3.49	76.2	237	1298	342
100	No	Yes	3	1	3.19	83.2	494	958	308
101	Yes	No	2	3	3.51	81.1	295	1156	330
102	Yes	No	4	8	3.14	85.9	444	961	369
103	Yes	No	4	3	3.37	84.8	60	1027	321
104	Yes	No	3	5	3.46	91.1	442	1428	379
105	No	No	5	4	3.4	88.4	150	1141	353
106	No	No	2	9	2.47	60.6	448	1020	312
107	No	No	1	9	2.89	74.3	279	1165	329
108	Yes	No	0	6	3.27	84.2	216	1063	311
109	No	No	4	1	3.41	89.8	65	1229	318
110	No	No	4	3	3.48	82.3	382	1268	360
111	Yes	No	4	4	3.45	87.5	351	1242	364
112	Yes	No	3	4	3.42	85.2	235	1332	349
113	No	No	3	3	3.45	81.9	80	1087	320
114	No	Yes	3	8	3.75	91.5	429	1195	408
115	Yes	No	3	4	3.21	88.4	241	1136	327
116	No	No	3	4	3.51	81.9	429	1245	361
117	Yes	No	4	0	4	100	290	1516	378
118	Yes	No	1	5	2.99	71.9	137	1026	288
119	No	No	3	5	4	95	282	1569	409
120	No	Yes	2	7	3.67	89.5	438	1193	385
121	Yes	No	3	6	3.4	90.6	384	1297	373
122	Yes	No	3	6	3.51	92.4	194	1377	371
123	Yes	No	2	4	3.36	76.8	260	985	318
124	Yes	No	4	3	2.87	86.1	328	1072	311
125	Yes	No	3	8	3.18	85	309	1122	361
126	No	No	4	6	3.2	87.2	192	1057	345
127	Yes	No	4	6	3.77	89	423	1544	420
128	Yes	No	0	10	3.09	78.5	484	1248	356
129	Yes	No	3	6	3.2	87.5	281	1255	351
130	No	Yes	2	10	3.09	75.2	291	1236	364
131	No	No	3	6	3.44	86.3	240	1415	371
132	Yes	No	1	5	2.85	78.6	286	1155	296
133	No	No	1	9	2.52	71.8	327	1111	306
134	No	No	2	8	3.32	89	304	1256	366
135	No	Yes	3	7	3.39	91.3	253	1118	363
136	No	Yes	6	3	3.36	89.6	190	1293	363
137	Yes	No	4	1	3.3	79.2	53	1197	309
138	No	No	3	8	3.08	82.4	223	1206	353
139	Yes	No	0	2	3.49	91.7	282	1342	312
140	No	No	2	2	3.67	87.4	256	1380	342
141	No	Yes	5	5	3.35	80.1	348	1290	376
142	No	No	1	11	3.7	94.9	425	1214	411
143	No	Yes	4	9	3.06	83.3	364	1207	379
144	No	No	5	8	3.58	96	407	1167	416
145	No	No	0	7	3.12	78.8	517	932	320
146	No	Yes	3	5	3.71	99.8	511	991	380
147	Yes	No	2	7	3.47	91.6	73	1299	354
148	No	Yes	3	8	3.6	93.9	186	1086	377
149	No	No	4	6	3.53	88.8	450	1531	406
150	Yes	No	4	2	3.4	95.7	279	1314	346
151	Yes	No	4	9	2.99	81.3	130	929	346
152	Yes	No	3	6	3.48	85	229	1129	358
153	Yes	No	2	8	3	77.1	308	1310	347
154	Yes	No	4	0	3.82	96	195	1502	358
155	Yes	No	6	3	3.42	87.4	140	1239	360
156	Yes	No	4	9	2.86	78.5	406	1055	360
157	Yes	No	1	5	3.07	77.7	236	1249	310
158	Yes	No	1	7	3.19	84.4	346	1039	331
159	No	Yes	3	10	3.43	85.2	340	1217	398
160	Yes	No	3	8	2.84	76.3	193	1219	336
161	No	Yes	3	13	3.13	79.1	286	1091	393
162	No	No	4	9	3.45	82	142	1309	393
163	Yes	No	3	9	3.28	85.8	225	1112	369
164	Yes	No	3	10	3.07	78.3	441	1137	377
165	No	No	0	6	3.29	89.6	401	949	320
166	Yes	No	4	7	3	81.6	350	1250	360
167	Yes	No	4	3	3.4	85.8	66	1322	338
168	Yes	No	2	5	3.36	87.8	377	1272	350
169	Yes	No	4	5	3.25	84.1	84	916	326
170	No	Yes	5	5	3.54	92	341	923	372
171	Yes	No	1	7	3.51	85.3	241	1185	351
172	No	No	6	1	3.45	86.4	290	1342	361
173	No	Yes	6	2	3.35	87.6	385	1150	360
174	Yes	No	2	9	3	81.4	408	1087	351
175	Yes	No	3	4	3.09	75.4	199	1063	312
176	No	No	2	10	2.88	81.8	533	1134	362
177	Yes	No	1	10	3	81.8	123	1155	335
178	Yes	No	2	7	3.35	83.8	282	1410	365