please use R (a) What is the population mean µ? (b) Using a for loop, simulate 1

Question

please use R

(a) What is the population mean µ?

(b) Using a for loop, simulate 10,000 data sets, and determine the proportion of times that the population mean is between ¯y B and ¯y + B. What can you say about the claim that these bounds capture the true population mean about 95% of the time?

Data:

4153 3855 4287 4401 4302
4379 4453 4282 4219 4395
4288 4388 4291 4292 4091
4322 4045 4154 4317 4208
4366 4141 4291 4081 4396
4220 4070 3999 4358 4181
4185 4439 4253 4159 4006
4280 4292 4180 4088 4398
3993 4216 4163 4157 4101
4418 4316 4306 3843 950
4269 4331 4253 4821 4208
4205 4189 4047 4085 7010
4163 4300 4134 4199 3945
4237 4162 4038 4250 4327
4063 4070 4277 4243 4163
4224 4111 4304 4157 4170
4313 4249 4209 4121 3878
4327 4243 4098 4218 4380
4138 4117 4196 4224 4145
4317 4237 4087 4150 4621
4126 4280 3804 4214 4321
4159 4181 4032 4172 4444
4385 4254 4216 4052 4270
4132 4229 4702 4231 4094
4100 4220 5542 4137 4138
3935 4058 4016 4222 4161
3797 4111 4317 3957 4116
4317 4106 4101 4094 4230
4171 4351 4222 3816 4002
4256 4293 4284 4177 4143
4209 4322 4234 4225 4263
4195 4411 4266 4210 4102
4098 4459 4081 4349 3945
4141 4244 4090 4090 4251
4335 4424 4505 4247 4192
4230 4954 4064 4232 4191
4314 4280 4192 4110 4208
5515 3872 4307 4384 4242
4300 4247 4169 4209 3804
4297 4185 4287 4197 4212
3890 4115 4081 4293 4065
4194 4302 4230 4073 4205
4355 4506 4307 4242 3954
4204 3760 4362 3877 4247
4581 4197 4139 4110 4301
4408 4188 4132 4065 3960
4041 4110 4217 4213 4304
4259 4184 4363 4100 4230
4171 4131 4186 4292 4559
4415 4301 4179 4188 4232
4290 4221 4129 4189 4272
4372 4278 4326 4588 4145
4304 4204 4124 4190 4231
4138 4247 4072 3822 4253
4297 4271 4524 4179 4117
4108 4174 4381 4101 4433
4281 4271 4485 4239 4133
4022 4755 4248 4150 4137
4048 4390 4109 4407 4112
4247 4149 4125 4565 1745
4332 4219 4367 4005 3659
4231 4143 4241 4261 3440
4127 4214 4140 4218 4423
4211 4161 4412 4185 4270
4020 4185 4313 4211 4501
4037 4212 4261 4154 4206
4245 3246 4017 4372 4729
3777 3888 3918 4140 4004
4108 4251 4191 4251 4452
4177 3878 4587 4222 4121
4300 4198 4286 4138 4279
4298 3950 4015 4380 4230
4130 4055 4422 4174 4045
4220 4238 4330 4263 4129
4223 4292 4202 4207 4358
4178 4205 3972 4203 4256
4182 4286 4249 4197 3630
4378 4261 4166 4270 4165
4170 4060 4227 4113 4161
4581 4177 4234 3963 4390
4337 4130 4274 4085 3904
4238 3952 4249 4201 3972
4237 4166 4200 4305 4138
3939 4024 4240 4222 4309
4101 4556 4247 4233 4005
4189 4193 4410 4299 4271
4020 4149 4288 4151 4114
4214 4133 4346 4145 4370
4235 4295 4076 4363 4329
4079 4322 4384 4138 4361
4210 4075 4147 4165 4100
4160 4311 3369 4227 4319
4133 4207 4224 4374 4032
4392 4250 4427 3937 4887
4131 3718 4281 4208 3934
4380 4353 4210 4166 4185
4101 4071 4303 4276 4265
4207 4043 4195 4703 4223
4305 4330 3802 4165 4590
4109 4204 4154 4145 4196
4222 3981 4187 4093 4274
4140 4053 4747 4237 4522
4187 4142 4198 3980 4235
4134 4324 4179 4258 4693
4430 4082 4892 4390 3935
4199 4258 3805 4124 4190
4053 4179 4412 4019 4173
4031 4215 4244 4312 4272
4079 4273 4195 4156 4278
4178 4232 2781 4393 4170
4185 4285 4257 4138 4137
4230 4029 4191 4275 4159
4177 4253 4019 4199 4158
4151 4088 3894 4212 3976
4157 4096 4174 4041 4254
4022 4243 4426 4140 4340
4080 4182 4281 3876 4221
4166 4241 4129 4181 4209
4093 4139 4192 4078 3647
4292 4300 4275 4068 4217
4202 4063 4313 3891 4277
4123 4198 4218 4443 4175
4292 3996 4201 4170 4306
4141 4644 4144 4254 4131
4196 4257 4010 4250 4212
3902 4239 4235 4208 4246
4228 4216 4207 4207 4232
4001 4018 4549 4346 3613
4120 4084 4105 4100 4399
4225 4195 4645 3734 4296
4432 3969 4217 4185 4150
4052 4169 4247 4142 4097
3993 4314 4234 4244 3975
4185 4065 4168 4175 4464
4216 4466 4324 4181 4226
4564 4181 4090 4066 3921
4309 4000 4386 4235 4351
4171 4372 4147 4190 4534
4077 4131 4166 4326 4518
4225 4108 4240 4242 3595
4262 4219 4450 4328 4234
4217 4170 4262 4433 4198
4196 3923 4268 4090 4459
4036 4201 4240 4188 4186
4254 4346 4178 4242 4400
4195 4213 5021 4230 4199
4268 4157 4217 3938 4129
3970 4222 4216 4431 4268
4264 4331 4335 4290 4304
4296 4202 3988 4245 4320
4105 4197 4186 4299 4199
6182 4319 4165 4203 4130
4046 4270 4088 4101 4317
4235 4102 4560 4104 4301
4294 4127 4078 4439 3634
4218 4156 4473 4152 4150
4203 4210 3764 4394 4634
4093 4297 3613 4301 4161
4255 4513 4257 4212 4411
4185 4549 3935 4240 4331
3754 4163 4256 3837 4152
4287 3912 4348 4053 4207
4093 4287 4208 4176 4194
4103 4236 4272 3829 4344
4277 4245 4210 4150 3990
4209 4267 4213 4131 4333
4376 4162 4312 4275 4219
4176 4214 4256 4236 4614
4101 4235 4388 4411 4175
4145 728 4062 4320 4252
4298 4184 4307 4248 3940
4137 4321 4266 4290 4251
4179 4135 4255 4435 4018
4433 4998 4213 4218 4196
4420 4182 4134 4208 4172
4448 4087 3959 3889 4090
4343 4537 4069 4309 4189
4331 4232 4092 4371 4117
3860 4008 4198 3520 4453
4221 4349 4130 4066 4344
4188 3991 4402 3650 4357
4170 4443 4163 3995 4521
4218 4189 3630 3372 4219
4181 3959 4220 4088 4042
4289 4109 4198 4050 4153
4053 4222 3749 4248 4063
4308 4070 4121 4166 4268
4120 4300 4087 4328 4220
4119 4193 4206 4294 3978
4036 4116 4510 4365 4224
4069 4203 4376 4263 4143
4178 4068 4171 4237 3984
4160 4156 4197 4155 4194
4048 4399 4263 4338 4335
4070 4112 5054 3835 4186
4216 4164 4155 4142 4094
4298 4205 4402 4255 4292
4224 4329 4330 4287 4243
4184 4363 4222 4112 4294
4318 4470 4085 4158 4631
4239 4149 4169 4015 4035
4101 4167 4157 4376 4242
4200 4630 4384 4228 4052
4018 4285 4187 4091 4014
4071 4176 4393 4177 4140
4166 4254 4174 4469 4026
4167 4238 4013 4231 4667
4241 4512 4212 4759 3582
4318 4039 4194 4242 4224
4093 4178 4138 4025 4325
4042 3864 4217 4183 4099
4033 4222 4287 4379 4239
4207 4142 4369 4095 4054
4176 4429 4245 4247 4374
4317 4394 4148 4090 4192
4096 3959 5065 4067 4321
4360 4173 4178 4264 4182
4152 4174 3671 4197 4346
3886 4456 4173 4107 5839
4052 4286 4091 4063 4344
4314 4234 4176 4061 4200
4012 4156 4314 4363 4254
4152 4215 4205 4214 4128
4205 4286 4003 4149 4509
4008 4150 4341 4042 4228
4287 4190 4181 4243 4141
4425 4191 4342 3711 4064
4162 4621 4067 4223 4415
4083 4304 4159 4184 4207
4315 4274 4175 4171 4326
4263 4274 4055 4124 4236
4138 4358 4199 4168 4255
4349 4264 4204 4363 4192
4346 3981 4264 4442 4036
4199 4351 4205 4184 4093
3981 4224 4082 5023 4264
4243 4238 4113 4248 4361
4078 4209 4296 4098 4280
4027 4175 4136 4018 4165
4176 4215 4043 4203 4249
4355 3781 4341 4013 4282
4187 4194 4605 4282 4196
4256 4296 4463 4287 4202
4346 3444 4194 4645 4048
4403 4336 4168 4221 3972
4309 4066 4272 4409 4231
4263 4263 4100 4496 4322
4241 4180 4332 4254 4088
4198 4109 4126 3996 4206
4175 4268 4160 4173 4381
4155 4243 4268 4029 4218
4060 4177 4512 4277 4182
4383 4244 4185 4245 4141
4227 4208 4204 4143 3853
4533 4319 4256 4204 4110
4060 4293 4231 4056 4214
4169 4256 3914 3917 4296
4290 4315 4092 4161 4354
4293 4200 4106 3613 4264
4269 4287 4308 4161 4224
4426 4217 4329 4199 4132
3987 4201 4019 4158 4426
4175 4245 4168 4380 4190
4256 4153 4034 3863 4061
4441 4292 4260 4126 4121
4420 4121 4379 4123 4351
4089 4159 4176 4134 4046
4091 4232 4557 4155 4091
4169 4061 4002 3940 4115
3914 4148 4241 4149 5089
4230 4139 4375 4243 4018
4226 4138 4253 4273 4152
4241 4224 4277 4371 4056
4291 4236 4240 4019 4220
4138 4321 4154 4243 4134
4097 4289 4830 4414 4109
4093 4351 4022 4081 4148
4817 4354 4144 4246 4297
4320 4125 4125 4318 4148
4478 4184 4223 4143 4254
4256 4193 4104 4298 4177
4306 4364 4202 4212 4164
4096 4062 4324 4098 4105
4149 4035 4135 4388 4203
4293 4333 4795 5634 4274
3337 4195 3788 4397 4219
4502 4165 4500 4269 3997
4297 4041 3997 4225 3561
4177 4183 4063 4241 4156
4096 4052 4284 4063 4297
4088 4304 4244 4223 4373
4265 3828 4120 4096 4281
4322 4148 4593 4033 4177
4235 4611 4242 4083 4219
4242 4150 3995 4281 4196
4282 4025 3628 4279 4199
4174 4108 4259 4443 4294
4153 4133 4074 4228 4142
4207 4060 4025 4230 4130
4142 4140 4276 4448 4073
4223 4367 3941 4225 3965
4189 4286 4204 4168 4200
4327 4440 4171 4236 4144
4302 4349 4234 4124 4071
4364 4169 4256 4191 4576
4252 4252 4260 4295 4115
4291 4198 4271 4038 4241
4009 4008 4231 4344 4243
3913 4208 4194 4108 4112
4114 4183 3978 4239 4367
4462 4303 4092 4089 4158
4222 4209 4069 4098 4264
4292 4152 4199 4301 4183
4091 4239 4134 4235 4270
4333 4013 4110 4183 4149
4202 4204 4255 4321 4325
4399 3995 3875 4255 4314
4216 4246 4268 4337 4228
4175 3560 4261 4201 4365
4259 4267 4153 4237 4292
4305 4198 4218 4165 4304
4190 4038 4079 4171 4263
4432 4344 4196 4241 4201
3998 3807 4155 4384 4164
4318 4093 4224 4301 4026
4189 4229 4237 4224 4399
4006 4183 4200 4210 4296
4288 4214 4196 4273 4213
4466 4217 4156 4320 4112
4115 4207 4189 4026 4213
4049 4284 3399 4227 4237
4204 4162 4289 4070 4035
3518 4217 4217 4224 4352
4174 4264 4206 4333 4272
4094 4037 4212 4346 3600
4017 4339 4244 4136 4236
4171 4154 3888 3957 4321
4175 4806 4414 4155 4172
4294 4360 4368 4276 3413
4412 4209 4258 4970 4239
4709 4166 4115 4237 4225
4170 4239 4087 4241 4323
4014 4288 4279 3960 4098
4260 4121 4313 4178 4212
3784 4188 4128 4177 3916
4184 4779 4240 4044 4116
4209 4464 4250 4111 4187
4215 4284 4088 4103 4227
4100 4200 4118 4275 4030
4449 4038 4191 4225 4223
4183 4272 3996 4106 4507
4057 4209 4223 4141 3851
4178 4116 4128 4078 4391
4282 4435 4158 4367 4068
4246 4259 4122 4151 4116
4143 4194 4197 4168 4344
4187 4245 4155 4209 4209
4376 4268 4029 4342 4508
4227 4312 4188 4404 4352
4405 4133 4234 4308 4364
4137 3197 4444 4567 4314
4223 3958 3914 4057 4290
4028 4268 4177 4268 4103
4329 4301 4198 4117 4586
4329 4050 4035 4250 4227
4235 4200 4571 4107 4147
4191 4156 4319 4025 4258
4125 4226 4096 4266 4257
4263 4201 4217 3996 4639
4212 4542 4103 4277 4202
4152 4221 4179 4400 4397
4382 4127 4112 4058 4057
4301 4230 4309 4133 4214
4269 4230 4063 4250 4112
4212 3982 4128 4117 4252
4173 4355 4339 4196 4078
4155 4065 3527 4269 4229
4460 4337 4171 4179 4273
4261 4270 3819 4064 4145
4251 3905 4319 3688 4370
4262 4158 6959 4144 4297
4568 4051 4380 4117 4144
4265 4242 4109 4252 4278
4213 4067 4429 4103 4267
4123 4041 4337 4214 4252
4196 4074 4398 4098 4183
4196 4074 4067 4266 4194
3789 4177 4240 5040 3902
4230 4122 4036 4101 4228
4065 4156 4026 4027 4562
4341 4275 4319 4178 3969
4180 4002 4085 4179 3905
4092 4074 4172 4270 4339
4054 4100 4100 4141 4269
4267 4081 4208 4426 4172
4249 4334 4161 4284 4305
4305 4263 3979 4235 4319
4240 4120 4195 4136 4395
4359 4059 4269 4231 4389
4556 4224 4252 4103 4223

Explanation / Answer

Hi There,

Thanks for the question. I am answering this question assuming that you know the fundamentals of R.

Here is the Code:

scores=c(4153,3855,4287,4401,4302,
4379,4453,4282,4219,4395,
4288,4388,4291,4292,4091,
4322,4045,4154,4317,4208,
4366,4141,4291,4081,4396,
4220,4070,3999,4358,4181,
4185,4439,4253,4159,4006,
4280,4292,4180,4088,4398,
3993,4216,4163,4157,4101,
4418,4316,4306,3843,950,
4269,4331,4253,4821,4208,
4205,4189,4047,4085,7010,
4163,4300,4134,4199,3945,
4237,4162,4038,4250,4327,
4063,4070,4277,4243,4163,
4224,4111,4304,4157,4170,
4313,4249,4209,4121,3878,
4327,4243,4098,4218,4380,
4138,4117,4196,4224,4145,
4317,4237,4087,4150,4621,
4126,4280,3804,4214,4321,
4159,4181,4032,4172,4444,
4385,4254,4216,4052,4270,
4132,4229,4702,4231,4094,
4100,4220,5542,4137,4138,
3935,4058,4016,4222,4161,
3797,4111,4317,3957,4116,
4317,4106,4101,4094,4230,
4171,4351,4222,3816,4002,
4256,4293,4284,4177,4143,
4209,4322,4234,4225,4263,
4195,4411,4266,4210,4102,
4098,4459,4081,4349,3945,
4141,4244,4090,4090,4251,
4335,4424,4505,4247,4192,
4230,4954,4064,4232,4191,
4314,4280,4192,4110,4208,
5515,3872,4307,4384,4242,
4300,4247,4169,4209,3804,
4297,4185,4287,4197,4212,
3890,4115,4081,4293,4065,
4194,4302,4230,4073,4205,
4355,4506,4307,4242,3954,
4204,3760,4362,3877,4247,
4581,4197,4139,4110,4301,
4408,4188,4132,4065,3960,
4041,4110,4217,4213,4304,
4259,4184,4363,4100,4230,
4171,4131,4186,4292,4559,
4415,4301,4179,4188,4232,
4290,4221,4129,4189,4272,
4372,4278,4326,4588,4145,
4304,4204,4124,4190,4231,
4138,4247,4072,3822,4253,
4297,4271,4524,4179,4117,
4108,4174,4381,4101,4433,
4281,4271,4485,4239,4133,
4022,4755,4248,4150,4137,
4048,4390,4109,4407,4112,
4247,4149,4125,4565,1745,
4332,4219,4367,4005,3659,
4231,4143,4241,4261,3440,
4127,4214,4140,4218,4423,
4211,4161,4412,4185,4270,
4020,4185,4313,4211,4501,
4037,4212,4261,4154,4206,
4245,3246,4017,4372,4729,
3777,3888,3918,4140,4004,
4108,4251,4191,4251,4452,
4177,3878,4587,4222,4121,
4300,4198,4286,4138,4279,
4298,3950,4015,4380,4230,
4130,4055,4422,4174,4045,
4220,4238,4330,4263,4129,
4223,4292,4202,4207,4358,
4178,4205,3972,4203,4256,
4182,4286,4249,4197,3630,
4378,4261,4166,4270,4165,
4170,4060,4227,4113,4161,
4581,4177,4234,3963,4390,
4337,4130,4274,4085,3904,
4238,3952,4249,4201,3972,
4237,4166,4200,4305,4138,
3939,4024,4240,4222,4309,
4101,4556,4247,4233,4005,
4189,4193,4410,4299,4271,
4020,4149,4288,4151,4114,
4214,4133,4346,4145,4370,
4235,4295,4076,4363,4329,
4079,4322,4384,4138,4361,
4210,4075,4147,4165,4100,
4160,4311,3369,4227,4319,
4133,4207,4224,4374,4032,
4392,4250,4427,3937,4887,
4131,3718,4281,4208,3934,
4380,4353,4210,4166,4185,
4101,4071,4303,4276,4265,
4207,4043,4195,4703,4223,
4305,4330,3802,4165,4590,
4109,4204,4154,4145,4196,
4222,3981,4187,4093,4274,
4140,4053,4747,4237,4522,
4187,4142,4198,3980,4235,
4134,4324,4179,4258,4693,
4430,4082,4892,4390,3935,
4199,4258,3805,4124,4190,
4053,4179,4412,4019,4173,
4031,4215,4244,4312,4272,
4079,4273,4195,4156,4278,
4178,4232,2781,4393,4170,
4185,4285,4257,4138,4137,
4230,4029,4191,4275,4159,
4177,4253,4019,4199,4158,
4151,4088,3894,4212,3976,
4157,4096,4174,4041,4254,
4022,4243,4426,4140,4340,
4080,4182,4281,3876,4221,
4166,4241,4129,4181,4209,
4093,4139,4192,4078,3647,
4292,4300,4275,4068,4217,
4202,4063,4313,3891,4277,
4123,4198,4218,4443,4175,
4292,3996,4201,4170,4306,
4141,4644,4144,4254,4131,
4196,4257,4010,4250,4212,
3902,4239,4235,4208,4246,
4228,4216,4207,4207,4232,
4001,4018,4549,4346,3613,
4120,4084,4105,4100,4399,
4225,4195,4645,3734,4296,
4432,3969,4217,4185,4150,
4052,4169,4247,4142,4097,
3993,4314,4234,4244,3975,
4185,4065,4168,4175,4464,
4216,4466,4324,4181,4226,
4564,4181,4090,4066,3921,
4309,4000,4386,4235,4351,
4171,4372,4147,4190,4534,
4077,4131,4166,4326,4518,
4225,4108,4240,4242,3595,
4262,4219,4450,4328,4234,
4217,4170,4262,4433,4198,
4196,3923,4268,4090,4459,
4036,4201,4240,4188,4186,
4254,4346,4178,4242,4400,
4195,4213,5021,4230,4199,
4268,4157,4217,3938,4129,
3970,4222,4216,4431,4268,
4264,4331,4335,4290,4304,
4296,4202,3988,4245,4320,
4105,4197,4186,4299,4199,
6182,4319,4165,4203,4130,
4046,4270,4088,4101,4317,
4235,4102,4560,4104,4301,
4294,4127,4078,4439,3634,
4218,4156,4473,4152,4150,
4203,4210,3764,4394,4634,
4093,4297,3613,4301,4161,
4255,4513,4257,4212,4411,
4185,4549,3935,4240,4331,
3754,4163,4256,3837,4152,
4287,3912,4348,4053,4207,
4093,4287,4208,4176,4194,
4103,4236,4272,3829,4344,
4277,4245,4210,4150,3990,
4209,4267,4213,4131,4333,
4376,4162,4312,4275,4219,
4176,4214,4256,4236,4614,
4101,4235,4388,4411,4175,
4145,728,4062,4320,4252,
4298,4184,4307,4248,3940,
4137,4321,4266,4290,4251,
4179,4135,4255,4435,4018,
4433,4998,4213,4218,4196,
4420,4182,4134,4208,4172,
4448,4087,3959,3889,4090,
4343,4537,4069,4309,4189,
4331,4232,4092,4371,4117,
3860,4008,4198,3520,4453,
4221,4349,4130,4066,4344,
4188,3991,4402,3650,4357,
4170,4443,4163,3995,4521,
4218,4189,3630,3372,4219,
4181,3959,4220,4088,4042,
4289,4109,4198,4050,4153,
4053,4222,3749,4248,4063,
4308,4070,4121,4166,4268,
4120,4300,4087,4328,4220,
4119,4193,4206,4294,3978,
4036,4116,4510,4365,4224,
4069,4203,4376,4263,4143,
4178,4068,4171,4237,3984,
4160,4156,4197,4155,4194,
4048,4399,4263,4338,4335,
4070,4112,5054,3835,4186,
4216,4164,4155,4142,4094,
4298,4205,4402,4255,4292,
4224,4329,4330,4287,4243,
4184,4363,4222,4112,4294,
4318,4470,4085,4158,4631,
4239,4149,4169,4015,4035,
4101,4167,4157,4376,4242,
4200,4630,4384,4228,4052,
4018,4285,4187,4091,4014,
4071,4176,4393,4177,4140,
4166,4254,4174,4469,4026,
4167,4238,4013,4231,4667,
4241,4512,4212,4759,3582,
4318,4039,4194,4242,4224,
4093,4178,4138,4025,4325,
4042,3864,4217,4183,4099,
4033,4222,4287,4379,4239,
4207,4142,4369,4095,4054,
4176,4429,4245,4247,4374,
4317,4394,4148,4090,4192,
4096,3959,5065,4067,4321,
4360,4173,4178,4264,4182,
4152,4174,3671,4197,4346,
3886,4456,4173,4107,5839,
4052,4286,4091,4063,4344,
4314,4234,4176,4061,4200,
4012,4156,4314,4363,4254,
4152,4215,4205,4214,4128,
4205,4286,4003,4149,4509,
4008,4150,4341,4042,4228,
4287,4190,4181,4243,4141,
4425,4191,4342,3711,4064,
4162,4621,4067,4223,4415,
4083,4304,4159,4184,4207,
4315,4274,4175,4171,4326,
4263,4274,4055,4124,4236,
4138,4358,4199,4168,4255,
4349,4264,4204,4363,4192,
4346,3981,4264,4442,4036,
4199,4351,4205,4184,4093,
3981,4224,4082,5023,4264,
4243,4238,4113,4248,4361,
4078,4209,4296,4098,4280,
4027,4175,4136,4018,4165,
4176,4215,4043,4203,4249,
4355,3781,4341,4013,4282,
4187,4194,4605,4282,4196,
4256,4296,4463,4287,4202,
4346,3444,4194,4645,4048,
4403,4336,4168,4221,3972,
4309,4066,4272,4409,4231,
4263,4263,4100,4496,4322,
4241,4180,4332,4254,4088,
4198,4109,4126,3996,4206,
4175,4268,4160,4173,4381,
4155,4243,4268,4029,4218,
4060,4177,4512,4277,4182,
4383,4244,4185,4245,4141,
4227,4208,4204,4143,3853,
4533,4319,4256,4204,4110,
4060,4293,4231,4056,4214,
4169,4256,3914,3917,4296,
4290,4315,4092,4161,4354,
4293,4200,4106,3613,4264,
4269,4287,4308,4161,4224,
4426,4217,4329,4199,4132,
3987,4201,4019,4158,4426,
4175,4245,4168,4380,4190,
4256,4153,4034,3863,4061,
4441,4292,4260,4126,4121,
4420,4121,4379,4123,4351,
4089,4159,4176,4134,4046,
4091,4232,4557,4155,4091,
4169,4061,4002,3940,4115,
3914,4148,4241,4149,5089,
4230,4139,4375,4243,4018,
4226,4138,4253,4273,4152,
4241,4224,4277,4371,4056,
4291,4236,4240,4019,4220,
4138,4321,4154,4243,4134,
4097,4289,4830,4414,4109,
4093,4351,4022,4081,4148,
4817,4354,4144,4246,4297,
4320,4125,4125,4318,4148,
4478,4184,4223,4143,4254,
4256,4193,4104,4298,4177,
4306,4364,4202,4212,4164,
4096,4062,4324,4098,4105,
4149,4035,4135,4388,4203,
4293,4333,4795,5634,4274,
3337,4195,3788,4397,4219,
4502,4165,4500,4269,3997,
4297,4041,3997,4225,3561,
4177,4183,4063,4241,4156,
4096,4052,4284,4063,4297,
4088,4304,4244,4223,4373,
4265,3828,4120,4096,4281,
4322,4148,4593,4033,4177,
4235,4611,4242,4083,4219,
4242,4150,3995,4281,4196,
4282,4025,3628,4279,4199,
4174,4108,4259,4443,4294,
4153,4133,4074,4228,4142,
4207,4060,4025,4230,4130,
4142,4140,4276,4448,4073,
4223,4367,3941,4225,3965,
4189,4286,4204,4168,4200,
4327,4440,4171,4236,4144,
4302,4349,4234,4124,4071,
4364,4169,4256,4191,4576,
4252,4252,4260,4295,4115,
4291,4198,4271,4038,4241,
4009,4008,4231,4344,4243,
3913,4208,4194,4108,4112,
4114,4183,3978,4239,4367,
4462,4303,4092,4089,4158,
4222,4209,4069,4098,4264,
4292,4152,4199,4301,4183,
4091,4239,4134,4235,4270,
4333,4013,4110,4183,4149,
4202,4204,4255,4321,4325,
4399,3995,3875,4255,4314,
4216,4246,4268,4337,4228,
4175,3560,4261,4201,4365,
4259,4267,4153,4237,4292,
4305,4198,4218,4165,4304,
4190,4038,4079,4171,4263,
4432,4344,4196,4241,4201,
3998,3807,4155,4384,4164,
4318,4093,4224,4301,4026,
4189,4229,4237,4224,4399,
4006,4183,4200,4210,4296,
4288,4214,4196,4273,4213,
4466,4217,4156,4320,4112,
4115,4207,4189,4026,4213,
4049,4284,3399,4227,4237,
4204,4162,4289,4070,4035,
3518,4217,4217,4224,4352,
4174,4264,4206,4333,4272,
4094,4037,4212,4346,3600,
4017,4339,4244,4136,4236,
4171,4154,3888,3957,4321,
4175,4806,4414,4155,4172,
4294,4360,4368,4276,3413,
4412,4209,4258,4970,4239,
4709,4166,4115,4237,4225,
4170,4239,4087,4241,4323,
4014,4288,4279,3960,4098,
4260,4121,4313,4178,4212,
3784,4188,4128,4177,3916,
4184,4779,4240,4044,4116,
4209,4464,4250,4111,4187,
4215,4284,4088,4103,4227,
4100,4200,4118,4275,4030,
4449,4038,4191,4225,4223,
4183,4272,3996,4106,4507,
4057,4209,4223,4141,3851,
4178,4116,4128,4078,4391,
4282,4435,4158,4367,4068,
4246,4259,4122,4151,4116,
4143,4194,4197,4168,4344,
4187,4245,4155,4209,4209,
4376,4268,4029,4342,4508,
4227,4312,4188,4404,4352,
4405,4133,4234,4308,4364,
4137,3197,4444,4567,4314,
4223,3958,3914,4057,4290,
4028,4268,4177,4268,4103,
4329,4301,4198,4117,4586,
4329,4050,4035,4250,4227,
4235,4200,4571,4107,4147,
4191,4156,4319,4025,4258,
4125,4226,4096,4266,4257,
4263,4201,4217,3996,4639,
4212,4542,4103,4277,4202,
4152,4221,4179,4400,4397,
4382,4127,4112,4058,4057,
4301,4230,4309,4133,4214,
4269,4230,4063,4250,4112,
4212,3982,4128,4117,4252,
4173,4355,4339,4196,4078,
4155,4065,3527,4269,4229,
4460,4337,4171,4179,4273,
4261,4270,3819,4064,4145,
4251,3905,4319,3688,4370,
4262,4158,6959,4144,4297,
4568,4051,4380,4117,4144,
4265,4242,4109,4252,4278,
4213,4067,4429,4103,4267,
4123,4041,4337,4214,4252,
4196,4074,4398,4098,4183,
4196,4074,4067,4266,4194,
3789,4177,4240,5040,3902,
4230,4122,4036,4101,4228,
4065,4156,4026,4027,4562,
4341,4275,4319,4178,3969,
4180,4002,4085,4179,3905,
4092,4074,4172,4270,4339,
4054,4100,4100,4141,4269,
4267,4081,4208,4426,4172,
4249,4334,4161,4284,4305,
4305,4263,3979,4235,4319,
4240,4120,4195,4136,4395,
4359,4059,4269,4231,4389,
4556,4224,4252,4103,4223
)
library(BSDA)
z.test(scores, sigma.x = 6.8)

RESULT: One-sample z-Test

data: scores
z = 27643, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
4202.910 4203.506
sample estimates:
mean of x
4203.208

INTREPRETATION: we are 95% confident that the mean falls between 4202.910 and 4203.506

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