Question

In the packaging process, equipment is set to fill boxes with a mean weight of 20.2 ounces. This is the standard for the population. Population standard deviation is not known.

1. Calculate a 95% confidence interval for the mean weight of the cereal boxes, using the sample data

Package Weights
21.295
21.296
19.816
19.885
20.211
20.212
20.304
20.306
20.313
20.465
20.468
20.468
20.722
20.723
20.724
20.483
20.484
20.485
20.64
20.64
20.646
20.649
19.199
19.988
20.213
20.225
20.245
20.222
20.6
20.599
20.61
20.94
20.94
20.153
20.16
20.939
20.943
20.819
20.816
20.819
20.226
20.227
20.223
20.591
20.592
20.592
20.593
20.588
20.315
20.311
20.316
20.316
20.251
20.33
20.331
20.332
20.424
20.425
20.425
19.228
19.695
20.894
20.898
20.896
20.243
20.244
20.244
20.642
20.642
20.643
20.643
20.333
20.334
20.352
20.942
20.947
20.948
20.949
20.419
20.421
20.421
20.423
20.423
20.746
20.747
20.748
20.386
20.382
20.612
20.611
20.613
20.609
19.232
19.983
19.984
20.374
20.374
20.384
21.145
21.172
21.177
21.178
20.618
20.619
20.364
20.368
20.434
20.435
20.566
20.584
20.589
20.903
20.469
20.469
20.471
20.905
20.906
20.445
20.443
20.949
20.951
20.969
20.369
20.37
20.37
20.37
20.834
20.845
20.847
20.84
20.666
20.666
20.667
20.428
20.431
20.635
20.636
20.636
20.637
20.529
20.533
20.534
20.23
20.233
20.242
20.243
19.985
19.187
19.986
19.987
20.463
20.465
20.792
20.799
20.806
20.818
20.463
20.525
20.529
20.529
19.989
20.151
20.151
20.654
20.655
20.657
20.661
20.354
20.36
20.362
20.5
20.508
20.512
20.876
20.876
20.867
20.495
20.497
20.498
20.162
20.162
20.164
20.99
20.21
20.629
20.63
20.632
20.493
20.492
20.491
20.246
20.258
20.259
20.26
20.447
20.527
20.519
20.527
20.526
20.528
20.48
20.481
20.723
20.726
20.725
20.451
20.453
20.456
20.462
20.623
20.624
20.625
20.626
20.517
20.527
21.289
21.29
20.651
20.652
20.541
20.634
20.635
20.499
20.499
20.719
20.719
20.72
20.721
20.493
20.494
20.494
20.496
20.495
20.627
20.511
20.524
20.526
20.628
20.629
20.59
20.534
20.535
20.536
20.537
20.878
20.878
20.878
20.881
20.652
20.653
20.727
20.482
20.483
20.48
20.729
20.729
20.621
20.622
20.622
20.595
20.597
20.598
20.644
20.64
19.696
19.697
20.882
20.88
20.639
20.246
20.258
20.259
20.639
20.785
20.787
20.565
20.556
20.558
20.567
20.651
20.637
20.638
20.638
20.487
20.486
20.776
20.777
20.778
20.778
20.65
20.651
20.397
20.397
20.399
20.419
20.418
20.667
20.668
20.67
20.371
20.373
20.372
20.736
20.74
20.74
20.742
20.656
20.653
20.653
20.665
20.847
20.849
20.526
20.526
20.517
20.874
20.671
20.672
20.672
20.674
20.673
20.676
20.702
20.514
20.509
20.515
20.426
20.422
20.426
20.426
20.426
20.427
20.617
20.62
20.428
20.429
20.435
20.437
20.437
20.442
20.713
20.753
20.75
20.64
20.641
20.73
20.91
20.911
20.934
20.936
20.73
20.388
20.389
20.594
20.594
20.395
20.396
21.061
20.887
20.883
21.179
21.18
20.738
19.698
19.812
20.717
20.718
20.695
20.696
20.697
20.769
20.769
20.771
20.922
20.909
20.919
20.771
20.774
20.224
20.229
20.231
20.231
20.682
20.694
20.694
20.904
20.912
20.907
20.448
20.45
20.452
20.884
20.885
20.888
20.79
20.791
20.352
20.353
20.355
20.787
20.788
20.944
20.941
20.942
20.152
20.153
20.659
20.653
20.658
20.822
20.823
20.789
20.789
21.287
21.288
20.162
20.821
20.821
20.302
20.304
20.301
20.831
20.832
20.833
21.144
21.146
21.145
20.824
20.83
20.938
20.939
20.755
20.937
20.928
20.929
20.857
20.875
20.876
20.673
20.361
20.362
20.677
20.675
20.679
20.877
20.878
20.907
20.907
20.908
20.26
20.261
20.261
20.886
20.893
20.892
21.292
21.293
21.294
20.749
20.23
20.91
20.761
20.762
20.768
21.063
21.065
21.143
20.756
20.757
20.759
20.981
20.987
20.988
20.664
20.662
20.975
20.977
20.541
20.542
20.543
20.544
20.989
20.731
20.732
20.733
20.679
20.68
20.681
21.283
20.709
20.712
20.713
20.974
20.976
19.813
19.814
21.053
21.063
20.759
20.76

2. Based on the confidence interval for the mean weight of cereal boxes, does this sample provide evidence that the packaging process is meeting the company standard? Explain.

Answer 1

1. For given data of size n=500, $\mu=20.6$ and $\sigma=0.30$

Margin of Error= $E=z*\frac{\sigma }{\sqrt{n}}=1.96*\frac{0.30}{\sqrt{500}}=0.03$

Hence $CI=\mu \pm E=20.6 \pm 0.03=(20.57,20.63)$

2. As 20.2 is not in range this is not meeting the company standard