Depaul University Confidence Interval Statistics Worksheet

MAT 137: Business StatisticsProject 3, Statistical Inferences
Due by 11:59 pm Monday November 7, 2022
******* Please submit the Excel worksheet and the written report.
Purpose: The purpose of this project is to help students synthesize the material from the
course and to make statistical inferences [confidence intervals and hypothesis tests] using
Excel.
Problem Description: In this project we will continue investigating the diamond data
from the last Excel project, but now from the point of view of inferential statistics. We
aim at drawing statistical inferences about mean carat and price of all round diamonds sold
in Singapore using a sample of n = 308 round diamonds
Data Set Background: Diamonds are categorized according to the “four Cs”: carats,
clarity, color, and cut. Each diamond stone that is sold on the open market is provided
a certificate by an independent diamond assessor that lists these characteristics. Data for
a sample of 308 Round diamond stones that were listed for sale in the February 18, 2000
edition of Singapore’s Business Times is given in the file diamond-data.xlsx [on D2L].
This time, we only focus on the two quantitative variable, “CARAT” and “Price”, the
number of carats (a unit of weight used to measure the sizes of gemstones), and the asking
price (in Singapore dollars), respectively. The other non-relevant columns have been
deleted from the original Excel file.
Answer the following questions in your written report (not point-by-point,
incorporate the answers into your report.):
I. CONFIDENCE INTERVAL:
• Calculate the sample mean x̄ and sample standard deviation s of “PRICE” data.
Using your sample mean and sample standard deviation. Construct a 95%
confidence interval for the mean price of all round diamonds sold in Singapore.
Include the following information in your report:
1. The confidence coefficient (that’s the confidence level) for the confidence interval.
2. The corresponding value of α.
3. The critical value zα/2 corresponding to an area of α/2 in the right tail of the
standard normal distribution.
4. The standard error (SE) for the confidence interval: 𝑆𝐸 = 𝑧∝ ∙
2
𝑠
√𝑛
Note: In SE, used s instead of the population standard deviation σ because σ is not given and
because the sample size n = 308 is large.
5. The lower and upper bounds of the confidence interval.
6. Conclusion of your study, i.e., “We are 95% confident that. ….”
7. If a certain diamond dealer claims that the mean price of all round diamonds sold
in Singapore is 5,350 Singapore dollars, does the 95% confidence interval found
earlier support the claim. Explain.
II. TEST of HYPOTHESIS:
• Now we will test some hypotheses of mean price of round diamonds sold in Singapore.
Do the data provide sufficient evidence showing that the mean price of all round
diamonds sold in Singapore is above 5,350 Singapore dollars? (That’s the claim.)
1. Formulate the null hypothesis (H0) and the alternative hypothesis (Ha) for this study.
Your alternative hypothesis in this situation should be of the upper-tailed type.
Calculate the test statistic z-stat
𝑥̅ − 𝜇0
𝑧=
𝑠
√𝑛
and explain why the z-test can be used.
2. Using the significance level α = 0.05, describe the rejection region, i.e., what
value must z be larger than in order for us to reject the null hypothesis?
3. Draw conclusion using the rejection region.
4. Calculate the attained (or observed) significance level, p-value, of the test.
5. Draw conclusion using the p-value at α = 0.05.
CARAT COLOR CLARITY CERT
0.76
D
IF
GIA
0.31
D
VS1
GIA
0.53
D
VS1
GIA
0.71
D
VS1
GIA
1
D
VS1
GIA
0.3
D
VS2
GIA
0.52
D
VS2
GIA
1
D
VVS1
GIA
1.01
D
VVS1
GIA
0.75
D
VVS2
GIA
0.63
E
IF
GIA
0.3
E
VS1
GIA
0.31
E
VS1
GIA
0.34
E
VS1
GIA
0.35
E
VS1
GIA
0.5
E
VS1
GIA
0.5
E
VS1
GIA
0.52
E
VS1
GIA
0.54
E
VS1
GIA
0.5
E
VS1
GIA
0.56
E
VS1
GIA
0.7
E
VS1
GIA
0.71
E
VS1
GIA
0.72
E
VS1
GIA
1
E
VS1
GIA
1.01
E
VS1
GIA
1.03
E
VS1
GIA
0.33
E
VS2
GIA
0.73
E
VS2
GIA
0.83
E
VS2
GIA
1
E
VS2
GIA
1.01
E
VS2
GIA
0.62
E
VVS1
GIA
0.46
E
VVS2
GIA
0.55
E
VVS2
GIA
1
F
IF
GIA
0.31
F
VS1
GIA
0.32
F
VS1
GIA
0.34
F
VS1
GIA
0.35
F
VS1
GIA
0.36
F
VS1
GIA
0.4
F
VS1
GIA
0.41
F
VS1
GIA
PRICE
9885
1641
3921
6372
11419
1302
3490
15582
16008
7368
6512
1510
1555
1693
1738
3501
3501
3635
3767
3501
3900
5800
5881
5961
10588
10692
10900
1327
5738
7156
9757
9853
5845
2942
4138
13913
1427
1468
1551
1593
1635
1911
1956
0.5
0.52
0.53
0.55
0.56
0.6
0.7
0.71
0.9
1
1.01
1.02
1.04
0.37
0.71
0.72
0.85
0.7
0.7
1
1.01
1.02
0.54
0.51
0.7
0.77
0.5
0.51
0.52
0.53
0.71
1
0.3
0.34
0.35
0.5
0.51
0.7
0.7
1
0.32
0.34
1
1
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
G
G
G
G
G
G
G
G
G
G
G
G
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS2
VS2
VS2
VS2
VS2
VS2
VS2
VS2
VS2
VVS1
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS2
VS2
VS2
VS2
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
3293
3418
3480
3605
3667
4291
5510
5586
7680
10713
10272
10372
10571
1420
5193
5263
6805
5122
5122
9480
9573
9666
4066
3851
6285
6919
3501
3567
3635
3701
5881
10588
1260
1410
1447
3016
3205
5122
5122
9619
1202
1269
9169
9203
1.06
0.3
0.64
0.5
0.31
0.48
0.53
0.55
0.57
0.59
0.63
0.74
1
1.02
0.55
0.6
0.34
0.37
0.4
0.89
0.73
0.73
0.78
1.01
0.31
0.34
0.35
0.84
0.74
1
1.01
1.06
1.1
0.66
0.57
0.72
0.36
0.43
0.7
0.71
0.86
0.7
0.71
1.04
G
G
G
G
G
G
G
G
G
G
G
G
G
G
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
I
VS2
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
IF
IF
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS1
VS2
VS2
VS2
VS2
VS2
VS2
VS2
VS2
VS2
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
IF
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
9743
1510
4759
3432
1427
2532
3407
3529
3651
3773
4401
5815
9896
10090
3605
4291
1316
1420
1525
6709
5030
5030
5386
9153
1126
1222
1255
5705
4585
8788
8873
9302
9646
4300
3415
5662
1485
1747
5122
5193
6882
5122
5193
9563
0.31
0.45
0.73
0.75
1
0.33
0.7
0.82
0.8
1
1.01
0.73
1.01
0.56
0.8
0.9
0.75
1.01
1.05
1.07
1
1.01
1
1.01
0.52
0.81
0.7
0.81
1
0.7
1
1.01
0.8
0.56
0.8
0.81
1.01
0.56
0.57
0.73
0.82
0.85
0.5
0.51
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
D
D
E
E
E
E
E
E
E
E
E
E
F
F
F
F
F
F
F
F
F
F
F
F
VS1
VS1
VS1
VS1
VS1
VS2
VS2
VS2
VS2
VS2
VS2
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
VS1
VS1
VS2
VS2
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
IF
VS1
VS1
VS1
VS1
VS2
VS2
VS2
VS2
VS2
VVS1
VVS1
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
GIA
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
1126
1572
4221
4355
8095
1098
3861
4948
4832
7818
7895
4727
8873
2892
5441
6682
4667
8455
8781
8945
13775
13909
10588
10692
3346
6988
6867
8715
14051
6285
11419
11531
8611
3667
6905
6988
10272
3202
3256
5333
6572
6805
3778
3851
0.53
0.85
1
1.01
0.53
1
1.02
0.71
0.6
0.81
0.6
1
1.01
0.5
0.7
0.73
0.85
1
0.55
0.7
0.8
0.82
1
1.01
0.58
0.81
0.8
1
1.01
0.7
0.86
1
1.02
0.52
0.57
0.6
0.66
0.72
0.74
1
0.61
0.64
0.71
0.8
F
F
F
F
F
F
F
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
VVS1
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
IF
VS1
VS1
VS2
VS2
VS2
VVS1
VVS1
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VVS2
VVS2
IF
IF
VS1
VS1
VS1
VS2
VS2
VS2
VS2
VVS1
VVS1
VVS1
VVS1
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
3995
8359
11696
11811
3701
10588
10796
6372
3925
6495
3421
9203
9293
3432
5800
6041
7711
10450
3529
5510
6905
7072
9896
9993
3792
7358
6051
9065
9153
4346
5835
8788
8959
3130
3415
3925
4300
5662
5815
9480
3616
3785
5193
6416
1.01
1.06
1
1.01
1
1
1.01
0.62
0.65
1
1.09
0.2
0.21
0.71
0.19
0.19
0.21
0.22
0.23
0.58
0.3
0.31
0.51
0.56
0.19
0.21
0.23
0.25
0.26
0.27
0.48
0.18
0.19
0.26
0.31
0.34
0.35
0.51
0.58
0.18
0.19
0.26
0.3
0.34
H
H
I
I
I
I
I
I
I
I
I
D
D
D
D
E
E
E
E
E
E
E
E
E
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
VVS2
VVS2
VS1
VS1
VS2
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VS1
VS1
VS1
VVS2
IF
IF
IF
IF
VVS1
VVS2
VVS2
VVS2
VVS2
IF
IF
IF
IF
IF
IF
VS1
VVS1
VVS1
VVS1
VVS1
VVS1
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VVS2
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
HRD
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
9433
9890
8095
8175
7818
8788
8873
3615
3643
8372
9107
880
919
6160
967
1050
1149
1198
1248
4831
1580
1628
3722
4070
967
1057
1147
1485
1539
1595
2383
823
863
1365
1628
1773
1821
3722
4209
765
800
1260
1459
1636
0.47
0.55
0.76
0.18
0.18
0.19
0.2
0.21
0.23
0.25
0.29
0.4
0.5
0.2
1.01
0.2
0.19
0.3
0.58
0.7
0.18
0.35
0.56
0.7
0.78
0.18
0.19
0.24
0.25
0.27
0.32
0.33
1.01
0.3
1
0.25
0.26
0.28
0.29
0.3
0.31
0.52
1.01
0.41
F
F
F
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
H
H
H
H
H
H
H
H
H
H
I
I
I
I
I
I
I
I
I
VVS2
VVS2
VVS2
IF
IF
IF
IF
IF
IF
IF
IF
IF
IF
VS1
VS1
VS2
VVS1
VVS1
VVS1
VVS1
VVS2
VVS2
VVS2
VVS2
VVS2
IF
IF
IF
IF
IF
IF
IF
VS2
VVS2
VVS2
IF
IF
IF
IF
IF
IF
IF
VS1
VVS1
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
IGI
2651
3706
6095
803
803
842
880
919
995
1283
1471
2276
3652
705
9713
638
800
1459
3821
5607
705
1540
3470
5326
5937
725
758
1108
1149
1233
1462
1503
8873
1218
9342
1082
1121
1199
1238
1299
1337
3095
8175
1616
0.41
I
VVS2
IGI
1506