Download Sample Take-Home test

Sample Final Exam
Data Sets
We observe a number of data sets, resulting in the following summaries:
Set #1
0.92154
0.95749
0.95969
0.96660
0.96837
0.98737
0.98897
1.03850
1.04007
1.04705
1.06594
1.06960
1.06964
1.08102
1.09246
1.09496
1.10273
1.14394
1.14707
1.15204
1.15276
Sum
Count
Sum of squares
1.15544
1.18723
1.1994
1.20593
1.21070
1.21613
1.30351
30.5261
28
33.5317
Set #2
0.90357
2.66148
3.54599
5.28769
6.93558
7.02943
7.04880
Sum
Count
Sum of Squares
7.24213
7.44643
7.72484
8.19351
8.65482
8.74767
8.82486
9.46781
10.1070
10.8687
11.5657
12.2587
12.7516
13.5501
14.6619
14.9376
15.5333
15.8570
16.8235
17.8379
18.5768
18.6690
18.8775
18.9959
19.2264
27.6086
40.1723
68.1279
496.722
35
11929.8
Set #3
We only report about the differences between the data in Sample 1 and this Sample. The
summaries for the differences are
Sum
−3.0456
Count
28
Sum of Squares
1.21637
Set #4
0.13234
0.15308
0.35685
0.40157
0.40893
0.43890
0.51273
0.54072
0.66466
0.71547
0.73718
0.78282
0.81730
0.83066
Sum
Count
Sum of Squares
0.92685
0.93011
0.97048
1.07575
1.1858
1.19442
1.21328
1.25213
1.37178
1.40444
1.49122
1.49174
1.8339
1.94618
25.7813
28
29.8625
Set #5
0.80214
0.82697
0.87572
0.89670
0.92259
0.93288
0.93394
0.93650
0.95775
0.97102
0.97303
1.00718
1.01767
1.03088
Sum
Count
Sum of Squares
1.04796
1.05381
1.05695
1.05709
1.06174
1.08236
1.09798
1.12141
1.13258
1.13815
1.16667
1.21023
1.26489
1.27568
28.8525
28
30.1252
1. Descriptive Statistics
Calculate the following statistics for data set #1 and data set #2
1. Sample mean
2. Sample Variance
3. Sample Standard Deviation
4. Median
5. Range
Optional (Extra Credit): If time allows, you may calculate these statistics for some other data
set as well
2. Confidence Intervals
Determine a confidence interval, at a confidence interval of your choice, for the true mean of
data set #1 and data set #2
Optional (Extra Credit)
• If time allows, you may calculate confidence intervals for other data sets as well
• If time allows you may calculate confidence intervals for the variance of one or more
data sets.
3. Testing
3.1. Testing for the mean
Data set #1 simulates measurements of a batch of resistors with nominal resistance of 1 kOhms.
Set up a two-tailed test (at a significance level of your choice) to verify that the resistors are up
to specification
1. (Optional - Extra Credit)
The data in Data set #2 simulates measurements of the voltage of a batch of nominal 9 V
batteries. Set up a two-tailed test (at a significance level of your choice) to verify that the
batteries are up to specification
2. (Optional - Very Extra Credit)
The resistors are sold as having a standard deviation from the nominal value of no more than
10% of their nominal specification. Set up a one-tailed test for their variance (which should be
no more than 0.01, as the square of a standard deviation of 0.1)
3.2. Testing Proportions
A coin is flipped 80 times, and this results in 36 “Heads”. Set up a test to determine whether the
coin is fair
3.2.1. (Optional - Extra Credit)
The experiment is repeated four more times, resulting in, respectively, 42 42, 41 and 48 “Heads”.
• Repeating the test for each experiment, what is the result?
• If you combine all five experiments (for a total of 400 tosses and 209 “Heads”), what is
the outcome of a statistical test now?
3.3. Paired Samples
Data set #3 is supposed to measure the resistance of the items from Sample #1 after a stress test.
Set up a paired sample test to verify whether the average resistance has changed or not.
3.4. Independent Samples
1. Unequal Variances
We compare sample #1 with sample #4, which comes from a different facility. We can no longer
assume that the variances are equal. Set up a test to verify that the two batches come,
nonetheless, from populations with the same “true” mean.
2. Equal Variances
3. We compare the resistors in sample #1 with a second batch from the same production line,
which we can assume produces items with the same variance as that of sample #1, resulting in
Sample #5. Set up a test to verify that the two batches have the same “true” mean
Remarks about the data sets
All data sets were simulated by computer
• Set #1 should look like a simple random sample from a normal distribution, with mean
1.06 and standard deviation 0.1
• Set #2 is a reduced sample from a fat-tailed distribution (a Cauchy distribution) that has
no true mean nor true variance (its median should be 9, but the data reported here came
from excluding negative values obtained in the simulation, including a whopping
¡26;035:8 - talk about really fat tails!)
• Proportions were all simulated assuming p = 0.52
• The second set of paired data (used for set #3) was obtained by adding to each element of
set #1 a random variable, uniformly distributed between -10 and 2, for an expected
difference of - 4
• Set #4 should look like a simple random sample from a normal distribution with mean
0.9 and standard deviation 0.5 (hence, different from set #1)
• Set #5 should look like a simple random sample from a normal distribution with mean
1.03 and standard deviation 0.1 (hence, with the same variance as set #1)