Download Solutions to the problems in the Sample Test

Sample Test #2 - Solutions
Descriptive Statistics
The following sample was observed, reported sorted with rank and percentile rank:
Rank
14.3470313645721 1
13.9511951251287 2
13.2779176069121 3
13.067373009224 4
12.9228692844372 5
12.8741441944157 6
12.6910633361323 7
12.6510189803734 8
12.5264890042628 9
12.4939617464683 10
12.334512558672 11
12.2565739164815 12
12.1923405114508 13
12.1707334449166 14
12.1155856376745 15
12.0387821210453 16
11.9756692824804 17
11.8845835411695 18
11.6922262079605 19
11.5295952141309 20
11.4899749544308 21
11.0564021006897 22
11.0538426539281 23
10.7488343272596 24
10.6074606195043 25
10.2807997575152 26
9.19230499721994 27
9.08190010947565 28
Percentile Rank
100.00%
96.30%
92.59%
88.89%
85.19%
81.48%
77.78%
74.07%
70.37%
66.67%
62.96%
59.26%
55.56%
51.85%
48.15%
44.44%
40.74%
37.04%
33.33%
29.63%
25.93%
22.22%
18.52%
14.81%
11.11%
7.41%
3.70%
0.00%
The following are statistics for this sample:
Mean
11.9466137717118
Standard Error
0.2315685387286
Median 12.1431595412955
1 quartile 11.3815817409955
3 quartile 12.6610300693131
Standard Deviation 1.22534552988502
Sample Variance 1.50147166760921
Estimating Proportions
A simple random sample of 500 individuals was tested for the presence of a specific gene. 150 turned out to be
positive. Compute a confidence interval for the proportion of people in the general population that carry that
gene. Be sure to specify what method you used in your calculation.
Solutions
The sample mean is
^p=
150
=0.3 . If we choose a 95% confidence level, applying our formulas we have
500
•
Using the “worst case” estimate
•
Using the “rough” estimate
σ=
1
, (0.25617387297117,0.34382612702883)
2
σ=√ ^p ( 1− ^p )≈0.458 … , (0.25983269109473,0.34016730890527)
Estimating the Mean
Determine a confidence interval, at a level of your choice, for the true mean of the distribution sampled
above
Confidence intervals, at various levels:
95% CI for the Mean from
to
11.4714743772917
12.4217531661319
90% CI for the Mean from
to
11.5521857553027
12.341041788121
98% CI for the Mean from
to
11.3740235291274
12.5192040142963
99% CI for the Mean from
to
11.3050107680508
12.5882167753729
* Estimating the Variance
Determine a confidence interval, at a level of your choice, for the true variance of the distribution
sampled above
We have that the sum of squares of the difference between the data points and the sample mean is equal
to 40.5397350254484.
Hence, using tables, we have the following confidence intervals
99%: (0.81659390052922,3.43336312652881)
95%: (0.93853904393575,2.78176561847303)
90%: (1.01063146769221,2.5099833725078)