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Answer to Exercises on Statistics Concepts
BSNS5230, J. Wang
1. Statistics is to investigate some characters of a population by using a sample from it.
a. True
b. False
2. A sample usually reflects the population perfectly.
a. True
b. False
3. Statistics and probability theory refer to a same subject of study.
a. True
b. False
4. Probability theory is _____________________.
a. a study of data processing
b. a study of chances
c. a field of statistics
d. another name of statistics
5. A key word of statistics is _________________.
a. mean
b. standard deviation
c. sample
d. accuracy
6. Statistics is synonymous with data processing.
a. True
b. False
7. Probability theory is used in statistics to tell the possible error rate of extrapolating a sample to
its population.
a. True
b. False
8. In general, the _________ the sample size, the more accurate the sample reflects the
population.
a. larger
b. smaller
9. If sample size equals to population size, then _____________________
a. the sample would be useless
b. the sample would perfectly reflect what the population is like
10. There is always possible to have errors when we use a sample to tell what its population is
like, unless the sample is the same as the population.
a. True
b. False
11. Suppose the mean of a sample is 264.83. Where is the mean of the population?
a. 264.83
b. Less than 264.83
c. More than 264.83
d. We do not know.
12. If we select two samples from a same population, then the means of the two samples must be
same.
a. True
b. False
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13. If we select 100 samples from a same population, then the 100 sample means are typically
different.
a. True
b. False
14. When we investigate the population mean from a sample, we can calculate the sample mean
but we never know where the population mean exactly is.
a. True
b. False
15. When we investigate the population mean from a sample, we figure out a range in which the
population mean is likely located.
a. True
b. False
16. Suppose 95% of confidence interval for the population mean is (14.7, 20.5). It implies that
there is ______ of chance the population mean is within that range, and there is _______ of
chance the population mean is not in that range.
a. 95%, 5%
b. 5%, 95%
17. If the confidence level for a confidence interval is 98%. then the error rate α is ______.
a. 98%
b. 2%
c. 95%
d. 5%
18. Suppose the sample mean is μ and the confidence interval for population mean is (μ-E, μ+E).
E will be getting smaller if:
(18.a) the sample size is getting ___________;
a. smaller
b. larger
(18.b) the standard deviation of the data in the sample is __________;
a. smaller
b. larger
(18.c) the allowed error rate α is ___________;
a. smaller
b. larger
(18.d) the confidence level (1-α) is ___________.
a. lower
b. higher.
19. The 99% confidence interval is _________ than 95% confidence interval in a same problem.
a. smaller
b. larger
20. Suppose 95% confidence interval of the population mean is (4.5, 9.7).
(20.a) The population mean must be in the range (4.5, 9.7).
a. True
b. False
(20.b) What is the chance of error if you say that the population mean is somewhere in the range
(4.5, 9.7)
a. 0.95
b. 0.05
c. 0
d. 0.5
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(20.c) How sure is it that population mean is within (4.5, 9.7)?
a. With 95% of sureness
b. With 5% of sureness
(20.d) Based on this confidence interval, can you infer with 95% of confidence that the
population mean is more than 5?
a. Yes
b. No
(20.e) Based on this confidence interval, can you infer with 95% of confidence that the
population mean is more than 4?
a. Yes
b. No
21. Given a sample of family incomes (in $) as below:
53,600
97,700
65,000
110,400
84,700,
38,800
76,200
120,000
(21.a) Calculate the sample mean.
mean = 76,975
47,300
63,400
77,000
89,600
(21.b) Calculate the sample standard deviation
SD = σ = 24,917.286
(21.c) Calculate 95% confidence interval for the population mean.
E = CONFIDENCE.T(0.05,24917.286,12) = 15,831.688
95% confidence interval = (76975-15831.688, 76975+15831.688) = (61143.31, 92806.69)
(21.d) If you want a narrower confidence interval, you need to ________ sample size.
a. increase
b. decrease
22. In a sample of 300 adults, 52 smoke.
(22.a) Calculate the proportion of smokers in the sample.
p = 52/300 = 0.1733
(22.b) When we use Excel function =CONFIDENCE.T(α,σ,n), we need the value σ. Calculate
that σ in this problem.
σ = √𝑝(1 − 𝑝) = √0.1733(1 − 0.1733) = 0.3785.
(22.c) Figure out the value E by using Excel function =CONFIDENCE.T(α,σ,n), with allowed
error rate 0.05.
E = CONFIDENCE.T(0.05, 0.3785, 300) = 0.043.
(22.d) What is the 95% confidence interval of the proportion of smokers in population?
(0.1733-0.043, 0.1733+0.043) = (0.1303, 0.2163)
23. A poll of 250 people shows the supporting rates of two candidates of the state congress: candidate A: 28.3%±3%; candidate B: 32.1%±3%. Do we conclude that candidate B’s
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supporting rate is higher than candidate A (or can we reject the hypothesis that the two
candidates’ supporting rates are same)? Why?
No, we cannot conclude that B’s supporting rate is higher than A, since the two
confidence interval overlap somewhat.
What if the supporting rates of a poll are 28.5%±2% for A and 32.7%±2%? Why?
Yes, we can conclude that B’s supporting rate is higher than A with 95% confidence,
since B’s confidence interval does not overlap A’s confidence interval at all.
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