Download Quiz #3

Statistics (I)
2011Fal
Quiz #3
Date: 12/08/2011
A.
MULTIPLE CHOICE QUESTIONS (30%)
1.
For a two tail test, the p-value is the probability of obtaining a value for the test statistic as
a. likely as that provided by the sample
b. unlikely as that provided by the sample
c. likely as that provided by the population
d. unlikely as that provided by the population
2.
The power curve provides the probability of
a. correctly accepting the null hypothesis
b. incorrectly accepting the null hypothesis
c. correctly rejecting the alternative hypothesis
d. correctly rejecting the null hypothesis
3.
Which of the following does not need to be known in order to compute the
p-value?
a. knowledge of whether the test is one-tailed or two-tailed
b. the value of the test statistic
c. the level of significance
d. None of these alternatives is correct.
4.
If a hypothesis is rejected at the 5% level of significance, it
a. will always be rejected at the 1% level
b. will always be accepted at the 1% level
c. will never be tested at the 1% level
d. may be rejected or not rejected at the 1% level
5.
If a hypothesis is rejected at 95% confidence, it
a.
b.
c.
d.
will always be accepted at 90% confidence
will always be rejected at 90% confidence
will sometimes be rejected at 90% confidence
None of these alternatives is correct.
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6.
In a two-tailed hypothesis test the test statistic is determined to be -2.5. The
p-value for this test is
a. -1.25
b. 0.4938
c. 0.0062
d. 0.0124
7.
For a one-tailed test (lower tail) at 89.8% confidence, z =
a. -1.27
b. -1.53
c. -1.96
d. -1.64
8.
When s is used to estimate , the margin of error is computed by using
a. normal distribution
b. t distribution
c. the mean of the sample
d. the mean of the population
9.
In order to determine an interval for the mean of a population with unknown standard
deviation a sample of 61 items is selected. The mean of the sample is determined to be 23.
The number of degrees of freedom for reading the t value is
a. 22
b. 23
c. 60
d. 61
10. In interval estimation, the t distribution is applicable only when
a. the population has a mean of less than 30
b. the sample standard deviation is used to estimate the population standard deviation
c. the variance of the population is known
d. the standard deviation of the population is known
11. As the sample size increases, the margin of error
a. increases
b. decreases
c. stays the same
d. increases or decreases depending on the size of the mean
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12. After computing a confidence interval, the user believes the results are meaningless because
the width of the interval is too large. Which one of the following is the best
recommendation?
a.
b.
c.
d.
Increase the level of confidence for the interval.
Decrease the sample size.
Increase the sample size.
Reduce the population variance.
13. Whenever using the t distribution for interval estimation (when the sample size is very small),
we must assume that
a. the sample has a mean of at least 30
b. the sampling distribution is not normal
c. the population is approximately normal
d. the finite population correction factor is necessary
14. A random sample of 64 students at a university showed an average age of 25 years and a
standard deviation of 2 years. The 98% confidence interval for the true average age of all
students in the university is
a. 20.5 to 26.5
b. 24.4 to 25.6
c. 23.0 to 27.0
d. 20.0 to 30.0
15. In a random sample of 100 observations, p = 0.2. The 95.44% confidence interval for p is
a.
b.
c.
d.
0.122 to 0.278
0.164 to 0.236
0.134 to 0.266
0.120 to 0.280
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B.
Problems (70%)
1.
(5%) In a random sample of 400 registered voters, 120 indicated they plan to vote for
Candidate A. Determine a 95% confidence interval for the proportion of all the registered
voters who will vote for Candidate A.
2.
(7%) A bank installed an ATM on NCKU campus. And this ATM is for exclusive use of the
university’s 30,000 staff and students. After several months of operation, a sample of 100
students and staff reveals the following use of the ATM by the people in a month.
Number of times ATM used
0
1
2
3
4
5
Frequency
25
30
20
10
10
5
a. (2%) What is the estimate of the proportion of staff and students who do not use ATM in a
month?
b. (5%) Develop a 95% confidence interval for this estimate. Can the bank be sure that at least
40 percent of the university staff and students will use the ATM.?
3.
(6%) Given that x  20 for a random sample of size 25 from the probability density function
2
e [( x  ) / 32]
f ( x) 
4 2
a. (3%) Please find a 90% confidence interval
b. (3%) If the resulting confidence interval in a. were to be half as long, how large a sample
would be needed?
4.
(8%) Let Y be B(100, p). To test H0:p0.08, Ha:p<0.08, we reject H0 and accept Ha if and only
if Y6.
a. (3%) Please determine the significance level () of the test.
b. (5%) Please find the probability of type II error if in fact p=0.04.
5.
(20%) The NCKU Co. Ltd. produces PC at 100 sets/hour. To meet the rising demand, this
company hires a new production manager to improve the production efficiency. After 6 months,
the general manager randomly sampled the production data for 100 hours and found that the
average production rate is 96 sets/hour with standard deviation 20 sets/hour.
a. (5%) Do you think the new production manager has improved the production efficiency
under the level of significance 0.05? Why?
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b. (5%) What is type II error if the actual mean of the production line is 94.71 sets/hour?
c. (5%) What would be the significance level if you want to set the type II error to 0.06 and no
changes on other condition?
d. (5%) What would the sample size be if you wish type I error=Type II error=0.10?
6.
(12%) Suppose that a sample Xi is drawn from population N(0, 02), and the null hypothesis is
considered as H0:0=0 and the population variance is 16 with sample size=25. Please answer
the following questions.
a. (4%) Please give a drawing to show where and what the significance level (=0.05),
rejection values, and the labels of x-axis are. Please also give and compute any information
needed.
b. (8%) Please give a drawing to show where and what the type II error, accepting values, and
the labels of x-axis are when the actual 0=1.5. Please also give and compute any
information needed.
7.
(4%) If one wants to evaluate the proportion of a proposal to be accepted by local residents, the
random samples of 112 residents are selected. Again if H0:p0.15 and Ha:p>0.15 are set and the
critical value p*=0.21 is also set. What is the Power of Test Function, PF(p)?
8.
(8%) You are given the following information obtained from a random sample of 4
observations.
25
47
32
56
You want to determine whether or not the mean of the population from which this sample was
taken is significantly different from 48. (Assume the population is normally distributed.)
a. (2%) State the null and the alternative hypotheses.
b. (3%) Determine the test statistic.
c. (3%) Determine the p-value; and at 95% confidence test to determine whether or not the
mean of the population is significantly different from 48.
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