Download Midterm Exam 2: Practice

Midterm Exam 2: Practice
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Please write LEGIBLY. Show all work but do not include irrelevant material. Good luck!
1. Assume that the length of time, X between charges of cellphone is normally distributed with
a mean of 10 hours and a standard deviation of 1.5 hours. Find the probability that the cell
phone will last between 8 and 12 hours between charges.
2. Suppose that a paint manufacturer has a daily production, X that is normally distributed with
a mean of 100,000 gallons and a standard deviation of 10,000 gallons. Management wants to
create an incentive bonus for the production crew when the daily production exceeds the 90th
percentile of the distribution in hopes that the crew will in return become more productive.
At what level of production should management pay the incentive bonus?
3. An experiment involves selecting a random sample of 256 middle managers at random for
study. One item of interest is their mean annual income. The sample mean is computed to
be $35,420 and the sample standard deviation is $2,050. What is the standard error of the
mean?
4. Assume that the population of human body temperature follows a normal distribution with a
mean of 98.60 F . Also, assume that the population standard deviation is 0.620 F . If a sample
of size 106 is randomly selected, can we claim that the probability of getting a mean of 98.20 F
or lower is almost zero?
5. The Central Limit Theorem states that if the sample size, N , is sufficiently large, the sampling
distribution of the means will be approximately normal regardless of the population being
normally distributed, skewed, or uniform: true or false?
6. Which statement is incorrect?
(a) The standard error of the mean decreases as the sample size increases.
(b) As the sample size increases, the sample mean will be a more precise estimate of the
population mean.
(c) The sample mean, X̄ = n−1 ni=1 Xi is not a random variable, but a statistic.
(d) The dispersion in the sampling distribution of the sample mean gets smaller as N increases.
7. Suppose that a researcher has divided a population into subgroups. Each subgroup can be
regarded as a smaller version of population, and the members in each subgroup are heterogeneous. Then, if a random sample is collected from a few selected subgroups, what type of
sampling is used?
(a) simple random sampling
(b) systematic random sampling
(c) stratified random sampling
(d) cluster sampling
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8. All possible samples of size N are selected from a population and the mean of each sample is
determined. What is the mean of the sample means?
(a) Exactly the same as the population mean.
(b) Larger than the population mean.
(c) Smaller than the population mean.
(d) Cannot be determined in advance. We need to know the estimated standard deviation
9. The SpringFargo Bank has 650 checking account customers at the North-Denton branch.
A recent sample of 50 of these customers showed 26 to have a Visa card with the bank.
Construct the 95% confidence interval for the proportion of checking account customers who
have a Visa card with the bank.
10. Suppose that an experimenter has prepared a new drug that she claims will induce sleep for
90% or more people suffering from insomnia. The drug is submitted to FDA for testing. If her
claim is true, the drug will be approved by FDA. Otherwise, the drug will not be approved
by FDA and has to be discarded.
In an attempt to verify her claim, FDA conducts a following clinical test: her drug is administered to twenty patients (or insomniacs), and we observe the number of insomniacs who fall
asleep due to the drug. FDA sets up the following set of hypotheses:
H0 : π ≥ 0.9
H1 : π < 0.9
And the decision rule is to approve the drug if FDA finds the drug is as effective as claimed:
that is, more than or equal to 18 insomniacs fall asleep. What is type I error in this context?
In other words, explain the type I error that FDA may make.
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11. Let X̄ ∼ N ormal(μ, σn ) and zα/2 be the (1 − α/2)th percentile in the standard normal
distribution. Then, show that from the distribution of X̄, we can derive the expression for
the confidence interval for μ: that is, to show the right-hand side of the equation is derived
from the left-hand side.
σ
σ
X̄ − μ
√ ≤ zα/2 = P X̄ − zα/2 √ ≤ μ ≤ X̄ + zα/2 √
.
P −zα/2 ≤
σ/ n
n
n
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12. Dole pineapple, Inc. is concerned that the weight of the sliced pineapple can is not equal to 16
ounces. The quality-control department took a random sample of 20 cans and found that the
arithmetic mean weight was 16.05 ounces, with a sample standard deviation of 0.03 ounces.
(a) The quality-control department wishes to test whether or not the weight of the can is
equal to 16 ounces. State the null hypothesis and the alternative hypothesis.
(b) Formulate and calculate the test statistic.
(c) Find an appropriate critical value(s) at α = .05.
(d) Based on the test statistic and the critical value, what would be the conclusion of the
hypothesis testing? Explain (you have to explain what it means to reject the null or not
to reject the null)!
13. A manufacturer of EVERRUNNING alkaline batteries wants to be reasonably certain that fewer
than 5% of its batteries are defective. Suppose that 300 batteries are randomly selected from a
very large shipment. Each is tested and 10 defective batteries are found. The manufacturer would
like to know if this provides sufficient evidence to conclude that the fraction defective in the entire
shipment is less than 0.05.
(a) State the null hypothesis and the alternative hypothesis.
(b) Formulate and calculate the test statistic.
(c) Find an appropriate critical value(s) at α = .05.
(d) Based on the test statistic and the critical value, what will the manufacturer conclude?
Explain (you have to explain what it means to reject the null or not to reject the null)!
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14. The Director of Transportation at UNT wants to compare the distance traveled to work by employees at their office in Denton with the distance for those in Dallas. A sample of 35 Denton
employees showed they travel a mean of 370 miles per month, with a standard deviation of 30
miles per month. A sample of 40 Dallas employees showed they travel a mean of 380 miles per
month, with a standard deviation of 26 miles per month. The Director wants to know if the two
traveling distances are statistically different.
(a) Set up the null and alternative hypotheses.
(b) Compute the value of the test statistic where we assume that the two population variances are the same.
(c) Find an appropriate critical value(s) at the 10% significance level.
(d) What is your conclusion regarding H0 ? Explain!
15. In a recent study, researchers have examined the sense of direction of 30 male and 30 female
students. After being taken to an unfamiliar wooded park, the students were given some spatial
orientation tests, including pointing to south, which tested their absolute frame of reference.
The students pointed by moving a pointer attached to a 360o protractor. From the data on the
absolute pointing errors, in degrees, of the participants, we have the sample mean of 37.6 and
the sample standard deviation of 38.5 for the male students; the sample mean of 55.8 and the
sample standard deviation of 48.3 for the female students. At the 1% significance level, do the
data provide sufficient evidence to conclude that males have a better sense of direction and, in
particular, a better frame of reference than females?
(a) Set up the null and alternative hypotheses.
(b) Compute the value of the test statistic where we do not assume the equality of the two
population variances.
(c) Find an appropriate critical value(s).
(d) What is your conclusion regarding H0 ? Explain!
16. The Roper Organization conducted identical surveys in 1995 and 2005. One question asked women
was “Are most men basically kind, gentle and thoughtful?” The 1995 survey revealed that, of the
3,000 women surveyed, 2,010 said that they were. In 2005, 1,530 of the 3,000 women surveyed
thought that men were kind, gentle, and thoughtful. At the 1% significance level, can we conclude
that women think men are less kind, gentle, and thoughtful in 2005 compared with 1995?
(a) Set up the null and alternative hypotheses.
(b) Compute the value of the test statistic.
(c) Find an appropriate critical value(s).
(d) What is your conclusion regarding H0 ? Explain!
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