Statistics Questions. Help! ASAP

Name: EC

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303 Foundations for Econometrics
Matriculation #: PS4
Tutorial Section (W#):

Week

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1: Problem Set 5

DUE: Monday 4 November, 12pm

• Please hand in to the economics office (AS2 – L6)

• Late assignments will not be accepted.

• Show your work to get full credit.

• Carry through fractions or exact decimals when possible, or round to
four decimal places. Please round final answers to four decimal places.

1. For a class project, a student wants to check whether final exam stress raises blood pressure
of the first year women in her class. When they are not under any stress, healthy 18-year-
old women have systolic blood pressures that average 120mm Hg with a standard deviation
of 12mm Hg. If she finds that the average blood pressure for the 50 students in her class
on the day of the final exam is 125.2, what should she conclude? Set up and test and
appropriate hypothesis.

2. If H0 : µ = µ0 is rejected in favor of H1 : µ > µ0, will it necessarily be rejected in favor of
H1 : µ 6= µ0? Assume that α remains the same.

3. Cell phones emit radio frequency energy that is absorbed by the body when the phone is
next to the ear and maybe harmful. The table in the next column gives the absorption rate
for a random sample of twenty cell phones. (The U.S. Federal Communication Commission
sets a maximum of 1.6 watts per kilogram for the absorption rate of such energy.) Construct
a 90% confidence interval for the true average cell phone absorption rate.

0.87 0.72 1.30 1.05
0.79 0.61 1.45 1.01
1.15 0.20 1.31 0.67
1.09 1.35 0.66 1.27
0.49 1.28 1.40 1.55

Source: reviews.cnet.com/cell-phone-radiation-levels/

4. NCT 9.11 A manufacturer of detergent claims that the contents of boxes sold weigh on
average at least 16 ounces. The distribution of weight is known to be normal, with a
standard deviation of 0.4 ounces. A random sample of 16 boxes yielded a sample mean
weight of 15.84 ounces. Test at the 10% significance level the null hypothesis that the
population mean weight is at least 16 ounces.

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EC2303 – Foundations for Econometrics Version: October 28, 2013

5. Creativity, as any number of studies have shown, is very much a province of the young.
Whether the focus is music, literature, science, or mathematics, an individual’s best work
seldom occurs late in life. Einstein, for example, made his most profound discoveries at the
age of 26; Newton, at the age of 23. The following are 12 scientific breakthroughs dating
from the middle of the 16th century to the early years of the 20th century. All represented
high-water marks in the careers of the scientists involved.

Discovery Discoverer Year Age, y
Earth goes around sun Copernicus 1543 40
Telescope, basic laws of astronomy Galileo 1600 34
Principles of motion, gravitation, calculus Newton 1665 23
Nature of electricity Franklin 1746 40
Burning is uniting with oxygen Lavoisier 1774 31
Earth evolved by gradual processes Lyell 1830 33
Evidence for natural selection controlling evolu-
tion

Darwin 1858 49

Field equations for light Maxwell 1864 33
Radioactivity Curie 1896 34
Quantum theory Planck 1901 43
Special theory of relativity, E = mc2 Einstein 1905 26
Mathematical conditions for quantum theory Schrödinger 1926 39

(a) What can be inferred from these data bout the true average age at which scientists
do their best work? Answer the question by constructing a 95% confidence interval.

(b) Before constructing a confidence interval for a set off observations extending over
a long period of time, we should be convinced that the yi’s exhibit no biases or
trends. If, for example, the age at which scientists made major discoveries decreased
from century to century, then the parameter µ would no longer be a constant, and
the confidence interval would be meaningless. Plot “date” versus “age” for theses 12
discoveries, putting “date” on the horizontal axis. Does the variability in the yi’s
appear to be random with respect to time?

6. NCT 9.45 Each day, a fast-food chain tests that the average weight of its “two-pounders”
is at least 32 ounces. The alternative hypothesis is that the average weight is less than
32 ounces, indicating that new processing procedures are needed. The weights of two-
pounders can be assumed to be normally distributed, with a standard deviation of 3
ounces. The decision rule adopted is to reject the null hypothesis if the sample mean
weight is less than 30.8 ounces.

(a) If random samples of n = 36 two-pounders are selected, what is the probability of a
Type I error, using this decision rule?

(b) If random samples of n = 9 two-pounders are selected, what is the probability of a
Type I error, using this decision rule? Why is your answer different from part (a)?

(c) Suppose that the true mean weight is 31 ounces. If random samples of 36 two-
pounders are selected, what is the probability of a Type II error, using this decision
rule?

(d) Suppose that the true mean weight is 31 ounces. If random samples of 9 two-pounders
are selected, what is the probability of a Type II error, using this decision rule?

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