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Unformatted text preview: average sleep 7.46 to 7.99 hours per night.
(b) Does your conﬁdence interval agree with the result of the hypothesis test from the previous
question?
Yes, we rejected the null hypothesis and the null hypothesized value is not in the interval. It
should be noted however that the upper bound of the interval is very close to 8.
14. It is believed that large doses of acetaminophen (the active ingredient in over the counter pain relievers
like Tylenol) may cause damage to the liver. A researcher wants to conduct a study to estimate the
proportion of acetaminophen users who have liver damage. For participating in this study she will
pay each subject $20.
9 (a) If she wants to limit the margin of error of her 98% conﬁdence interval to 2%. What is the
minimum amount of money she needs to set aside to pay her subjects?
We are asked to solve for the sample size required to achieve a 2% margin of error. Since we are
not given information from a previous study to use as an estimate for p, we use 0.5 as it will
ˆ
yield the most conservative estimate.
p(1 − p)
ˆ
ˆ
0.5 ∗ 0.5
∗
ME = z
→ 0.02 = 2.326
n
n
0.5 ∗ 0.5
0.022 = 2.3262
n
2.3262 ∗ 0.5 ∗ 0.5
n=
0.022
n = 3381.423 ≈ 3382
She needs a minimum of 3382 subjects and therefore needs to set aside a minimum of 3, 382×$20 =
$67, 640.
(b) The amount you calculated in part (a) is way over her budget so she decides to use fewer subjects.
How will this aﬀect the width of her conﬁdence interval, i.e. how will the precision of her
conﬁdence interval change? (Hint: You do not need to calculate the interval to answer this
question.)
Decreasing the sample size would increase the margin of error hence make the interval wider, i.e.
the interval would lose precision.
15. According to a report on sleep deprivation by the Centers for Disease Control and Prevention the
proportion of California residents who reported insuﬃcient rest or sleep during the on each of the
preceding 30 days is 8.0% while this proportion is 8.8% for Oregon residents. A random sample of
11,545 California and 4,691 Oregon residents were surveyed. We are interested in ﬁnding out if there
evidence to suggest that the rate of sleep deprivation is diﬀerent for the two states.
(a) What kind of study is this?
This is an observational study.
(b) What statistical method is appropriate for testing if the rate of sleep deprivation is diﬀerent for
the two states?
Two proportion ztest.
(c) What are the hypotheses?
Let p1 denote the proportion of sleep deprived California residents and p2 denote the proportion
of sleep deprived Oregon residents.
H 0 : p1 = p2
HA : p1 = p2
(d) Are the assumptions and conditions for inference satisﬁed?
1. Independence Assumption:
• Random Sampling Condition: We are told that both samples are random.
10 • 10% Condition: 11,545 < 10% of all Californians and 4,691 < 10% of all Oregonians.
Since we have a random sample and the 10% condition is satisﬁed, we can assume that how
much one Califor...
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This document was uploaded on 12/04/2013.
 Winter '13

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