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Unformatted text preview: nian sleeps is independent of how much another Californian sleeps and how
much one Oregonian sleeps is independent of how much another Oregonian sleeps.
2. Nearly Normal Condition: First we must check if the success failure condition is met.
n1 p1 ≥ 10 → 11, 545 ∗ 0.08 = 923.6 > 10
ˆ
n2 p2 ≥ 10 → 4, 691 ∗ 0.088 = 412.8 > 10
ˆ and
and n1 q1 ≥ 10 → 11, 545 ∗ 0.92 = 10621.4 > 10
ˆ
n2 q2 ≥ 10 → 4, 691 ∗ 0.912 = 4278.2 > 10
ˆ 3. Independent Groups: The Californians and the Oregonians are independent of each other.
(e) Calculate the test statistic.
success1 = n1 ∗ p1 = 11, 545 ∗ 0.08 = 923.6 ≈ 924
ˆ success2 = n2 ∗ p2 = 4, 691 ∗ 0.088 = 412.8 ≈ 413
ˆ
success1 + success2
924 + 413
1, 337
ppooled =
ˆ
=
≈ 0.082
=
n1 + n2
11, 545 + 4, 691
16, 236
(ˆ1 − p2 )
p
ˆ
(0.08 − 0.088)
z=
=
= −1.68
ppooled qpooled
ˆ
ˆ
p
ˆ
q
ˆ
0.082∗0.918
∗0.
+ 0.082691918
+ pooled pooled
11,545
4,
n
n
1 2 (f) Find the pvalue.
pvalue = 2 ∗ P (z < −1.68) = 2 ∗ 0.0465 = 0.093
(g) What do you conclude? Interpret your conclusion in context.
Since pvalue > α (use α = 0.05 since not given), we fail to reject the null hypothesis and conclude
that there is no evidence to suggest that the rate of sleep deprivation is diﬀerent for the two states.
(h) Does this imply that the rate of sleep deprivation is equal in the two states? Explain.
No, this does not imply it; though there is support for that statement. We cannot infer causation
based on an observational study.
(i) What type of error might we have committed?
Since we failed to reject the null hypothesis, we may have committed a Type II error.
(j) Would you expect a conﬁdence interval for the diﬀerence between the two proportions to include
0? Explain your reasoning.
Yes, since we failed to reject the null hypothesis, it is possible that the two population proportions
are equal to each other and hence the diﬀerence between them could be 0.
(k) Construct a 95% conﬁdence interval for the diﬀerence between the population proportions. In 11 terpret the conﬁdence interval in context.
p1 q 1 p2 q 2
ˆˆ
ˆˆ
∗
(ˆ1 − p2 ) ± z
p
ˆ
+
n1
n
2
0.08 ∗ 0.92 0.088 ∗ 0.912
= (0.08 − 0.088) ± 1.96
+
11, 545
4, 691
= −0.008 ± 0.009
= (−0.017, 0.001) We are 95% conﬁdent that the diﬀerence between the proportions of Californians and Oregonians
who are sleep deprived is between 1.7% and 0.1%. In other words, we are 95% conﬁdent that the
proportion of Californians who are sleep deprives is 1.7% less to 0.1% more than the proportion
of Oregonians who are sleep deprived.
(l) Does the result of your hypothesis test agree with the interpretation of the conﬁdence interval?
If not, why might that be the case?
This conﬁdence interval includes 0 and we failed to reject the null hypothesis which set the two
population proportions equal to each other, in other words sets the diﬀerence between the two
population proportions...
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This document was uploaded on 12/04/2013.
 Winter '13

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