Final_Review_1_Solutions

# 08 0088 z 168 ppooled qpooled p q 00820918 0

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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 &gt; 10￿ ˆ n2 p2 ≥ 10 → 4, 691 ∗ 0.088 = 412.8 &gt; 10￿ ˆ and and n1 q1 ≥ 10 → 11, 545 ∗ 0.92 = 10621.4 &gt; 10￿ ˆ n2 q2 ≥ 10 → 4, 691 ∗ 0.912 = 4278.2 &gt; 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 p-value. p-value = 2 ∗ P (z &lt; −1.68) = 2 ∗ 0.0465 = 0.093 (g) What do you conclude? Interpret your conclusion in context. Since p-value &gt; 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.

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