Question

According to the 2010 US Census, the average number of residents per housing unit for the n=87 counties in Minnesota was 2.10, and the standard deviation was 0.38. Test whether the true mean number of residents per housing unit in Minnesota in 2010 is less than the national value of 2.34 at the level α = 0.05.

a. Show all five steps of this test.

b. What type of error could we be making in this context?

c. What is the minimum average number of residents per household needed in order to fail to reject H0? Assume the sample standard deviation is the same.

d. Suppose the true number of residents per household in Minnesota is normally distributed with a mean of 2.0 and standard deviation of 0.4. Suppose we reject null hypothesis if the sample mean number of residents is less than 2.27. What is the probability of making a type II error?

Answer 1

Solution Given data A = 2.34 OT 0.38 o 0.05 (a) Noco to state the null and altranative hyPothesis Nuil hythesis Ho 2.34 Attemative hyfothesis Ha : < 34 Hese o- 0s then Zcoitical = -G45 Noc to find the test-statistic,Z:- We kmo that The famula of test- statistic, z is given by X-A Z = 2.1o- 2.34 Z= 0-38 V8 0.24 0.0407 Z -5.897 Now to find the Pyalue e(ze-sa) P-Value P-value Decision O.0000 PValue< <O.05 O-0000

Thesefae a rejet the mull byPothesisCH) at O.OS level of sigmihance Con lusion Thene mean Conclude that tue is Sufficient evidence to in mumbea of Pen housiy less ta the mational Value in Mimmesota Unit 9tesidents of .34 Context in this ) what tfe of Tmoking the type-I eana since Could e be we Snejert We Could Cans) the null Commit hy fothesis(Ho) Nos to find ample mean Cut off Sample Cutoff mean Vn *Citial 0.38 a.34 -1-645 X- 3 1-45x x0.040) = 2.34 -0.0669 2.23 man (utt Sample (di Noc to find the tear given that stando deviation (-) = o.4 L(Tyfe eon) :- mean () =2.0 s 2 0l o. 0.0429

3 6.29 Noco P(Type I enna) Z> 1-P (Z<6.29) P(TyfeT en Thesrefae the Jieguined robability a0000 O.0000