Question

The Arizona Department of Transportation (ADoT) is considering changing statewide speed limits from 65 mph to 75 mph. They wish to perform analysis on ?, the proportion of Arizona residents that favor this change. In order to prove that a majority of residents favor the change, ADoT wishes to test the following hypotheses using a level of significance of 0.05.

H0 : p = 0.5

H1 : p > 0.4

ADoT wishes to design the test to have a power of 90% if the true population proportion is equal to 0.55.

a.) How many residents should be surveyed? You may use this sample size for the remainder of this problem. (Note: You should get a value between 800 and 900 on this part. If you did not, then choose a value between 800 and 900 and use it to work the remainder of the problem.)

b.) Suppose the sample size found in part a is implemented and 461 of the surveyed residents favor the change. Construct a point estimate on the proportion of interest.

c.) Identify the test statistic and critical region for the test. State the conclusion for this test.

d.) What is the P-value for this test? State the conclusion for this test.

e.) Construct an appropriate confidence interval to verify your conclusion from part c.

f.) We wish to use a 95% lower confidence interval to estimate p. How many residents should be surveyed so that the error in our estimate is less than 4%.

Answer 1

Solution bove pooportion pa hypothesized 0.55 pooportion po = = 0.5 d= = 0.050 power 1-ß -0.9 B = 0.1 Za = 1.645 [excel function : - := norm sin va] ] z ß = 1.282 [ excel fumeion ! = norm sinu (B) ] M = po* (1-p0 ) [26] + 2(B)]') 6-polz = 0.5 X05 (1.282 +1-645)°/055-0.512 056-38 sample size =857 (b) point estimale sample proportion = 8:a/n 2 .461 and n = 857 8 = 46/57 = 0.5379 a Standard error SE = <p co-P) n

0.017) SE- V PCI-P)/n (Rp) ISE z fest statistic - = 6.5379 -0.5 9-05/0 0.0171 z static - 2.220 critical zvalue = 1.645 [ Excel fimchiom = NOR MSINVG)] reject Ho if z > 1.645 decision : rject ho ld) p-value = 0.6132 Decision! rejed null hepothesis (0 ) number af items of inbrest x=461 n=057 d= 0.05 Zales oy 6.5379 zvalue - 2212 = 1.960 [ Exul formele = Norm SimV 64/2)] Stemoland Error - 00110 morgin error E= '2*BE 0.0334 = confidence in tervel is - interval Lower limit - Es 0-53 79- 0.0334 0.5045 inter val upper Limit = TE =0.5379 +0.00334 = 0.5713