Previously, the U.S. Census reported that the average family consists of 3.14 pe

Question

Previously, the U.S. Census reported that the average family consists of 3.14 persons with a standard deviation of 1.37 persons. In order to find out if the average has changed from the previous results, a sample of 5, 000 families was collected with an average of 3.10 persons. The U.S. Census wants to be ninety-eight percent confident when they report if the average is now lower or if there has not been any significant change. a. Write the Null and Alternate Hypothesis b. What is alpha (alpha) in this scenario? c. Would you conduct a 1 -tailed test (upper or lower) or a 2-tailed test? d. What critical value will you use to identify the rejection regions? e. Write the test statistic formula you should use f. Calculate the test statistic g. State your conclusion - Reject or fail to reject the null hypothesis h. What should the U.S. Census report per the results of this test? i. What is the p-value? j. To what should you compare the p-value to make a conclusion on this hypothesis test? k. What's your conclusion based on the p-value and how did you decide?

Explanation / Answer

a) Null hypothesis H0: mu = 3.14

Alternate H1: mu <=3.14

b) Alpha = 0.02, as we want to be 98% confident

c) We will conduct an one tailed test as we want to check only one side of the inequality i.e. if mu is lesser than 3.14. We will conduct a two tailed test when we want to check mu1 != mu2

d) Z crit at 0.02 = -2.05375

Rejection region is left of -2.05375

e) Test statistic, z = (mu1 – mu)/ (sd/sqrt(n) )

f) Z = (3.1-3.14) /[ 1.37/sqrt(5000) ]

Z = -2.06455

Z = -2.06455 is in the left of zcrit (ie. -2.05375)

Pvalue = norm.s.dist (-2.06455, true) = 0.019

g) Reject the null hypothesis as z value < z crit and is in the rejection region

h) Should report that the average of the family is less than 3.14

i) P-value = 0.019

j) We should compare it to 0.02, i.e alpha

k) Reject the null hypothesis as p-value is less than 0.02

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