Question

Examnle 7 IFRQ1203M] Independent random samples of 500 households were taken from a large metropolitan area in the United States for the years 1950 and 2000 Histograms of household size (number of people in a household) for the years are shown below. SocS 120 t00 100 80 80 6o 5 20 8 10 12 14 1 0 268 12 416 Houschold Stze in 1950 Houschold Size in 2000 A researcher wants to use these data to construct a confidence interval to estimate the m change in mean household size in the metropolitan area from the year 1950 to the year 2000. Carutate a) State the type of confidence interval for estimating the change in mean household size (m, σ*) from the year 1950 to the year, 2000 and explain whether the conditions for inference are met. b.) Find the sample means and standard deviations for both years. Year Mean Standard Deviation 1950 2000

Answer 1

b.

a. Here we can use Large sample Estimation for difference in two sample mean.

Conditions - 1. Size of both sample is > 30

                  2. Both the samples are almost normally distributed ( we can see from histogram), if means   distribution of (mean1 - mean2) is normally distibuted

c.

95% confidence interval is =

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=(27.55 - 23.76) +/- 1.96*√(49.12/50 + 50.85/50)

= 3.79 +/- 2.77

Change in means is 1.02 to 6.56

d .

Null Hypothesis Ho : There is significant change in means u1 - u2 = 0

Alternative Hypothesis Ha : There is no significant change in means u1 - u2 != 0

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Z = [ (27.55 - 23.76) - 0 ] / √(49.12/50 + 50.85/50)

Z = 2.67

p-value = Z(>2.67) + Z(<-2.67)

p-value = 0.0076

p-value > alpha ( 0.005)    

Hence Null hypothesis cant be rejected

:- There is significant change in means

Note: - Sample variance is taken as estimated population variance in formula