Question

A 1997 survey of 100 randomly selected teenage girls between the ages of 13 and 15 showed that 32 of them had smoked in the past 12 months. A similar survey of 100 in 2013 showed that 62 of the girls had smoked in the past 12 months. (Use subscript 1 for 1997 and 2 for 2013). We want to test, at 5% level of significance, the hypothesis that proportion of smokers in 1997 is smaller than that of the 2013.

a) Write down the null and alternative hypotheses.b) Write down the rejection region. c) Calculate the statistic which determines acceptance or rejection.

b)Write down the rejection region.

c) Calculate the statistic which determines acceptance or rejection.

d) Write your findings in words.

Answer 1

let p1 be proportion of smokers in 1997 and p2 be the proportion of smokers in 2013

(A) Null hypothesis: p1 = p2

Alternate hypothesis: (it is the claim, we have to test it)

(B) Significance level is given as 5%, so the z critical value for two tailed hypothesis at 0.05 alpha level are -1.96 and 1.96

So, rejection region is

Any value between -1.96 and 1.96 will result in Null hypothesis acceptance

(C) we have p1 = 32/100=0.32, n1 = 100, p2 = (62/100)=0.62, n2 = 100

Using the z statistic formula

z statistic =

setting the given values, we get

z statistic =

this gives

z statistic =

It is clear that -4.46 is less than -1.96, so we can reject the null hypothesis.

(D) Since the null hypothesis is rejected, we can conclude that there is enough evidence to support the claim that proportion of smoker in 1997 is smaller than that of 2013.