TAKE IT HOME When we think about the differences in sample proportions, we often
ID: 3174772 • Letter: T
Question
TAKE IT HOME When we think about the differences in sample proportions, we often assume that there is no difference in population proportions. We can use what we know about the normality of the distribution of p_1^cap - p_2^cap, to calculate the probability of observing particular values of the sample proportions. In this way we can draw conclusions about the reasonableness of the assumption that the true proportions are equal, or that p_1 - p_2 = 0. In a recent Gallup poll, researchers asked husbands and wives who pays the bills in their household. Let's assume that 50% of husbands pay the bills in their homes (p_1 = 0.50) and 50% of wives pay the bills in their homes (p_2 = 0.50). The article on Gallup's website reported that 35% of husbands say they pay the bills in their home and 45% of wives say they pay the bills in their home. Suppose this poll was based on sample sizes of n_1 = 15 for the husbands and n_1 = 25 for the wives. Would a normal distribution be an appropriate model for the differences in sample proportions? Why or why not? Suppose this poll was based on sample sizes of n_1 = 120 for the husbands and n_2 = 100 for the wives. Would a normal distribution be an appropriate model for the differences in sample proportions? Why or why not? Suppose 35% of the 120 husbands sampled pay the bills in their home, and 45 % of the 100 wives sampled say they pay the bills in their home. What is the value of p_1^cap = p_2^cap?Explanation / Answer
Result:
1).
The assumption is that both np10 and n(1p)10.
n1p=15*0.35 =5.25 < 10 and n1*(1-p)= 15*0.65=9.75 < 10
n2p=25*0.45 =11.25> 10 and n2*(1-p)= 25*0.55=13.75 > 10
Normal approximation is not appropriate because first group violates to apply central limit theorem.
2).
n1p=120*0.35 =42> 10 and n1*(1-p)= 120*0.65=78> 10
n2p=100*0.45 =45> 10 and n2*(1-p)= 100*0.55=55 > 10
Normal approximation is appropriate because sample size is enough to apply central limit theorem.
3).
Value of p1-p2=0.35-0.45
= - 0.10
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