Question

8.13 The following results show the relationship between smoking in mothers and the survival of newborns past their first year. Use the bino- mial test to determine if the proportion of newborns that died in the first year is lower in nonsmokers than smokers. Non-smoker Smoker Died in 1st year Alive at 1 year 28 5606 583

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Answer 1

and variance Pii P242 Therefore, our critical region(s) can be established by using the standard normal variable (P - P2) - (p1 - P2) When Ho is true, we can substitute p1p2-p and qi - q (where p and q are the common values) in the preceding formula for Z to give the form P1 - P2 To compute a value of Z, however, we must estimate the parameters p and q that appear in the radical. estimate of the proportion p is Upon pooling the data from both samples, the pooled ni +n2 where 2,1 and 2,2 are the numbers of successes in each of the two samples. Substi- tuting p for p and q 1-p for q, the z-value for testing P1-P2 is determined from the formula P1-P2 (1/m + 1/n2) The critical regions for the appropriate alternative hypotheses are set up as before using critical points of the standard normal curve. Hence, for the alternative P1メP2 at the a-level of significance, the critical region is z <-za/2 or z > /2 For a test where the alternative is P1 < P2, the critical region is z <-,, and when the alternative is P1 > P2, the critical region is z >

Ho : p1 = p2

Ha: p1 < p2

Sample   X     N Sample p
1       28 5606 0.004995
2        7   583 0.012007

Difference = p (1) - p (2)
Estimate for difference: -0.00701221
95% upper bound for difference: 0.000567364
Test for difference = 0 (vs < 0): Z = -2.15 P-Value = 0.016

p-value = 0.016 < 0.05

hence we reject the null hypothesis

we conclude that there is sufficient evidence that proportion is lower in non-smoker than smoker

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