Question

1. Suppose the annual incidence of asthma in the general population among children 0-4 years of age is 1.4% for boys.

   1. If 10 cases are observed over one year among 500 boys 0-4 years of age with smoking mothers, then test if there is a significant difference in asthma incidence between this group and the general population. [6]

      1. STEP 1: STATE THE APPROPRIATE NULL AND ALTERNATIVE HYPOTHESES

      2. STEP 2: DEFINE THE CRITICAL REGION

      3. STEP 3: COMPUTE THE APPROPRIATE TEST STATISTIC

      4. STEP 4: STATE YOUR STATISTICAL DECISION

      5. STEP 5: STATE YOUR PRACTICAL CONCLUSION

      6. STEP 6: REPORT THE P-VALUE

2. 2. Construct and practically interpret a 95% CI for the true incidence rate of asthma in this group. [8]

Answer 1

Solution:

We are given that: the annual incidence of asthma in the general population among children 0-4 years of age is 1.4% for boys. That is: p = 0.014

Sample size = n = 500

x = 10 cases are observed over one year among 500 boys 0-4 years of age with smoking mothers.

We have to test if there is a significant difference in asthma incidence between this group and the general population.

Step 1: state the appropriate null and alternative hypotheses:

       Vs

Step 2: define the critical region:

Level of significance is not specified , so we assume Level of significance =

Since this is two tailed test, critical region would be in two tails.

Thus critical region is:

Reject H0, if z test statistic value < - z critical value or z test statistic value > z critical value.

then

and

Look in z table for Area = 0.0250 area and find z value.

z critical value for left tail = -1.96

then z critical value for right tail = +1.96

Thus critical region is:
Reject H0, if z test statistic value < - 1.96 or z test statistic value > 1.96 .

Step 3: compute the appropriate test statistic

where

Thus

Step 4: state your statistical decision

Since z test statistic value is neither < -1.96 , nor > 1.96, we fail to reject null hypothesis H0.

Step 5: state your practical conclusion

There is not significant difference in asthma incidence between this group and the general population.

Step 6: report the p-value

p-value = 2 x P(Z > z test statistic value)

p-value = 2 x P(Z > 1.14)

p-value = 2 x [ 1 - P(Z < 1.14) ]

Look in z table for z = 1.1 and 0.04 and find area

From z table , we get P( Z < 1.14) = 0.8729

Thus

p-value = 2 x [ 1 - P(Z < 1.14) ]

p-value = 2 x [ 1 - 0.8729 ]

p-value = 2 x 0.1271

p-value = 0.2542

Construct and practically interpret a 95% CI for the true incidence rate of asthma in this group

where

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Thus

Interpretation:

Thus a 95% confidence interval for the true incidence rate of asthma in this group is between 0.77% to 3.23%.

That is: we are 95% confident that the true incidence rate of asthma in this group is between 0.77% to 3.23%.