Exploring the role of sample size: a. Construct 1,000 confidence intervals with displaystyle{n}={10},{p}={0.3}. What proportion of the 95% confidence

snowlovelydayM 2020-12-22 Answered
Exploring the role of sample size:
a. Construct 1,000 confidence intervals with n=10,p=0.3. What proportion of the 95% confidence intervals included the population proportion, 0.3?
b. Construct 1,000 confidence intervals with n=40,p=0.3. What proportion of the 95% confidence intervals included the population proportion, 0.3?
c. Construct 1,000 confidence intervals with n=100,p=0.3. What proportion 95% confidence intervals include the proportion of the population, 0.3?
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Expert Answer

ensojadasH
Answered 2020-12-23 Author has 100 answers
Step 1
The provided information is:
It is given that:
Sample mean (p^)=0.3
Level of significance (L)=0.05
The critical calue of zc at 0.05 level of significance woult be:
zc=zL2
=1.96
Step 2
a)
The confidence interval the population proportion (p), at 0.05 level of significance, can be calculated as:
Sample size (n)=10
p(p^±zcp^(1p^)n)
(0.3±1.96×0.3(10.3)10)
in (0.0159, 0.5840)
Thus, the confidence interval is (0.0159, 0.5840). Therefore, the population proportion 0.3, lies in the confidence interval.
The confidence interval would be same, 1000 times.
Step 3
b)
Again, the confidence interval for 40 different samples, the population proportion (p), at 0.05 level of significance, can be calculated as:
Sample size (n)=40
p(p^±zcp^(1p^)n)
(0.3±1.96×0.3(10.3)40)
in (0.1579, 0.4420)
Thus, the confidence interval is (0.1579, 0.4420). Therefore, the population proportion 0.3, lies in the confidence interval.
The confidence interval would be same, 1000 times.
Step 4
c)
The confidence interval the population proportion (p), at 0.05 level of significance, can be calculated as:
Sample size (n)=100
p(p^±zcp^(1p^)n)
(0.3±1.96×0.3(10.3)100)
(0.2102,0.3898)
Thus, the confidence interval is (0.2102, 0.3898). Therefore, the population proportion 0.3, lies in the confidence interval.
The confidence interval would be same, 1000 times.
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