Respuesta :
Answer:
(1) The sample size required is 97.
(2) The 95% confidence interval for the proportion of children not overdue for vaccination is (0.15, 0.30).
Step-by-step explanation:
(1)
Let the proportion be, p = 0.50.
The margin of error formula is:
[tex]MOE=z\times \sqrt{\frac{p(1-p)}{n}}[/tex]
The critical value of z for 95% confidence level is,
z = 1.96 (Use the standard normal table for z values)
Compute sample size n as follows:
[tex]n=(\frac{z}{MOE})^{2}\times p(1-p)\\=(\frac{1.96}{0.10} )^{2}\times 0.50\times (1-0.50)\\=96.04\\\approx97[/tex]
Thus, the sample size required is 97.
(2)
The confidence interval for population proportion is:
[tex]\hat p\pm z\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex]
The sample size is, n = 120.
The number of children who were not overdue for vaccination, X = 27.
The sample proportion is:
[tex]\hat p=\frac{X}{n}= \frac{27}{120}= 0.225[/tex]
The critical value of z for 95% confidence level is z = 1.96.
The 95% confidence interval is:
[tex]\hat p\pm z\sqrt{\frac{\hat p(1-\hat p)}{n} }=0.225\pm1.96\times \sqrt{\frac{0.225(1-0.225}{120} }\\=0.225\pm 0.075\\=(0.15, 0.30)[/tex]
Thus, the 95% confidence interval for the proportion of children not overdue for vaccination is (0.15, 0.30).
