Answer:
The 95% confidence interval for the population proportion of females is (0.693, 0.939)
Step-by-step explanation:
We have to calculate a 95% confidence interval for the proportion.
If the sample collected, of size n=38, has 7 males and 31 females, the sample proportion is p=0.8158.
[tex]p=X/n=31/38=0.8158[/tex]
The estimated standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.8158*0.1842}{38}}\\\\\\ \sigma_p=\sqrt{0.004}=0.063[/tex]
The critical z-value for a 95% confidence interval is z=1.96.
The margin of error (MOE) can be calculated as:
[tex]MOE=z\cdot \sigma_p=1.96 \cdot 0.063=0.123[/tex]
Then, the lower and upper bounds of the confidence interval are:
[tex]LL=p-z \cdot \sigma_p = 0.8158-0.123=0.693\\\\UL=p+z \cdot \sigma_p = 0.8158+0.123=0.939[/tex]
The 95% confidence interval for the population proportion of females is (0.693, 0.939), estimated from this sample.
