Answer to Question #175855 in Statistics and Probability for Denisse Bisuña

Question

A. Find the sample size for each of the following:

a. With 90% confidence, E = 0.08 and p = 0.38.

b. With 95% confidence, E = 0.05 and p = 0.38.

c. With 95% confidence, E = 0.12 and p = 0.38.

d. With 99% confidence, E = 0.20 and p = 0.38.

e. With 99% confidence, E = 0.25 and p = 0.38.

1.What is the advantage of the sample size formula?

2.What is the effect of the level of confidence on the confidence interval?

3.A particular brand of coffee contains an average of 112 mg of caffeine per cup with a standard deviation of 20 mg. Joshua wants to investigate the same to estimate the true population mean caffeine content correct to within 5 mg adopting 95% confidence. How many cups of the same brand of coffee does he need for a sample?

Answer 1

$E=Z_{\alpha}\sqrt{\frac{p(1-p)}{n}}$

$n=\frac{Z^2_{\alpha}p(1-p)}{E^2}$

A. a)

$n=\frac{1.65\cdot0.38(1-0.38)}{0.08^2}=60.74=61$

b)

$n=\frac{1.96\cdot0.38(1-0.38)}{0.05^2}=184.71=185$

c)

$n=\frac{1.96\cdot0.38(1-0.38)}{0.12^2}=32.07=32$

d)

$n=\frac{2.58\cdot0.38(1-0.38)}{0.2^2}=15.2=15$

e)

$n=\frac{2.58\cdot0.38(1-0.38)}{0.25^2}=9.73=10$

1. The sample size formula allows to calculate minimal sample size for research. Larger sample sizes provide more accurate mean values, identify outliers that could skew the data.

2.If the level of confidence is smaller than the confidence interval is larger.

3.

$n=(\frac{\sigma Z_{\alpha/2}}{SE})^2=(\frac{20\cdot1.96}{5})^2=61.46=61$