Question

1. The Acme Company just bought a new machine that makes markers. In a random sample of 250 markers from the new machine, they find that 234 markers work. Find the error bound for a 95% confidence interval for the true proportion of working markers from the new machine.

2. Find the required sample size for calculating a 95% confidence interval for a population mean  with a standard deviation of 26.2 and an error bound of 9.

3. Find the required sample size for calculating a 96% confidence interval for p with an error bound of 0.15.

Answer 1

Solution :

Given that,

1) Point estimate = sample proportion = = x / n = 234 / 250 = 0.936

1 - = 1 - 0.936 = 0.064

Z/2 = Z_0.025 = 1.96

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

E = 1.96 (((0.936 * 0.064) / 250)

E = 0.030

2) Z_/2 = Z_0.025 = 1.96  

sample size = n = [Z_/2* / E] ²

n = [1.96 * 26.2 / 9 ]²

n = 32.55

Sample size = n = 33

3) Z_/2 = Z_0.02 = 2.054

=  1 - =   0.5

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (2.054 / 0.15)² * 0.5 * 0.5

= 46.87

sample size = n = 47