Question

Solving Problems Involving Sample Size Determination

Apply the knowledge you obtained from the discussion by solving the given problems below.

1. A 95% confidence interval has to be created to estimate the mean volume of contents in the

bottles of Honey-Z. Assume that the population standard deviation is 30 ml. Calculate the sample

size needed if the margin of error is no more than:

a. 6 ml

b. 5 ml

c. 4ml

d. 3 ml

2. How large sample size is required to determine the mean IQ of seventh graders of Luis Palma

National High School within +5 points, population standard deviation of 25, and with the following

confidence level?

a. 98%

b. 95%.

c. 90%.

d. 80%.

Answer 1

$1:\\\frac{\sigma}{\sqrt{n}}z_{\frac{1+\gamma}{2}}\leqslant E\Rightarrow n\geqslant \frac{\left( \sigma z_{\frac{1+\gamma}{2}} \right) ^2}{E^2}=\frac{\left( 30\cdot 1.960 \right) ^2}{E^2}=\frac{3457.44}{E^2}\\a:n\geqslant \frac{3457.44}{6^2}=96.04\Rightarrow n=97\\b:n\geqslant \frac{3457.44}{5^2}=138.298\Rightarrow n=139\\c:n\geqslant \frac{3457.44}{4^2}=216.09\Rightarrow n=217\\d:n\geqslant \frac{3457.44}{3^2}=384.16\Rightarrow n=385\\2:\\\frac{\sigma}{\sqrt{n}}z_{\frac{1+\gamma}{2}}\leqslant E\Rightarrow n\geqslant \frac{\sigma ^2}{E^2}{z_{\frac{1+\gamma}{2}}}^2=\frac{25^2}{5^2}{z_{\frac{1+\gamma}{2}}}^2=25{z_{\frac{1+\gamma}{2}}}^2\\a:n\geqslant 25\cdot 2.3263^2=135.3\Rightarrow n=136\\b:n\geqslant 25\cdot 1.960^2=96.04\Rightarrow n=97\\c:n\geqslant 25\cdot 1.645^2=67.64\Rightarrow n=68\\d:n\geqslant 25\cdot 1.282^2=41.06\Rightarrow n=42$