Answer to Question #320140 in Statistics and Probability for Ali De Austria

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

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