Question

We examine the effect of different inputs on determining the sample size needed. Find the sample size needed to give, with \(95 \%\) confidence, a margin of error within ±3 , if the estimated standard deviation is \(\tilde{\sigma}=100\). If the estimated standard deviation is \(\tilde{\sigma}=50\). If the estimated standard deviation is \(\tilde{\sigma}=10 .\) Comment on how the variability in the population influences the sample size needed to reach a desired level of accuracy.

Short answer

The sample sizes required for the 95% confidence level with a margin of error of ±3 are: 427 when the estimated standard deviation is 100, 107 when it's 50, and 5 when it's 10. As the standard deviation (population variability) increases, the sample size required to achieve the same level of accuracy also increases.

Step-by-step solution

Find Z score from confidence level

A 95% confidence level corresponds to a \(Z_{score}\) of 1.96. This can be found using a standard normal distribution table or a Z-score calculator available online.

Apply the formula for \(\tilde{\sigma}=100\)

\(n=(Z_{score}*100/3)^2=(1.96*100/3)^2=426.247\) (rounded to 427). Therefore, a sample size of 427 is needed for a 95% confidence level with a margin of error of ±3 when the estimated standard deviation is 100.

Repeat step 2 with \(\tilde{\sigma}=50\)

\(n=(Z_{score}*50/3)^2=(1.96*50/3)^2=106.562\) (rounded to 107). Therefore, a sample size of 107 is needed for a 95% confidence level with a margin of error of ±3 when the estimated standard deviation is 50.

Repeat step 2 with \(\tilde{\sigma}=10\)

\(n=(Z_{score}*10/3)^2=(1.96*10/3)^2=4.265\) (rounded to 5). Therefore, a sample size of 5 is needed for a 95% confidence level with a margin of error of ±3 when the estimated standard deviation is 10.

Comment on the findings

As observed from the results, there is an inverse relationship between the estimated standard deviation of the population (variability of the population) and the required sample size to achieve a certain confidence level with the specified margin of error. As the variability increases, represented by an increase in the standard deviation, a larger sample size is needed to achieve the same level of accuracy and confidence.