Question

Suppose you wish to estimate a population mean based on a random sample of \(n\) observations, and prior experience suggests that \(\sigma=12.7\). If you wish to estimate \(\mu\) correct to within 1.6 , with probability equal to .95, how many observations should be included in your sample?

Short answer

Answer: To estimate the population mean to within 1.6 with a 95% confidence level, the required sample size is 245 observations.

Step-by-step solution

Find the z-score for 95% confidence level

Using a z-table or calculator, find the z-score that corresponds to the 95% confidence level. The z-score represents the number of standard deviations away from the mean that encompasses 95% of the area under the standard normal distribution curve. For a 95% confidence level, the z-score is 1.96.

Plug in the known values of σ and E into the formula

σ is given as 12.7, and the desired margin of error, E, is 1.6, so we can plug these values into the sample size formula: n = (Z * σ / E)^2 n = (1.96 * 12.7 / 1.6)^2

Calculate the sample size

Now, calculate the sample size by performing the operations inside the parentheses and then squaring the value: n = (1.96 * 12.7 / 1.6)^2 = (15.646)^2 = 244.79636

Round the calculated sample size up to the nearest whole number

The calculated sample size is 244.79636. Since we can't have a fraction of an observation, round up the value to the nearest whole number, which is 245. So, to estimate the population mean (μ) to within 1.6 with a 95% confidence level, you would need to include 245 observations in your sample.