A population has a standard deviation of 15.2. How large a sample must be drawn so that a 95% confidence interval for the population mean will have a margin error equal to 3.5?
To determine the sample size for a desired margin of error in estimating the population mean with a specified confidence level, you can use the following formula:
n = (Z * σ / E)^2
Where:
n = sample size
Z = z-score corresponding to the desired confidence level
σ = standard deviation of the population
E = desired margin of error
In this case, the standard deviation of the population (σ) is given as 15.2 and the desired margin of error (E) is 3.5.
The z-score corresponding to a 95% confidence level can be obtained from a standard normal distribution table. For a 95% confidence level, the z-score is approximately 1.96.
Now, we can plug in these values into the formula to calculate the required sample size:
n = (1.96 * 15.2 / 3.5)^2
n = (30.032 / 3.5)^2
n = 8.58^2
n ≈ 73.77
Therefore, you would need to draw a sample size of at least 74 to obtain a 95% confidence interval for the population mean with a margin of error equal to 3.5. Note that since the sample size must be whole number, rounding up is often followed. In this case, rounding up to 75 would be the appropriate choice.
Use a formula to find sample size.
Here is one:
n = [(z-value * sd)/E]^2
...where n = sample size, z-value = 1.96 for 95% confidence, sd = 15.2, E = 3.5, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculations. Round your answers to the next highest whole number.
I'll let you take it from here.