The population standard deviation of people who shop online is 3.1. If you want to be 75% confident that the sample mean is within 10% of the population mean, how many people must be surveyed?
To determine the sample size required to estimate the population mean within a certain range with a specified confidence level, we can use the formula for sample size determination:
n = (Z * σ / E)²
Where:
n = sample size
Z = Z-value for the desired confidence level (in this case, for 75% confidence level)
σ = population standard deviation
E = margin of error (as a proportion of the population mean)
First, let's calculate the Z-value for a 75% confidence level. The Z-value corresponds to the area under the standard normal curve; for a 75% confidence level, we need to find the Z-value that encloses 75% of the area under the curve.
Looking up the Z-value in a standard normal distribution table or using a statistical software, we find that the Z-value for a 75% confidence level is approximately 0.674.
Now, we can use the formula to calculate the required sample size:
n = (0.674 * 3.1 / 0.1)²
n = (2.0874 / 0.1)²
n = 20.874²
n ≈ 436.19
Since sample sizes must be whole numbers, we round up to the nearest whole number:
n ≈ 437
Therefore, approximately 437 people must be surveyed to be 75% confident that the sample mean is within 10% of the population mean.