The image of the public library is constantly changing, and their online services continue to grow. Usage of the library's home page grew by 17% during the past 12 months. It has been estimated that the current average length of a visit to the library's homepage is approximately 20 minutes. The library wants to take a sample to statistically estimate this mean. How large will the sample need to be to estimate the mean within 0.2 of 1 standard deviation with 0.95 confidence? (Round you answer up to the nearest whole number.)
To determine the sample size needed to estimate the mean within a certain range with a certain level of confidence, we can use the formula for sample size calculation:
n = (Z * σ / E)²
Where:
n = sample size
Z = z-value corresponding to the desired level of confidence (in this case, 0.95, which corresponds to a z-value of approximately 1.96)
σ = population standard deviation (unknown in this case)
E = maximum acceptable error (0.2 times 1 standard deviation)
Since the population standard deviation is unknown, we can use the sample standard deviation as an estimate. However, since we don't have any information about the sample standard deviation, we need to assume a value.
Let's assume a conservative value of 0.5, which is commonly used when the population standard deviation is unknown. You can adjust this value if you have more specific information about the data.
n = (1.96 * 0.5 / (0.2 * 1))²
n = (0.98 / 0.2)²
n = (4.9)²
n = 24.01
Round up the sample size to the nearest whole number:
n ≈ 25
Therefore, the sample size needed to estimate the mean within 0.2 of 1 standard deviation with 0.95 confidence is 25.