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 0.4 of 1 standard deviation with 0.95 confidence?

To determine the sample size needed to estimate the mean within 0.4 of 1 standard deviation with 0.95 confidence, we can use the following formula:

Sample size (n) = (Z * σ / E)²

Where:
- Z is the z-score corresponding to the desired confidence level (0.95)
- σ is the standard deviation of the population
- E is the desired margin of error (0.4 * standard deviation)

Since we don't have the population standard deviation (σ), we can estimate it using the sample standard deviation (S) based on the available data. However, since we are trying to estimate the mean visit length, we don't have the individual visit lengths in the sample. So, we'll have to make an assumption about the population standard deviation.

A common rule of thumb is to assume that the population standard deviation is similar to the sample standard deviation. Given that the mean visit length is estimated to be 20 minutes, we can make an assumption and estimate σ as a percentage (by taking the coefficient of variation) of the mean visit length.

Coefficient of variation (CV) = (S / Mean) * 100%

From the given data, we can calculate the sample standard deviation (S) and use it to estimate the population standard deviation (σ).

Next, we can calculate the sample size using the formula above, replacing σ with the estimated value.

Keep in mind that this estimation method assumes the visit lengths follow a normal distribution, and the sampled data is representative of the population.

Please provide the available data or assumptions made about the standard deviation, so I can proceed with the calculation.