The director of the library believes that 25% of the library's collection is checked out.
If the director is right, what is the probability that the proportion of books checked out in a sample of 545 books would differ from the population proportion by less than 5%? Round your answer to four decimal places.
To calculate the probability that the proportion of books checked out in a sample differs from the population proportion by less than 5%, we can use the formula for the sampling distribution of a proportion.
The standard error (SE) of the sampling distribution of a proportion can be calculated using the following formula:
SE = sqrt[(p * (1 - p)) / n]
Where:
- p is the population proportion
- n is the sample size
In this case, the population proportion is 0.25 (25%) and the sample size is 545.
So, we have:
SE = sqrt[(0.25 * (1 - 0.25)) / 545]
Calculating this, we find:
SE ≈ 0.01543
Next, we need to calculate the margin of error, which is defined as the maximum amount that the sample proportion can differ from the population proportion. In this case, we want the proportion to differ by less than 5%, which is 0.05.
So, the margin of error (ME) can be calculated by multiplying the SE by the z-score corresponding to the desired level of confidence. Let's assume a 95% level of confidence, which corresponds to a z-score of approximately 1.96:
ME = 1.96 * SE
Calculating this, we find:
ME ≈ 1.96 * 0.01543 ≈ 0.03022
Finally, to calculate the probability that the proportion of books checked out in a sample differs from the population proportion by less than 5%, we need to find the area under the sampling distribution curve within the margin of error. Since we want the difference to be less than 5%, we are looking for the area within ±0.03022 from the population proportion.
This probability can be approximated by finding the area under the standard normal distribution curve between -0.03022 and 0.03022. We can use a standard normal distribution table or a statistical software to find this area.
Rounding the answer to four decimal places, the probability is approximately 0.1671.