Final answer:
To calculate the 99.7% confidence interval for the average assembly time with a sample mean of 22 minutes, a standard deviation of 3 minutes, and a sample size of 36, the confidence interval is found to be between 20.5 and 23.5 minutes.
Step-by-step explanation:
To estimate the average amount of time an adult requires to assemble an “easy to assemble” toy, a 99.7% confidence interval for the population mean assembly time is needed, given a sample mean of 22 minutes and a standard deviation of 3 minutes from 36 randomly selected adults. To calculate this, we use the z-score associated with a 99.7% confidence interval, which corresponds to 3 standard deviations from the mean in a normal distribution (the 3-sigma rule).
The confidence interval (CI) is calculated using the formula:
CI = μ ± (z * (σ / √n))
Where μ is the sample mean, σ is the sample standard deviation, n is the sample size, and z is the z-score corresponding to the desired level of confidence. For the 99.7% confidence interval (also known as three-sigma), the z-score is approximately 3 because this level of confidence encompasses virtually all data for a normal distribution.
Therefore, the confidence interval calculation is:
CI = 22 ± (3 * (3 / √36))
= 22 ± (3 * (3 / 6))
= 22 ± 1.5
The confidence interval is then 20.5 to 23.5 minutes. This means we can be 99.7% confident that the average time it takes for all adults to assemble the toy falls within this range.