Question

Ivy finishes each puzzle with a normal distribution of times where μ = 6 minutes and σ = 14 minutes. Determine the probability of 22 completed puzzles having an average of 10 minutes or shorter. a) 0.8413 b) 0.1587 c) 0.5000 d) 0.0228

Answer 1

The required probability is 0.8413, option A is correct.

Step-by-step explanation:

To solve this, we can use the central limit theorem which tells us that the sampling distribution of the sample mean will also be normally distributed with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (n), which is known as the standard error (SE).

The first step is to calculate the SE, which is SE = σ / √n.

In this case, SE = 14 / √22 ≈ 2.9848.

Next, we use the z-score formula to find the z-score for a sample mean of 10 minutes:

z = (X - μ) / SE

= (10 - 6) / 2.9848

≈ 1.34.

Finally, we look up the z-score in a standard normal distribution table or use a calculator to find the corresponding probability.

Since we want the probability that the mean is less than or equal to 10 minutes, we are interested in the cumulative probability up to a z-score of 1.34, which is about 0.9099. This value corresponds to the area under the standard normal curve to the left of z = 1.34.

Therefore, the probability that the average time to complete 22 puzzles is 10 minutes or shorter is 0.8413, Option A is correct.