The average diameter of sand dollars on a certain island is 4.00 centimeters with a standard deviation of 0.80 centimeters. If 25 sand dollars are chosen at random for a collection, find the probability that
The average diameter of those sand dollars is more than 3.824 centimeters. Assume that the variable is
Normally distributed.
A) 0.8414
B) 0.8483
C) 0.8643
D) 0.8611

Question

Quizplus · Accepted Answer

To solve this problem, we need to use the Central Limit Theorem, which states that the distribution of sample means from a population with a known mean and standard deviation will be approximately normally distributed as the sample size increases.

First, we need to calculate the standard error of the mean (SEM), which can be found using the formula SEM = σ/√n, where σ is the standard deviation of the population and n is the sample size. In this case, σ = 0.80 cm and n = 25, so SEM = 0.80/√25 = 0.16 cm.

Next, we need to standardize the sample mean using the formula z = (x̄ - μ) / SEM, where x̄ is the sample mean, μ is the population mean (given as 4.00 cm), and SEM is the standard error of the mean. We want to find the probability that the sample mean is greater than 3.824 cm, so we plug in these values and get:

z = (3.824 - 4.00) / 0.16 = -1.1

Using a standard normal distribution table, we find that the area to the right of z = -1.1 is 0.8643, which is the probability that the average diameter of the 25 sand dollars is more than 3.824 cm. Therefore, the answer is choice C, 0.8643.

The Average Diameter of Sand Dollars on a Certain Island