A large school district in southern California asked all of its eighth-graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed, with a mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth-graders from this population are randomly selected. Approximately what is probability that the sample mean foot length is less than 23 cm?

Question

Quizplus · Accepted Answer

To find the probability that the sample mean foot length is less than 23 cm, we use the Central Limit Theorem since the sample size is greater than 30. The standard error (SE) of the mean is calculated as the population standard deviation divided by the square root of the sample size (n), which is $SE = \frac{1.7}{\sqrt{25}} = \frac{1.7}{5} = 0.34$. We then calculate the z-score for a mean of 23 cm: $z = \frac{23 - 23.4}{0.34} = \frac{-0.4}{0.34} \approx -1.18$. Looking up this z-score in a standard normal distribution table or using a calculator gives a probability of approximately 0.12, indicating that there's about a 12% chance the sample mean foot length is less than 23 cm.

A Large School District in Southern California Asked All of Its