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?

The probability of the sample mean foot length less than 23 cm is 0.120

Step-by-step explanation:

* Lets explain the information in the problem

- The eighth-graders asked to measure the length of their right foot at

the beginning of the school year, as part of a science project

- The foot length is approximately Normally distributed, with a mean of

23.4 cm

∴ μ = 23.4 cm

- The standard deviation of 1.7

∴ σ = 1.7 cm

- 25 eighth-graders from this population are randomly selected

∴ n = 25

- To find the probability of the sample mean foot length less than 23

∴ The sample mean x = 23, find the standard deviation σx

- The rule to find σx is σx = σ/√n

∵ σ = 1.7 and n = 25

∴ σx = 1.7/√25 = 1.7/5 = 0.34

- Now lets find the z-score using the rule z-score = (x - μ) / σx

∵ x = 23, μ = 23.4, σx = 0.34

∴ z-score = (23 - 23.4) / 0.34 = - 1.17647 ≅ - 1.18

- Use the table of the normal distribution to find P (x < 23)

- We will search in the raw of - 1.1 and look to the column of 0.08

∴ P (X < 23) = 0.119 ≅ 0.120

* The probability of the sample mean foot length less than 23 cm is 0.120