9.3 Central Limit Theorem: mean

Lab 9.3.1

Two bell curves over screw length in inches, both centred on the population mean of 5. The dashed red one is the population itself, standard deviation 1 inch, and never changes. The solid dark blue one is the distribution of the mean of a sample of 4 screws, already half as wide because its spread is 1 divided by the square root of the sample size. A green vertical line marks 6 inches: one standard deviation out on the dashed curve but two on the solid one, and larger samples pull the solid curve in until almost none of it reaches the line. Adjustable parameter: Sample size n (n) = 4. Viewing window: x from 1.75 to 8.25, y from -0.41 to 3.61. — The lab's population - screw lengths, mean 5 inches, standard deviation 1 inch - is the dashed curve and never moves. The solid curve is the sampling distribution of the mean of n screws, with n on a slider, and the green line marks the 6 inches the lab asks about. The figure computes no probability: it shows why the two questions have such different answers. At n = 1 the curves coincide and 6 inches is only one standard deviation out. Drag n up and the solid curve pulls in around 5, because its standard deviation is 1/sqrt(n) - 0.5 at n = 4, 0.316 at the lab's n = 10 - until almost nothing is left beyond the green line. Read the picture first, then compute both probabilities and check that the numbers say what the shapes say.

The lengths of metal screws are normally distributed, with a mean of 5 inches, and standard deviation of 1 inch.

If 10 screws are chosen at random, what is the probability that their mean length is less than 6 inches?
If 10 screws are chosen at random, what is the probability that their mean length is between 5 and 6 inches?

Adapted from Python for Introductory Statistics: Student's Lab Workbook, by Simon Aman (Truman College, City Colleges of Chicago), licensed under CC BY 4.0. Changes were made: reformatted as an accessible XYZ web edition with live in-browser code cells. License: CC-BY-4.0.