Analyzing Variability in a Sample?

📌 CONCEPT: The teacher can use the one-sample z-test to determine if the school administration's claim is valid by comparing the sample mean to the population mean claimed by the administration.

📐 RULE / FORMULA: The z-test statistic is calculated using the formula z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean claimed, σ is the sample standard deviation, and n is the sample size.

💡 WORKED EXAMPLE: Suppose the teacher wants to test if the average height of 165 cm is significantly different from the claimed average height of 170 cm. The sample standard deviation is 10 cm and the sample size is 25. Using the formula, z = (165 - 170) / (10 / √25) = -5 / 1.77 ≈ -2.82. This z-score indicates that the sample mean is 2.82 standard errors away from the claimed population mean.

⚠️ COMMON MISTAKE: Students often forget to check the assumptions of the z-test, such as normality of the sample data and independence of observations.