Hypothesis Testing Dilemma?

📌 CONCEPT: Hypothesis testing is a statistical method used to test a claim or hypothesis about a population based on a sample of data, determining whether the observed data is consistent with the null hypothesis or not.

📐 RULE / FORMULA: We use the Z-test to compare the sample mean to the population mean, given by the formula Z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean, σ is the sample standard deviation, and n is the sample size.

💡 WORKED EXAMPLE: Suppose we want to test the claim that the average height of students is not more than 160 cm. The sample mean height is 161.5 cm, the sample standard deviation is 4 cm, and the sample size is 50. We can calculate the Z-score as Z = (161.5 - 160) / (4 / √50) = 1.5 / 0.4 = 3.75. Since the Z-score is greater than the critical value of 1.96 (for a 95% confidence level), we reject the null hypothesis that the average height is less than or equal to 160 cm.

⚠️ COMMON MISTAKE: Students often forget to check the assumptions of the Z-test, such as normality of data and equal variances, which can lead to incorrect conclusions.