Summary
Highlights
The video introduces hypothesis testing for the population mean when the variance is known. It defines hypothesis testing as a procedure used by statisticians to decide whether to accept or reject a statement about a population. The Z-formula for calculating the test value is presented: Z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. This formula is applicable when the sample size (n) is greater than or equal to 30 and the population variance is known.
An example problem is introduced: A leader claims the average daily take-home pay of jeepney drivers is 400 pesos. A sample of 100 drivers shows an average of 425 pesos. With a 0.05 level of significance and a population variance of 8,464 pesos, the goal is to determine if the average daily take-home pay is different from 400 pesos. Step 1 involves stating the null (H₀) and alternative (H₁) hypotheses. Based on the claim, H₀: μ = 400, and because the question asks if the pay is 'different from' 400 pesos, H₁: μ ≠ 400.
Step 2 focuses on setting the level of significance (alpha, α) and determining the critical value. The given α is 0.05. Since the alternative hypothesis (μ ≠ 400) indicates a two-tailed test, alpha is divided by 2 (α/2 = 0.025). To find the critical Z-value, 0.5 (half of the normal distribution curve) is subtracted by α/2 (0.5 - 0.025 = 0.475). Consulting a Z-table for the area 0.4750 yields a Z-value of 1.96. Thus, the critical values for this two-tailed test are ±1.96.
Step 3 is to compute the test value using the Z-formula. First, the standard deviation (σ) needs to be calculated from the given population variance (σ² = 8,464). The square root of 8,464 is 92, so σ = 92. Plugging the values into the Z-formula: Z = (425 - 400) / (92 / √100) = 25 / (92 / 10) = 25 / 9.2 = 2.7173... which, when rounded to two decimal places, gives a test value of 2.72.
Step 4 involves deciding whether to accept or reject the null hypothesis by comparing the computed test value to the critical values. The critical values are -1.96 and +1.96. The calculated test value is 2.72. Since 2.72 is greater than 1.96, it falls within the rejection region on the right tail of the curve. Therefore, the decision is to reject the null hypothesis and accept the alternative hypothesis. Step 5 is to formulate the conclusion: Since the null hypothesis is rejected and the alternative is accepted, it is concluded that the average daily take-home pay of jeepney drivers is not equal to 400 pesos.