Brief Summary
This video provides a comprehensive guide to hypothesis testing for large samples, focusing on population means. It explains how to set up null and alternative hypotheses, calculate the test statistic, and interpret the results using critical values. The video includes detailed examples and step-by-step instructions, making it easy to follow along and understand the concepts.
- Introduction to hypothesis testing for large samples.
- Detailed explanation of the formula for calculating the test statistic (z-score).
- Step-by-step examples of hypothesis testing for population means.
Introduction to Hypothesis Testing for Large Samples
The video introduces hypothesis testing, specifically for large samples, and mentions that the chapter is divided into two parts: testing for large samples and testing for small samples. A large sample is defined as a sample size of 30 or more. The video outlines five topics to be covered within large sample hypothesis testing: population mean, difference of population means, population standard deviation, population proportion, and difference of population proportions. The first topic, "Test of Hypothesis About Population Mean," will be the focus of the current class.
Formula for Population Mean
The video introduces the formula for calculating the standard normal variate (z) when testing hypotheses about a population mean: z = |x̄ - μ| / (standard error of the mean). Here, x̄ represents the sample mean, μ represents the population mean, and the standard error of the mean is calculated as either √(σ²/n) or √(s²/n), where σ is the population standard deviation, s is the sample standard deviation, and n is the sample size. The presenter emphasizes that x̄ and μ will typically be provided in the problem, while the standard error of the mean needs to be calculated.
Example 1: Testing Population Mean
The video presents an example where the mean height of a random sample of 100 students is 64 inches, with a standard deviation of 3 inches. The task is to test the statement that the mean height of the population is 67 inches at a 5% level of significance. The null hypothesis (H0) is set as μ = 67, and the alternative hypothesis (H1) is μ ≠ 67 (two-tailed). After calculating the standard error of the mean as 0.3 and the z-value as 10, it's compared to the critical value of 1.96 at a 5% significance level for a two-tailed test. Since the calculated z-value (10) is greater than the critical value (1.96), the null hypothesis is rejected, concluding that the mean height of the population cannot be 67 inches.
Example 2: Determining if a Sample Comes from a Population
In this example, a random sample of 400 male students has a mean height of 171.5 cm. The question is whether this sample can be reasonably regarded as coming from a large population with a mean height of 171.17 cm and a standard deviation of 3.30 cm, with alpha = 0.05. The null hypothesis (H0) is set as μ = 171.17, and the alternative hypothesis (H1) is μ ≠ 171.17 (two-tailed). The standard error of the mean is calculated, and the audience is prompted to calculate the z-value and determine whether to accept or reject the null hypothesis based on the critical value of 1.96 at a 5% significance level.
Example 3: Verifying a Stenographer's Claim
A stenographer claims to take dictation at a rate of 120 words per minute. Based on 100 trials, she demonstrates a mean of 116 words with a standard deviation of 15 words. The null hypothesis (H0) is set as μ = 120, and the alternative hypothesis (H1) is μ ≠ 120 (two-tailed). Given a 5% significance level, the standard error of the mean and the z-value are calculated. Comparing the calculated z-value to the critical value of 1.96, the conclusion is made to reject the stenographer's claim if the calculated z-value exceeds the critical value.
Example 4: Testing an Educator's Claim about IQ
An educator claims that the average IQ of government college students is not more than 110. A random sample of 150 students is taken, and their average IQ is found to be 111.2, with a standard deviation of 7.2. With a significance level of 0.01 (1%), the null hypothesis (H0) is set as μ ≤ 110, and the alternative hypothesis (H1) is μ > 110 (one-tailed). The standard error of the mean and the z-value are calculated. The audience is asked to determine whether to accept or reject the educator's claim by comparing the calculated z-value to the critical value of 2.33 for a one-tailed test at a 1% significance level.
