Brief Summary
This video provides a comprehensive guide to hypothesis testing for small samples, focusing on the T-test and its application to population means. It explains the differences between large and small samples, introduces the T-test, Fisher Z-test, and F-test, and details the steps for conducting a T-test, including setting up null and alternative hypotheses, calculating the T-statistic, determining degrees of freedom, and making a decision based on the table value.
- Focuses on hypothesis testing for small samples using the T-test.
- Explains the formula for calculating the T-statistic and modified standard deviation.
- Provides a step-by-step guide to conducting a T-test with practical examples.
Introduction to Hypothesis Testing for Small Samples
The video introduces hypothesis testing for small samples, contrasting it with previous lessons on large samples. Small samples are defined as those with a sample size less than 30. The session will cover three tests for small samples: the T-test, Fisher Z-test, and F-test, with a focus on the T-test initially. The video outlines the structure of the T-test, which includes four topics, and begins with the first topic: hypothesis testing about the population mean (µ). The presenter promises to clarify the concepts and encourages viewers to take notes and practice the questions.
Formula for Test of Hypothesis About Population Mean
The video explains the formula for conducting a hypothesis test about a population mean using the T-test. The formula for the T-statistic is t = (x̄ - µ) / (s / √n), where x̄ is the sample mean, µ is the population mean, s is the modified standard deviation, and n is the sample size. The video details how to calculate 's' using three different formulas, depending on the information available in the problem. The first formula is s = √[Σ(x - x̄)² / (n - 1)], the second is s = √[Σd² - (Σd)²/n / (n - 1)], where d = x - a, and the third is s = √(n / (n - 1)) * s, where 's' is the standard deviation. The video also explains the terms "deviation from mean" (x - x̄) and "deviation from assumed mean" (x - a).
Question 1: Applying the T-Test Formula
The video presents the first question, involving a group of five patients treated with a medicine. The question asks to discuss whether the mean weight of the population is 48 kg, given the sample data. The presenter identifies this as a small sample T-test question focused on the population mean. The steps include defining the null hypothesis (H0: µ = 48), alternative hypothesis (H1: µ ≠ 48), and alpha (α = 0.05). The T-statistic formula is introduced, and the presenter explains how to choose the correct formula for calculating the modified standard deviation (s) based on whether the sample mean is an integer or a decimal. A table is created to calculate the values needed for the formula, and the T-statistic is calculated.
Determining Degrees of Freedom and Table Value
After calculating the T-statistic, the video explains how to determine the degrees of freedom (df = n - 1), which in this case is 4. The presenter then discusses how to find the table value of t for α = 0.05 and df = 4, which is given as 2.776. The calculated T-value (0.53) is compared to the table value, and since it is less, the null hypothesis is accepted. The conclusion is that the mean weight of the population can be 48 kg. The presenter also explains how to find the table value using a T-distribution table, emphasizing the importance of knowing the alpha level, degrees of freedom, and whether the test is one-tailed or two-tailed.
Question 2: Calculating T-Statistic with Decimal Mean
The video proceeds to the second question, which involves a random sample of nine boys with given heights. The task is to discuss whether the mean height of the population is 47.5 inches, with a provided table value of 2.306 at 8 degrees of freedom and a 5% level of significance. The presenter sets up the null hypothesis (H0: µ = 47.5) and the alternative hypothesis (H1: µ ≠ 47.5), and then calculates the sample mean (x̄), which comes out to be a decimal (49.1). Because the mean is a decimal, the presenter uses the second formula for calculating the modified standard deviation (s), which involves finding the assumed mean (a) and calculating d = x - a.
Calculating Modified Standard Deviation and Concluding the Test
The video continues with the calculation of the modified standard deviation (s) using the formula s = √[Σd² - (Σd)²/n / (n - 1)]. The presenter explains how to find the assumed mean (a) by either taking the closest integer to the sample mean or by taking the middle value of the data set. The values for d, d², and Σd are calculated, and then the modified standard deviation is found to be 2.62. The T-statistic is calculated as 1.83. The degrees of freedom are determined to be 8, and the table value is given as 2.306. Since the calculated T-value is less than the table value, the null hypothesis is accepted, and the conclusion is that the mean height of the population is 47.5 inches.
Question 3: Using Sum of Squares of Deviation
The video tackles the third question, where a random sample of nine values from a normal population shows a mean of 41.5 inches, and the sum of squares of deviation from the mean equals 72 inches. The task is to assess whether the assumption of a mean of 44.5 inches in the population is reasonable. The presenter sets up the null hypothesis (H0: µ = 44.5) and the alternative hypothesis (H1: µ ≠ 44.5). The presenter identifies that the sum of squares of deviation from the mean is given, so the first formula for calculating the modified standard deviation (s) can be used directly.
Calculating T-Statistic and Comparing with Table Value
The video proceeds with the calculation of the modified standard deviation (s) using the given sum of squares of deviation from the mean. The T-statistic is calculated to be 3. The degrees of freedom are determined to be 8. Since the table value is not provided, the presenter demonstrates how to find it using the T-distribution table, with α = 0.05 and df = 8, which gives a table value of 2.306. The calculated T-value (3) is greater than the table value, so the null hypothesis is rejected. The conclusion is that the mean is not 44.5.
Question 4: Determining Difference from Intended Weight
The video presents the fourth question, where 16 oil tins are taken at random from an automatic filling machine. The mean weight of the tins is 14.5 kg, with a standard deviation of 0.40. The task is to determine whether the sample mean differs significantly from the intended weight of 16 kg. The presenter sets up the null hypothesis (H0: µ = 16) and the alternative hypothesis (H1: µ ≠ 16). The presenter identifies that the standard deviation is given, so the third formula for calculating the modified standard deviation (s) can be used.
Calculating T-Statistic and Concluding the Test
The video continues with the calculation of the modified standard deviation (s) using the formula s = √(n / (n - 1)) * s. The T-statistic is calculated to be 14.6. The degrees of freedom are determined to be 15. The presenter demonstrates how to find the table value using the T-distribution table, with α = 0.05 and df = 15, which gives a table value of 2.131. The calculated T-value is greater than the table value, so the null hypothesis is rejected. The conclusion is that the sample mean is significantly different from the intended mean of 16 kg.
Question 5: Testing Manufacturer's Claim
The video presents the fifth and final question, where a consumer testing agency examines a new automobile for gasoline mileage performance. Twelve readings of miles covered per gallon result in an average of 16 miles per gallon, with a standard deviation of 1.8. The task is to determine whether the sample results support the manufacturer's claim that the new automobile gives a performance of more than 15 miles per gallon. The presenter sets up the null hypothesis (H0: µ = 15) and the alternative hypothesis (H1: µ > 15).
Calculating T-Statistic and Concluding the Test
The video continues with the calculation of the modified standard deviation (s) using the formula s = √(n / (n - 1)) * s. The T-statistic is calculated. The degrees of freedom are determined to be 11. The presenter demonstrates how to find the table value using the T-distribution table, with α = 0.10 and df = 11, which gives a table value of 1.363. The calculated T-value is compared to the table value, and a conclusion is drawn based on whether the null hypothesis is rejected or accepted.
Alternative T-Statistic Formula
The video concludes by presenting an alternative formula for calculating the T-statistic directly when the standard deviation is given: t = (x̄ - µ) / (s / √(n - 1)). The presenter notes that this formula can be used to calculate the T-statistic without first calculating the modified standard deviation. The presenter encourages viewers to calculate the T-statistic using this formula and confirm that it gives the same result. The presenter also notes that some instructors may prefer the modified standard deviation formula because it shows a more complete understanding of the concepts.
