Chapter 5: The Hypothesis with means of samples PDF

Chapter 5: The Hypothesis with means of samples Psyc115: Psychological Statistics Ms. Rosian Mae S.P. Bermudo, RPm Objectives: At the end of this lesson, you will be going to learn the following:  The Distribution of Means  The Z test Determining the Charac...

Chapter 5: The Hypothesis with means of samples Psyc115: Psychological Statistics Ms. Rosian Mae S.P. Bermudo, RPm Objectives: At the end of this lesson, you will be going to learn the following:  The Distribution of Means  The Z test Determining the Characteristics of a Distribution of Means The three key characteristics of the comparison distribution that you need to determine are: 1. Its mean. 2. Its spread (which you measure using the variance and standard deviation). 3. Its shape. 1. Mean Identify the Population: Determine the population you are comparing your sample to. Calculate the Mean: If you have the entire population data, calculate the mean using the formula: where X represents each value and N is the number of values. 2. Spread: Calculate Variance: Use the formula: Here, 𝑋 is each value, 𝜇 is the mean, and 𝑁 is the number of values. Calculate Standard Deviation: The standard deviation is the square root of the variance:  3. Shape Determine Distribution Type: Identify the shape of the distribution (e.g., normal, skewed, uniform). This can be done through visual inspection using histograms or by calculating skewness and kurtosis. Use Visualizations: Plotting the data can help you see the shape. A bell curve indicates a normal distribution, while skewed shapes indicate positive or negative skewness. Statistical Tests: You can also use tests (example: Shapiro-Wilk test) to assess normality. You can also use Excel in generating your date, you should learn both, please see the step-by-step process for Excel. Please listen to your teacher as she demonstrate it:  Input Your Data: Open Excel and enter your data in rows or columns. Select the Data: Highlight the cells that contain your data. Insert a Chart: Go to the "Insert" tab on the ribbon. Choose the type of chart you want to create (e.g., Line, Scatter, Bar). Click on your desired chart type, and Excel will generate the chart based on the selected data. Customize Your Chart: Use chart tools to modify titles, axes, and styles. Means of Sample Rule 1: The mean of a distribution of means is the same as the mean of the population of individuals. There are two means those are: - M is the mean (average) of the sample means you get when you take multiple samples from a population. - μ (mu) is the mean of the entire population of individuals. When you take a sample, it’s made up of randomly selected individuals. This means that sometimes the average of a sample will be higher than the population mean, and sometimes it will be lower. Because you are randomly selecting individuals and taking many samples, the higher and lower sample means will balance each other over time. More Samples, Better Accuracy This is a fundamental idea in statistics that helps us understand how sample averages relate to the true population average. In short, taking lots of random samples helps ensure that the average of those samples gets closer to the average of the whole population. Rule 2A: The variance of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample. A distribution of means will be less spread out than the distribution of individuals from which the samples are taken. If you are taking a sample of two scores, it is less likely that both scores will be extreme. Example with Balls: Imagine you have balls numbered from 1 to 9. In the population, you have many 1s and 9s, creating a widespread. If you take two balls at a time, it's much more common to get an average around 5 (like a 1 and a 9) than to get both balls as 1s or both as 9s. Rule 2B: The standard deviation of a distribution of means is the square root of the variance of the distribution of means The rule says that if you have the variance of the distribution of means, you can find the standard deviation by taking the square root of that variance. In short, if you know how spread out the means are (variance), you can find out how spread out they are in a more interpretable way (standard deviation) by taking the square root. The standard deviation of the distribution of means also has a special name of its own. The standard error of the mean (SEM), or the standard error (SE). Rule 3: The shape of a distribution of means is approximately normal if Conditions for Normality: Condition (a): If each sample has 30 or more individuals, the distribution of the sample means will usually be approximately normal. This is true even if the population itself isn’t normal. Condition (b): If the population of individuals is already normally distributed, then the distribution of sample means will also be normal, regardless of the sample size. With a sample size of 30 or more, the percentages you find using the normal distribution will be very accurate. Even larger samples improve this approximation, but 30 is generally sufficient for practical purposes. If the original population is already normally distributed, the distribution of means will also be normal, no matter how many individuals are in each sample. Rule 1: The mean of the distribution of means is the same as the population mean. For this example, both are 200. Rule 2a: The variance of the distribution of means is the population variance divided by the sample size. Given a population variance of 2,304 and a sample size of 64, the variance of the distribution of means is 36. Rule 2b: The standard deviation of the distribution of means (standard error) is the square root of the variance. Thus, the standard deviation is 6 (since 36=6 36​=6). Rule 3: The distribution of means is approximately normal if the sample size is 30 or more, or if the population distribution is normal. Here, the sample size is 64, and the population is normal, so the distribution of means will also be normal. Distribution of a Population of Individuals: This represents all the individual data points in a specific group. It shows how values are spread out across the entire population. For example, if you were looking at the test scores of all students in a school, this distribution would include every student's score. Distribution of a Particular Sample: This distribution includes data from a smaller subset (sample) taken from the larger population. It provides a snapshot of that specific group. For instance, if you randomly selected 30 students from the school and recorded their test scores, the distribution of those scores represents the sample distribution. Distribution of Means: This distribution shows the means (averages) of multiple samples taken from the population. It summarizes how the average scores from different samples are distributed. If you took several samples of 30 students each and calculated the average score for each sample, the distribution of those sample means would be represented here. What is a Z Score? A Z score tells you how many standard deviations a particular value (in this case, a sample mean) is from the mean of the distribution. Steps to Calculate the Z Score of a Sample Mean: 1. Identify the Sample Mean (𝑋ˉ): This is the average of the sample you're working with. 2. Find the Population Mean (𝜇): This is the mean of the entire population from which the sample is drawn. 3. Calculate the Standard Error of the Mean (SEM): The SEM is the standard deviation of the distribution of means and is calculated using: 4. Calculate the Z score Example Calculation: Suppose: Population Mean (μ) = 200 Population Standard Deviation (σ) = 48 Sample Size (N) = 64 Sample Mean (Xˉ) = 210 A Z score of 1.67 means that the sample mean of 210 is approximately 1.67 standard deviations above the population mean of 200. The 95% and 99% Confidence Intervals A 95% confidence interval is narrower and gives you a decent level of certainty. A 99% confidence interval is wider, giving you more certainty but less precision. As you increase your confidence level, the interval gets wider because you want to be surer that it includes the true mean.In summary, confidence intervals help you estimate where the true average lies based on your sample, and the percentages indicate how certain you are about that estimate. The 95% and 99% Confidence Intervals A 95% confidence interval is narrower and gives you a decent level of certainty. A 99% confidence interval is wider, giving you more certainty but less precision. As you increase your confidence level, the interval gets wider because you want to be surer that it includes the true mean.In summary, confidence intervals help you estimate where the true average lies based on your sample, and the percentages indicate how certain you are about that estimate.  For 99% Confidence Interval: Z-score for 99%: The Z-score used is 2.58. 1.Calculate Distance: 1.Multiply the Z-score by the standard error: 2.58 X 1.40= 3.61 This number (3.61) tells you how far to go above and below the mean. 2.Upper Limit: 1.Add this distance to the sample mean: 48=3.61= 51.61 Lower Limit: 2.Subtract this distance from the sample mean: 1. 48-3.61= 44.39 Conclusion Confidence Interval: Based on the sample, you can say you are 99% confident that the true population mean (the average for all students) falls between 44.39 and 51.61. Imagine you conducted a study with 25 students to see how many "fillers" (like "um" or "like") they used in a speech. Sample Mean (M): 48 fillers Sample Standard Deviation (A): 7 fillers Sample Size (N): 25 students Conclusion You can say that you are 99% confident that the true average number of fillers used by all students (the population mean) falls between 44.39 and 51.61 fillers. Thank you!