Transcript Powerpoint
PSY 307 – Statistics for the Behavioral Sciences Chapter 9 – Sampling Distribution of the Mean Random Sampling Mean = m Population Sample 1 Mean = x1 Repeated Random Sampling Population Sample 4 Mean = x4 Sample 2 Sample 3 Mean = x2 Mean = x3 Sample 1 Mean = x1 All Possible Random Samples Population Sample 1 Sample 3 Sample 3 Sample 3 Sample 3 Sample 3 Sample 3 Sample 3 Sample n Mean = mx Mean = m Sampling Distribution of the Mean Probability distribution of means for all possible random samples of a given size from some population. Used to develop a more accurate generalization about the population. All possible samples of a given size – not the same as completely surveying the population. Mean of the Sampling Distribution Notation: x = sample mean m = population mean mx = mean of all sample means The mean of all of the sample means equals the population mean. Most sample means are either larger or smaller than the population mean. Standard Error of the Mean A special type of standard deviation that measures variability in the sampling distribution. It tells you how much the sample means deviate from the mean of the sampling distribution (m). Variability in the sampling distribution is less than in the population: sx < s. Central Limit Theorem The shape of the sampling distribution approximates a normal curve. Larger sample sizes are closer to normal. This happens even if the original distribution is not normal itself. Demo Central Limit Theorem: http://onlinestatbook.com/stat_sim/sam pling_dist/index.html Why the Distribution is Normal With a large enough sample size, the sample contains the full range of small, medium & large values. Extreme values are diluted when calculating the mean. When a large number of extreme values are found, the mean may be more extreme itself. The more extreme the mean, the less likely such a sample will occur. Probability and Statistics Probability tells us whether an outcome is common (likely) or rare (unlikely). The proportions of cases under the normal curve (p) can be thought of as probabilities of occurrence for each value. Values in the tails of the curve are very rare (uncommon or unlikely). Z-Test for Means Because the sampling distribution of the mean is normal, z-scores can be used to test sample means. To convert a sample mean to a z-score, use the z-score formula, but replace the parts with sample statistics: Use the sample mean in place of x Use the hypothesized population mean in place of the mean Use the standard error of the mean in place of the standard deviation Z-Test To convert any score to z: z=x–m s Formula for testing a sample mean: z=x–m sx Formula Aleks refers to sx or sM. This is the standard error of the mean. It is easiest to calculate the standard error of the mean using the following formula: Step-by-Step Process State the research problem. State the statistical hypotheses using symbols: H0: m = 500, H1: m ≠ 500. State the decision rule: e.g., p<.05 Do the calculations using formula. Make a decision: accept or reject H0 Interpret the results. Decision Rule The decision rule specifies precisely when the null hypothesis can be rejected (assumed to be untrue). For the z-test, it specifies exact z-scores that are the boundaries for common and rare outcomes: Retain the null if z ≥ -1.96 or z ≤ 1.96 Another way to say this is retain H0 when: -1.96 ≤ z ≤ 1.96 Compare Your Sample’s z to the Critical Values a = .05 .025 .025 COMMON -1.96 1.96 Assumptions of the z-test A z-test produces valid results only when the following assumptions are met: The population is normally distributed or the sample size is large (N > 30). The population standard deviation s is known. When these assumptions are not met, use a different test.