Lab 5 worksheet Sampling Distributions

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Apr 3, 2024

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Sherley Cambizaca Lab 5: The Distribution of Sample Means Psych248 Part A. Welcome to Inferential Statistics! This worksheet is shorter but the concepts are deep. The Distribution of Sample Means is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population. Let’s construct the distribution of sample means of age for all possible samples with a size of two (n = 2) from a statistics class that has five students. Their ages are: 18, 18, 19, 20, and 21. 1. For this population (i.e., the class of five students), what is the mean age? 19.2 Complete the table below by filling in all possible samples with a size of two that can be derived from our population. Then, calculate the sample mean for each possible sample. n 1 n 2 Sample Mean 18 18 18 18 19 18.5 18 20 19 18 21 19.5 18 19 18.5 18 20 19 18 21 19.5 18 20 19.5 18 21 20 18 21 20.5 Copy this table into Excel, save the file as a .csv, and open the file in JASP. Click Descriptives, transfer the Sample Mean variable to the Variables box, click Plots, and check off Distribution plots and Display density. 2. What is the mean of age in this histogram (i.e., the distribution of sample means for age)?_ 19.2 3. Does the mean age from this distribution of sample means match the mean age of the population that you calculated in #1 above? Why or why not? The mean age population does match because the data or calculations are the same. Using the same population, construct the distribution of sample means for all possible samples of size four. n 1 n 2 n 3 n 4 Sample Mean 18 18 19 20 18.75 18 18 19 21 19 18 18 20 21 19.25 18 19 20 21 19.5 18 19 20 21 19.5 4. What is the mean of age in this distribution of sample means? 19.2_ ____ 5. Compared to the sample means from the first table, does the individual sample means from the second table better approximate the population mean? Why or why not? T he sample means from the second table is a better approximate from the population mean because each sample has a larger size (n=4 > n=2) which means it has more data which then provides an accurate approximation. Part B: Let’s visualize the distribution of sampling means with larger samples. Open the following URL: http://onlinestatbook.com/stat_sim/sampling_dist/index.html Click Begin on the top left of the web page to get started. Click the Animated button. Five scores from a normal distribution (first graph) will be sampled and plotted in a histogram (second graph). The mean of the sample will be computed and plotted in

another histogram (third graph). Repeat this 2 or 3 times. The third graph with blue histogram is the distribution of sample means . Click the 10,000 button to draw 10,000 random samples from the population with n=5 scores in each sample. Then answer questions about what you found. 6. Does the shape of the distribution of sample means look approximately normal? Yes 7. Record the value of the mean (3 rd graph) 16.0 and the population mean (top graph) _ 16 _? 8. What are the values for skew = __ -0.02_ ______ and kurtosis = _ -0.01 __ in 3 rd graph? 9. In sum, we see only a small difference between M of sampling dist and pop M, and measures of normality like skew and kurtosis are near zero _which indicates no bias in sample M distrib. A sample statistic is unbiased if the average value for the statistic is approximately equal to the population parameter and is normally distributed (M is about same as μ, and ND so M is unbiased). Is variance unbiased? In the 4 th or bottom graph select Variance, N=5, & click 10,000 samples. 10.What is the shape of the distribution of variances (bottom graph ) ? Positive skew 11. So is the variance a biased or unbiased statistic? __ Biased_ . Do you remember that we calculate SD using n-1 as a correction for this bias The standard deviation of a sampling distribution of M’s is the standard error (SEM) . Standard error estimates the average distance between a sample statistic and population parameter for a particular sample size. In other words, how close is a sample M to the real pop μ? To see the effects of sample size on SEM record the standard error of the mean (sd of 3 rd graph) for each of the sample sizes below, after you draw 10,000 samples. (You can change sample sizes by clicking on the N= drop-down menu below Mean . Draw 10,000 samples for each N.) 12. N=2 ___ 3.50_ _____ N=10 __ 2.25 _______ N=16 __ 1.26_ ____ N=25 _ 0.99 ____ 13.What happens to the standard error of the mean as sample size increases? The standard error of the mean will decrease as the sample size increase. ________ Now, let’s try sampling from populations that are not normally distributed. According to the Central Limit Theorem, the Distribution of Sample Means is almost perfectly normal if the population from which the samples were selected is normal and/or if the number of scores in each sample is ≥30. Change the shape of the parent population (first graph) by clicking on the drop-down menu that says Normal. Look at the shape of the sampling dist of M’s from these populations varying N. For a uniform distribution: 14.Is the sample mean still an unbiased estimate of the population mean? Yes_ ____ 15.Does it look more normal with N = 25 than with N = 2? __ N=25 __ For a skewed distribution: 16.What is the direction of skew for the parent population shown? N=25 ___ 17.According to CLTheorem, how large a sample is needed for the distribution to be normal? 30+

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Lab 5 worksheet Sampling Distributions

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