BIO 206 Spring 2014 PAGE BIO 206: BIOSTATISTICS A4. SAMPLING DISTRIBUTIONS AND STANDARD ERROR The Sampling Distribution and Standard Error Imagine rolling a die an infinite number of times and then...

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BIO 206 Spring 2014 PAGE BIO 206: BIOSTATISTICS A4. SAMPLING DISTRIBUTIONS AND STANDARD ERROR The Sampling Distribution and Standard Error Imagine rolling a die an infinite number of times and then averaging the outcomes of those rolls. What mean would you anticipate? ___3.5_____ This value is your true population mean (). You will now obtain a sample average (y (bar)) to estimate this value. Part 1: Roll the die 4 times and record your observations. Use these values to compute the statistics listed below: Trial roll observation yi deviation yi – y (bar) deviation2 (yi – y (bar))2 1 2 3 4 y (bar) = ( = sample mean: y (bar) = _____________ record this value with Dr. Soderstrom variance of your sample: s2 = ((yi – y (bar))2 / (n -1) = ______________ standard deviation of your sample: s = √ s2 ______________ standard error of the mean: SE = s / √ n __________________ Find the 95% CI for  Did your 95% CI include the true population mean ()?_________________ Out of 40 students in the class, how many do you predict should have CIs that don’t include ? ______________ How many people in the class actually have 95% confidence intervals that don’t include the true population mean? ______________ Part 2: What happens if you create a second sample, but change the sample size? Obtain your sample mean (y (bar)) when n = 8, to get an estimate of the true population mean () and record your statitics below: Trial roll observation yi deviation yi – y (bar) deviation2 (yi – y (bar))2 1 2 3 4 5 6 7 8 y (bar) = ( = sample mean: y (bar) = _____________ record this value with Dr. Soderstrom variance of your sample: s2 = ((yi – y (bar))2 / (n -1) = ______________ standard deviation of your sample: s = √ s2 ______________ standard error of the mean: SE = s / √ n __________________ Find the 95% CI for  Did your 95% CI include the true population mean ()?_________________ Out of 40 students in the class, how many do you predict should have CIs that don’t include ? ______________ How many people in the class actually have 95% confidence intervals that don’t include the true population mean? ______________ Part 3: Using the sample means (y(bar))s provided by 40 students, Dr. Soderstrom will create frequency distributions meant to simulate the sampling distributions we could create had we run these experiments an infinite number of times (rather than 40). Use these distributions (one for n=4 and one for n=8) to answer the following questions: 1. Compare the sampling distribution means (Y)s when n = 4 and n=8. How do they compare to the true population distribution mean ()? Had we been able to run this experiment an infinite number of times, what value would you expect for Y when n=4? What about when n=8? 2. Compare the sampling distribution standard deviations (Y)s when n = 4 and n=8. How do they compare to the true population distribution standard deviation ()? Had we been able to run this experiment an infinite number of times, what value would you expect for Y when n=4? What about when n=8? 3. Note the shape of each sampling distribution. Did your estimates of standard error (SE) when n=4 versus when n=8 predict the change in shape observed? Explain your answer.