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Chapter 9 Day 2 Warm-up ο If students picked numbers completely at random from the numbers 1 to 20, the proportion of times that the number 7 would be picked is 0.05. When 15 students picked a number βat randomβ from 1 to 20, 3 of them picked the number 7. Identify the parameter and accompanying statistic in this situation. Homework Solutions 1. parameter: ΞΌ = 2.5003; statistic: π₯ = 2.5009 2. Statistic: π = 7.2% 3. Parameter: Ο = 52%; statistic: π = 48% 4. Statistic for both: π₯ = 335 and π₯ = 289 5. We will finish this together! Example ο ο ο Letβs simulate drawing simple random samples from the population of all U.S. residents. Suppose that in fact 60% of the population find clothes shopping timeconsuming and frustrating. Then the true value of the parameter we want to estimate is Ο = .6 We can simulate the population by using our calculators letting 0 to 5 stand for people who find shopping frustrating and 6 to 9 stand for those who do not. Use your calculator to find a simple random sample of 50 people and determine the statistic π. Repeat 10 times. ο Make a histogram of the 10 values of π you found. ο Describe the overall shape of the distribution. ο Now, letβs use the classβ data to find 100 SRSs. ο We will make the histogram together. ο Describe the overall shape of the distribution. How does this compare to your earlier histogram? The Bias of a Statistic ο The statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. ο Find the mean of the 100 observations of π. Mark the mean on the histogram to show its center. Does the statistic π appear to have a large or small bias as an estimate of the population proportion? The variability of the statistic ο The variability of a statistic is described by the spread of its sampling distribution. This spread is determined by the sampling design and size of the sample. Larger samples give smaller spread. The variability of the statistic ο As long as the population is much larger than the sample (say, at least 10 times as large), the spread of the sampling distribution is approximately the same for any population size. ο So in other words... A scoop of 50 M&Ms would have the same distribution regardless of if it was a sample for a truckload of M&Ms or a bucket of M&Ms Bias vs. Variability RESULTS - Summary ο If n is βlargeβ enough (n is approaching N, n is getting bigger) ο Then three things occurβ¦ ο ο ο 1. The shape of the distribution become approximately normal (bell shaped) 2. The mean of the sampling distribution equals the mean of the population. 3. The standard deviation of the sampling distribution will become smaller (determined by your choice of n)