It turns out that the mean, or the average, of all the sample proportions in the histogram above is 0.10. You might recall that 0.10 is p, or the population proportion. All _o_t: the sample proportions in the distribution average to a value that is equal to the claimed population proportion. This is an important characteristic of a sampling distribution. The standard deviation of the above distribution of sample proportions is equal to approximately 0.0424. We explain in our Chapter 18 coverage that the formula you would use to determine the standard deviation of the sampling distribution is: ram-p) T] Here, we know our sample size is n = 50 and the claimed population proportion is p = 0.10, so the standard deviation of the sampling distribution is: /(o.10)(1 0.10) _ 0.09 N What happens to the distribution of sample proportions when we increase our sample size? In other words, how does sample size affect the resulting sampling distribution? Below, we have attempted to compare three sampling distributions. Each distribution is based on taking samples of a particular size from the population of M&M candies and then computing the proportion of green candies in each sample. We start with a smaller sample of size of n = 50 and then increase the sample sizes rst to n = 70 and then to n = 100. Sample size Distribution of Sample Proportions Mean and Standard Deviation n = 50 20000 Mean = 0.10 15000- Standard Deviation = 0.0424 Frequency - 10000 5000 0.0 0.1 0.2 Sample Proportion p n = 70 15000 Mean = 0.10 Frequency 10000- Standard Deviation = 0.0358 0.0 0.1 Sample Proportion p n = 100 15000- Mean = 0.10 10000- Frequency Standard Deviation = 0.0299 500 0.0 0.1 Sample Proportion p 15. As you look at the three distributions presented in the image above, we hope you are noticing that they all have similar shapes, and they all have the same mean. However, the standard deviation is different for each distribution of sample proportions. How is the standard deviation changing as the sample size increases, and why do you think it changes in this way? 16. Let's focus specifically now on the distribution of sample proportions based on samples of size n = 100. Because that distribution is approximately Normal, with a mean of 0.10 and a standard deviation of 0.0299, we can apply the Empirical Rule. Use that rule to fill in the blanks below. 7A. Approximately 68% of the sample proportions of green candies based on samples of size n = 100 are between and B. Approximately 95% of the sample proportions of green candies based on samples of size n = 100 are between and C. Approximately 99.7% of the sample proportions of green candies based on samples of size n = 100 are between and 17. Think about some of the examples we worked through in our lecture coverage of Chapter 18 that involved converting a sample statistic to a z-score and then using Table B. You can even nd a similar example in your textbook, in Chapter 18 (see Example 5 in Chapter 18). Based on what you learned from that coverage, what is the probability of obtaining a sample proportion of green candies that is 0.06 or smaller from a sample size of n = 100? As you answer this, remember, as shown previously in this lab activity, that the sampling distribution for samples of size n = 100 has a mean of 0. 10 and a standard deviation of 0.0299. Please show your work below as you attempt to answer this question. 18. What is the probability of obtaining a sample proportion of green candies that is 0.12 or larger from a sample of size n = 100? Again, please show all work below as you are guring out the answer to this W use the mean and standard deviation given in Question 17 to help you work through the calculations