Lab Activity 6 Sampling DistributionsData Set: None but Software is RequiredNumber of Questions: 41 Sampling Distributions of Sample Proportions According to a 2015 Gallup poll 48% of American adults said they were dieting to lose weight. In a sample of 501 students enrolled in World Campus sections of STAT 200, 203 reported dieting to lose weight. a.  Compute the sample proportion, , for this sample of n = 501, STAT 200 students. Round your answer to 4 decimal places. Show all work!b. Compute the standard error of the sample proportion. Round your answer to four decimal places. Remember to show all your work!c.  Verify that it is appropriate to use normal approximation methods with the data from this study.         i.  Write the assumptions that need to be met.Hint: Review the online notes section 6.2        ii.  Verify the assumptions are met in this case.d. If we were to randomly select 501 World Campus students, what is the probability that more than 55% of students in this sample are dieting to lose weight? 2 Sampling Distribution of Sample Mean – Standard ErrorFor the following questions, assume a normally distributed population.a.  Given μ = 200, σ = 20, and n = 100, compute the standard error of the mean. Round your answer to two decimal places. Show all work!b. Given μ = 400, σ = 25, and n = 100, compute the standard error of the mean. Round your answer to two decimal places. Show all work!c.  Given μ = 400, σ = 20, and n = 100, compute the standard error of the mean. Round your answer to two decimal places. Show all work!d. Given μ = 400, σ = 20, and n = 900, compute the standard error of the mean. Round your answer to two decimal places. Show all work!e.  How does the standard error of the mean change when the sample size, population mean, and population standard deviation change?       i.  How does standard error of the mean change when population mean changes?      ii.  How does standard error of the mean change when sample size changes?     iii.  How does standard error of the mean change when population standard deviation changes?3 Central Limit Theorem Using StatKey you are going to construct a sampling distribution for a mean. Select a population distribution that is NOT normal (e.g., the built in Hollywood Movies, Rock Bands, or Baseball Players datasets, or a skewed distribution of your own). Using the same population distribution for each, construct the distribution of sample means for N = 3 and N = 50. Take at least 5,000 samples. a.  Include a screen shot of your parent population and your two sampling distributions here.b. How are your two distributions of sample means similar? How are they different?c.  Describe how your results relate to the Central Limit Theorem.4 Sampling Distributions of Sample Means In the population weights are normally distributed with a mean of 170 pounds and a standard deviation of 42 pounds. Suppose that we are taking a random sample of 500 students from Penn State World Campus STAT 200 in Spring 2021 semester. a.  Calculate the standard error of the mean. Round your answer to 4 decimal places. Show all work!b. If we were to repeatedly pull samples of 100 individuals from the population, the distribution of sample means would have a mean of _____ and a standard deviation of ______.c.  Given the values from part b, 95% of random samples of n = 100 will have sample means between ______ and ______.d. What is the probability that you would pull a random sample of 100 students from the and they would have a sample mean of greater than 175?