Answer the following questions showing all work.

1. Using http://onlinestatbook.com/stat_sim/sampling_dist/index.html, select a population distribution that is NOT normal (i.e., heavily skewed, uniform, bimodal). Using the same population distribution for each, construct the distribution of sample means for N=2 and N=25.  Take at least 10,000 samples.

a. Include a screen shot of your parent population and your two distributions of sample means 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.

2. For the following questions, assume a normally distributed population. 

a. Given μ=10, σ=2, and n=8, compute the standard error of the mean.

b. Given μ=10, σ=2, and n=200, compute the standard error of the mean.

c. Given μ=10, σ=1, and n=8, compute the standard error of the mean.

d. Given μ=5, σ=2, and n=8, compute the standard error of the mean.

e. How does the standard error of the mean change when the population mean, population standard deviation, and sample size change?

3. Vehicle speeds at a certain highway location are normally distributed with a mean of 55 mph and standard deviation of 10 mph. For each of the following questions, fill in the blank with the appropriate speeds. You may apply the Empirical Rule where appropriate.

Hint: Pay careful attention to whether you are dealing with a distribution of individual observations or a distribution of sample means. The standard deviation that you use will differ for each (i.e., standard deviation versus standard error of the mean).

a. One vehicle is randomly selected; there is about a 68% chance that the vehicle's speed will be between ___ and ___.

b. One vehicle is randomly selected; there is about a 99.7% chance that the vehicle's speed will be between ___ and ___.

c. The speeds of randomly selected samples of 25 vehicles will be recorded. For samples of n=25 vehicles, there is about a 68% chance that a sample's mean speed will be between ___ and ___.

d. The speeds of randomly selected samples of 25 vehicles will be recorded. For samples of n=25 vehicles, there is about a 99.7% chance that a sample's mean speed will be between ___ and ___.

4. ACT scores have a mean (μ) of 21 and a standard deviation (σ) of 5. Suppose that we are taking a simple random sample of 40 students from one high school.

a. Calculate the standard error of the mean.

b. If we were to repeatedly pull samples of 40 individuals from the population of all ACT test takers, the distribution of sample means would have a mean of ____ and a standard deviation of ____.

c. Given the values from part (b), 95% of samples of n=40 will have sample means between ___ and ___.

d. What is the probability that you would pull a random sample of 40 individuals from the population of all test takers and they would have a sample mean of 20 or higher?

e. Suppose that the high school in question boasts that their students (i.e., the population of all of their students) have an average ACT score above the national average of 18. In your sample of 40 students from that school, you compute a sample mean of 20. Is it likely that the mean in the population of all students at this school is less than 18? In other words, do you think these sample results (x-bar = 20) could be due to sampling error? Or, do you think there is evidence to state that the mean ACT score of students at this high school is above 18? Explain your reasoning. Hint: Refer back to your answer from part (d).

5. World Campus wants to estimate the proportion of its students who are over the age of 30. In a sample of 100 World Campus students, 31 students were over the age of 40.

a. Compute the sample proportion (p-hat).

b. Compute the standard error of the sample proportion. Use the sample proportion as an estimate of the population proportion because at this point we do not have a population parameter.

c. Verify that it is appropriate to use normal approximation methods with the data from this study.

d. An article online claims that 25% of all World Campus students are over the age of 30.  What is the probability that a population where p=.25 would produce a random sample of n=100 with a sample proportion as high (or higher) than the one that you computed in part a?

In other words, given that the population proportion (p) is .25, what proportion of samples of n=100 would have a sample proportion greater than the one that you observed in your sample?

You will need to compute the standard error again because now we have a population parameter: p=.25

e. Given your results from part (d), do you think that the article's claim that the population proportion is .25 could be accurate? Or, do you think that their claim is an underestimate of the true proportion of World Campus students who are over the age of 30? Explain your reasoning.