Lesson 6: Sampling Distributions | Complete Solution

Answer the following questions showing all work. If you use Minitab Express include the appropriate output (copy + paste) along with an explanation. Round all answers to 3 decimal places. If you have any questions, post them to the discussion board. You may work together and discuss questions on the lab assignments but do not post solutions on the discussion board.   Grading will be done based on completeness.   If a sincere and ingenuous attempt is made to have all questions answered – your score is 100; if any problems are unanswered (and/or software output missing) – your score is 0.  If you answer “I don’t know” (or any variation), it is the same as leaving the question blank and you would earn a 0.

If you have questions, please post them to the Lesson 6 Discussion Board.

1. Vehicle speeds at a certain highway location are believed to be approximately normally distributed with a mean of 60 mph and standard deviation of 6 mph. For each of the following questions, fill in the blank with the appropriate speeds. Apply the Empirical Rule.
a. One vehicle is randomly selected; there is about a 68% that the vehicle’s speed will be between ___ and ___.

b. One vehicle is randomly selected; there is about a 95% chance that the vehicle’s speed will be between ___ and ___.

c. The speeds of randomly selected samples of 16 vehicles will be recorded. For samples of  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 36 vehicles will be recorded. For samples of  vehicles, there is about a 95% chance that a sample’s mean speed will be between ___  and ___.

e. The speeds of one randomly selected sample of 36 vehicles were recorded. The sample mean    was 57 mph. What is the probability, given   and  , that a sample of   would have a mean of 57 mph or less?

f. Given your results from part (e), do you think it is likely that this sample of 36 vehicles came from a population with  ? Why or why not?

2. Suppose that a team of Penn State medical researchers wanted to estimate the true proportion of all teenagers with high blood pressure whose blood pressure decreases after taking calcium supplements. To test this, they plan a clinical trial in which 200 teenagers with high blood pressure were given calcium supplements.  Of the 200 participants, 120 experienced a decrease in blood pressure.
a. Compute   

b. Compute   using   as an estimate of p

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

d. A pharmaceutical company claimed that calcium supplements decrease blood pressure in 64% of all teenagers with high blood pressure.  What is the probability that a population where   would produce a random sample of   with a sample proportion as low (or lower) than the one found in this sample?

e. Given your results from part (d), do you think that the pharmaceutical company’s claim that the population proportion is .64 could be accurate? Or, do you think that their claim is an overestimate of the true effectiveness of calcium supplements for reducing blood pressure in teens? Why?

3. Use the “Grocery Shopping” datafile located in the ‘Data Set’ folder in the main lesson area of the course.   Make certain to include output in your lab activity submission.
Do each of the following:
a.      Use Minitab Express to find the summary statistics of Spending (Stat > Basic Statistics > Display Descriptive Statistics).  Record the sample size, mean, and standard deviation.
 
b.      Calculate the estimated standard error of the mean.  In many situations, such as this, we do not know the population standard deviation, so we use the sample standard deviation instead:   
 
c.       Now assume that the true average spending is   and standard deviation  .  Using this information, follow these steps to finding   when n = 50.
         i.  What is the standard error of the sample mean?

ii.   Fill in the blanks:  The sample mean follows a _____________________ distribution with mean ________ and standard deviation _________. (Hint: you should be using the value from part (i)).
 
iii.  What are the z-scores for 25 and for 28? (You should have two values, one for each.)
 
iv.  Recall that P(a < Z < b) = P(Z < b) - P(Z < a).  Using this and the values you calculated in part (iii), find  in the Standard Normal Table.

v.  Interpret the value in part (iv).

vi.  Does the sample mean value in your dataset (from part (a)) seem unusual to you based on this information?  Why or why not?
 d.      Now suppose that we take a sample of size n = 100 from the population described in part (d).  Compared to when n = 50, would you expect the sample mean to be closer to or farther from the true population mean?  Why?

 
4.       Suppose that at a blood drive a total of n = 300 participants donate blood, and the blood type for each donor is recorded.  A “universal” donor is one who has a blood type of O negative. Though their goal was to have at least 30 universal donors during the drive, a total of just 25 people donated O negative blood.
 
a.      What is the point estimate for the population proportion of universal donors, based on those who actually donated O negative blood? Recall that a point estimate for the population proportion is the sample proportion (p-hat).
 
 

b.      According to the American Red Cross, the true proportion of universal donors in the U.S. population is .07.

i.  Using this, calculate standard error of the sample proportion,

 
ii.  Verify that normal approximation methods would be appropriate for estimating the distribution of the sample proportion (state the assumptions and check if they hold).
 
 

iii.  Find    and interpret. (Hint: the steps are similar to those we took in question 1 part (d).)
 

 
iv.   Given that there were 300 participants in the blood drive, is it surprising that they didn’t meet their goal of at least 30 universal donors?  Why or why not?       
            

5. Using http://onlinestatbook.com/stat_sim/sampling_dist/index.html, select a population distribution that is not normal (i.e., skewed, uniform, or custom). 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 two distributions of sample means here.

b. How are two distributions similar? How are they different?

c. How do your results support the Central Limit Theorem?