Use Minitab to find the summary statistics of Spending (Stat > Basic Statistics > Display Descriptive

Lab Activity 5 - Sampling Distributions

Use “Grocery Shopping” data

1. (9 points)

a. Use Minitab to find the summary statistics of Spending (Stat > Basic Statistics > Display Descriptive Statistics). Record the sample size, mean, and standard deviation of our sample data. Copy and paste your computer output.

Descriptive Statistics: spending

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum

spending 50 0 25.84 2.28 16.15 2.32 14.19 20.73 34.04 69.49

b.Calculate by hand the estimated standard error of the sampling distribution of the sample mean. In many situations, such as this, we do not know the population standard deviation, so we use the sample standard deviation instead. Show all calculations.

c. Now assume that the true average spending is μ = 28 and standard deviation σ = 10. Using this information, follow the steps below to findP(25 << 28) when n = 50.

i. What is the standard error of the sample mean sampling distribution? Show calculations.

ii. Fill in the blanks: The sampling distribution of the sample mean,, follows a normal_________ distribution with mean and standard deviation (called SE in sampling distributions)(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.). Show all calculations.

iv. Recall that P(a<Z<b) = P(Z<b) - P(Z<a). Using this and the values you calculated in part (iii), find P(25 << 28) 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 (c). Compared to when n = 50, would you expect the sample mean to be closer to or farther from the true population mean? Why?

2. (5 points)

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). Show calculation.

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

1. Using this value (0.07), calculate the standard error of the sample proportion sampling distribution. Show all calculations.

.

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

iii. Find P(phat> 0.10) and interpret. (Hint: the steps are similar to those we took in question 1 part (c).) Show all calculations.

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?

3. (For practice)

To get a better understanding of the Central Limit Theorem as discussed in the lecture notes you can visit and review a simulation program a (this program is also the one in the lecture notes):