Download Confidence Intervals Act 13 Exploring Sampling

Confidence Intervals Act 13
Exploring Sampling Variability for 2 Population Mean or
2 Population Percentage with a Sampling Distribution
Part I – Columbian coffee verses Brazilian coffee: Difference between two sample means
Our goal is to understand how random samples can estimate the population mean difference
between two groups. To explore this topic we will look at random sample means from two
populations (Columbian mild and Brazilian coffee price data).
Normally we never know the difference between two population means, but for this exercise
we have the population census data. The population mean for the Columbian mild coffee
was 134.338 cents per pound and the population mean for the Brazilian natural coffee was
111.617 cents per pound. If we subtract the Columbian population mean minus the Brazilian
population mean we get 134.338 – 111.617 = +22.721 cents per pound. Our goal is to
compare random sample mean differences with the population mean difference of +22.721
cents per pound.
Open the Sampling Distribution II data on the statistics / data sets page on
www.matt-teachout.org . We have taken 60 random samples from the Columbian coffee
data and 60 random samples from the Brazilian coffee data. Each person will compute 2
sample mean differences. For example, the first person in class will calculate the sample
mean from “Columbian 1” and the sample mean from “Brazilian 1”. Then subtract
Columbian mean – Brazilian mean. Repeat for “Columbian 2” and “Brazilian 2”. Etc. Once
everyone in class has calculated two sample mean differences, go up to the board and put up
magnets for each of your sample mean differences.
Answer the following questions.
1. Describe how the sampling data was collected. Were we comparing means from two
independent groups or was it matched pair? How do you know? Suppose that instead of
randomly selecting 30 prices from the Columbian data and 30 prices from the Brazilian data, we
had the computer randomly select months and then subtracted the prices in those months.
Would that be independent groups or matched pairs? Explain.
2. What were your two sample means? Was your Columbian sample mean or the Brazilian
sample mean more expensive? How much more expensive? When you subtracted
Columbian – Brazilian, did you get a positive or negative value? Remember, the population
difference when we subtract Columbian minus Brazilian is +22.721 . How far was your sample
difference from +22.721? What does this tell us about sampling variability when trying to
estimate a population difference. (Repeat these questions twice for each sample difference
calculated.)
3. In general, what does a positive difference tell us? What does a negative difference tell us?
If the difference came out to be exactly zero, what would that tell us?
4. Now look at the sampling distribution on the board. Each person in class has put up two
magnets (or drawn two dots). (Remember, the population difference when we subtract
Columbian minus Brazilian is +22.721) . Were all the sample differences the same or was there
a lot of variability in sample mean differences? Were all the sample differences exactly the
same as the population value 22.721 or was there a lot of variability from the actual population
mean difference. Discuss the implications on the difficulty of finding an unknown population
mean difference when all you have is one sample from each of your two groups.
5. What is 95% of the 60 dots on the board? Find two values that 95% of the dots fall in
between.
6. Find the shape of the sampling distribution and estimate the center and standard deviation
of the sampling distribution. This is “Standard Error”. (Hint: If you look at the two numbers in
question #5, find the difference and divide by 4 to find an approximate standard deviation.)
7. Does your sample data meet the assumptions for estimating the difference between two
population means? Explain how.
Part II – Flipping a coin with the Left hand verses Right hand: Difference between two sample
percentages.
Our goal is to understand how random samples can estimate the population percentage
(proportion) difference between two groups. To explore this topic we will look at random
sample percentages from two populations, coin flips with the left hand verses coin flips with
the right hand.
Normally we never know the difference between two population percentages, but for this
exercise we do. The population probability of getting tails with the left hand is 0.5 and the
population probability of getting tails with the right hand is also 0.5 . Hence the population
mean difference is 0.
Just because the population difference is zero, does not mean that two random sample
percentage when subtracted will always get zero. Flip the coin 20 times with the right hand
counting how many times you got tails. Calculate the sample percentage of getting tails with
right handed coin tosses. (Leave your answer in decimal form.) Now flip the coin 20 times
with the left hand counting how many times you got tails. Calculate the sample percentage
of getting tails with left handed coin tosses. (Leave your answer in decimal form.) Now
subtract your right handed sample percent minus your left handed sample percent. Put your
difference on the board with a magnet or drawn dot. Repeat this process twice so that you
will calculate two sample percent differences.
Answer the following questions.
8. Describe how the sampling data was collected.
9. What were your two sample percentage? Was your right handed sampl e percent or your
left handed sample percent greater? How much more? When you subtracted
right handed sample percent minus left handed sample percent, did you get a positive value, a
negative value or zero? Remember, the population difference was zero. How far was your
sample difference from zero? What does this tell us about sampling variability when trying to
estimate a population difference. (Repeat these questions twice for each sample difference
calculated.)
10. In general, what does a positive difference tell us? What does a negative difference tell us?
If the difference came out to be exactly zero, what would that tell us?
11. Now look at the sampling distribution on the board. Each person in class has put up two
magnets (or drawn two dots). (Remember, the population difference should be zero.) Were all
the sample differences the same or was there a lot of variability in sample percent differences?
Were all the sample differences exactly the same as the population value of zero or was there a
lot of variability from the actual population percent difference. Discuss the implications on the
difficulty of finding an unknown population percentage difference when all you have is one
sample from each of your two groups.
12. Count the dots on the board. What is 95% of the number of dots on the board? Find two
values that 95% of the dots fall in between.
13. Find the shape of the sampling distribution and estimate the center and standard deviation
of the sampling distribution. This is “Standard Error”. (Hint: If you look at the two numbers in
question number 12, find the difference and divide by 4 to find an approximate standard
deviation.)
14. Does your sample data meet the assumptions for estimating the difference between two
population percentages? Explain how.