Posner – Statistics Activities

Random Rectangles Worksheet

1. Look at the following page for a few seconds. What is your best guess for the average area of the rectangles?

_______

2. Look at the following page again. Choose five rectangles that you think are representative. Write down their ID#. Calculate the area for each one.

Then calculate the average area.3.

Rectangle 1 Rectangle 2 Rectangle 3 Rectangle 4 Rectangle 5ID#Area

_______ = Average Area of the five rectangles

4. Look at the following page again. Select five random rectangles using the following table of random digits [or using a computer program]. Write down their ID#. Calculate the area for each one. Then calculate the average area.

Rectangle 1 Rectangle 2 Rectangle 3 Rectangle 4 Rectangle 5ID#Area

_______ = Average Area of the five rectangles

5. Join together with two other students to form groups of three. Write down the average of the five random rectangles each group member has calculated. Take the average of these averages to get the average area for 15 rectangles.

Member 1 Member 2 Member 3Average

_______ = Average Area of the four group members

Posner – Statistics Activities

Stars in the Sky Activity

“Star light, star bright, first star I see tonight,” …but the first star of how many? How many stars are there in the sky? The figure on the next page is a star map and is assumed to represent a grid of stars observed in the sky at a given time. While you can just count the starts, let’s employ statistics to help us out with the task. We are going to take a sample of five squares and use those to extrapolate about the entire population. The grid can be read so that each square gets a number from 00-99 with the tens digit corresponding to the row and the ones digit corresponding to a column. Thus, for example, the second square down from the upper left hand corner is #80.

Consider a convenience sample of the first five squares (starting in the upper left hand corner and going across).

a. How many stars does each square have?b. What is the total number of stars in your sample?c. Multiply this number by 100/5 (or 20) to estimate the total number of stars in the sky.

Now consider a systematic sample of five squares.d. What is the value of k?e. Use Excel to identify the first square. What function did you use? Which square did

you choose?f. Now, choose every kth square from there so a total of five squares are obtained.

Which squares did you choose?g. How many stars does each square have?h. What is the total number of stars in your sample?i. Using this number, estimate the total number of stars in the sky.

Now consider a simple random sample of five squares.j. Use Excel to identify five squares at random. Which squares did you choose?k. How many stars does each square have?l. What is the total number of stars in your sample?m. Using this number, estimate the total number of stars in the sky.

Lastly, consider a stratified random sample of five squares, where the strata are “dense” vs. “sparse” squares (by visual examination only). For simplicity sake, identify 40 squares that look dense (lots of stars) and 60 squares that appear to be sparse (not as many squares).

n. List the 40 squares that you considered to be the most dense.o. Use Excel to choose dense squares for your sample. Which squares did you choose?p. How many stars does each square have?q. Use Excel to choose sparse squares for your sample. Which squares did you choose?r. How many stars does each square have?s. What is the total number of stars in your sample?t. Using this number, estimate the total number of stars in the sky.

Posner – Statistics Activities

From Petocz (1990), Sampling Space: Practical Experiments for Teaching Sampling

Posner – Statistics Activities