Download Pi-Related Stats Problems

Pi Day Review Packet
Pi is the irrational constant that equals the ratio of the circumference over the diameter of any
Euclidean circle. Measure the circumference (using the yarn as needed) as well as the diameter of at
least 2 different pi day treats. Divide the two to find a numerical approximation of π*, record these
values on the whiteboard.
1) Using the entire class’s data, create a stemplot of the approximations for π. Describe the
shape, center, and spread.
2) Using the entire class’s data, run the appropriate hypothesis test to determine whether our
approximation for the ratio of circumference and diameter is different from π. Carry out all of
the appropriate steps.
3) Construct a 95% confidence interval for the ratio of circumference over diameter. Follow the
steps for a confidence interval (you may skip the conditions since you already checked them in
#1). Does this provide evidence that our circular food really is approximately circular (does the
value of π fall within your interval)?
*Note: You probably noticed that Greek letters are usually used to note population values (µ for mean, σ for deviation…),
and you may have wondered what happened to population proportion (which is p) – statisticians decided that using the
Greek letter for p (π) would be too confusing since it is almost exclusively used to refer to the value of 3.14159…
4) Pi is an irrational number (one which runs on forever without repeating in any form of cyclical
pattern), in essence just as random as our random-numbers table we use in statistics.
a. If we treat the digits of pi as a discrete random variable, complete the probability
distribution table below.
X
P(X)
0
1
2
3
4
5
6
7
8
9
b. Calculate the expected value and deviation for the distribution below.
c. If a relative frequency histogram was created to represent this distribution, what shape
would it have (normal, skewed left, skewed right, uniform, bimodal)
5) “Buffon’s Needle” is a famous problem in the world of probability. If a needle of length n units is
dropped onto lined paper (where the lines are spaced a distance of n units apart), the P(the
2
needle will intersect a line) = 𝜋. You plan to drop 5 needles onto such a surface and count how
many needles cross a line.
a. Describe how this satisfies all the requirements of a binomial experiment
b. Calculate the mean and deviation for this binomial distribution.
c. What is the probability that exactly 2 needles cross the line? More than 3 cross the line?
d. What change(s) would need to be made to the above scenario to make it a geometric
experiment instead of a binomial.
6) Enter the values below into your calculator lists
8
92
8
94
a. Sketch the scatterplot below. Describe the relationship between the variables
8
96
(form, direction, strength)
8
98
8
100
9
110
10
110
10
112
10
114
b. Calculate the correlation coefficient. What does this tell you about the
10
116
relationship between the variables?
10
118
10
120
14
152
14
154
14
156
14
158
15
161
c. Calculate the coefficient of determination. What does this tell you about the
16
170
relationship between the variables?
16
172
16
174
16
176
16
178
4
59
5
68
5
70
d. Create a residual plot and sketch below. What does this tell you about the
6
82
relationship between the variables?
8
104
10
125
12
144
14
162
16
180
18
199
20
220
6
80
8
102
10
123
12
142
14
160
16
178
18
197
20
218