Question

The student will compare and contrast empirical data and a theoretical distribution to determine if Terry Vogel's lap times fit a continuous distribution.

Directions :

Round the relative frequencies and probabilities to four decimal places. Carry all other decimal answers to two places.

Collect the Data

1. Use a stratified sampling method by lap (races 1 to 20) and a random number generator to pick six lap times from each stratum. Record the lap times below for laps two to seven.

135, 130, 131, 129, 126, 133

131, 132, 128, 127, 135, 134

130, 133, 126, 131, 128, 129

132, 135, 134, 128, 130, 131

133, 127, 125, 124, 129, 135

128, 131, 132, 129, 135, 130

Construct a histogram. Make five to six intervals. Sketch the graph using a ruler and pencil. Scale the axes.

3. Calculate the following:

a. x ¯ = _______

b. s = _______

4. Draw a smooth curve through the tops of the bars of the histogram. Write one to two complete sentences to describe the general shape of the curve. (Keep it simple. Does the graph go straight across, does it have a v-shape, does it have a hump in the middle or at either end and so on?)

Analyze the Distribution

Using your sample mean, sample standard deviation, and histogram to help, what is the approximate theoretical distribution of the data?

• X ~ _____(_____,_____)

• How does the histogram help you arrive at the approximate distribution?

Describe the Data

Use the data you collected to complete the following statements.

• The IQR goes from __________ to __________.

• IQR = __________. (IQR = Q3 - Q1)

• The 15th percentile is _______.

• The 85th percentile is _______.

• The median is _______.

• The empirical probability that a randomly chosen lap time is more than 130 seconds is _______.

• Explain the meaning of the 85th percentile of this data.

Theoretical Distribution

Using the theoretical distribution, complete the following statements. You should use a normal approximation based on your sample data.

• The IQR goes from __________ to __________.

• IQR = _______.

• The 15th percentile is _______.

• The 85th percentile is _______.

• The median is _______.

• The probability that a randomly chosen lap time is more than 130 seconds is _______.

• Explain the meaning of the 85th percentile of this distribution.

Answer 1

	X	X2
	135	18225
	130	16900
	131	17161
	129	16641
	126	15876
	133	17689
	131	17161
	132	17424
	128	16384
	127	16129
	135	18225
	134	17956
	130	16900
	133	17689
	126	15876
	131	17161
	128	16384
	129	16641
	132	17424
	135	18225
Sum =	2615	342071

4.

Interquartile Range (IQR): 4.5

1st Quartile (Q₁): 128.5

2nd Quartile (Q₂): 131

3rd Quartile (Q₃): 133

IQR = Q₃ - Q₁ = 133 - 128.5 = 4.5

15^th percentile = 127.5

Solution:

Step 1. Arrange the data in ascending order: 126, 126, 127, 128, 128, 129, 129, 130, 130, 131, 131, 131, 132, 132, 133, 133, 134, 135, 135, 135

Step 2. Compute the position of the p^th percentile (index i):

i = (p / 100) * n), where p = 15 and n = 20

i = (15 / 100) * 20 = 3

Step 3. The index i is an integer ⇒ the 15^th percentile is the average of the values in the 2^th and 3^th positions (127 and 128 respectively)

Answer: the 15^th percentile is (127 + 128) / 2 = 127.5

85^th percentile = 134.5

Solution:

Step 1. Arrange the data in ascending order: 126, 126, 127, 128, 128, 129, 129, 130, 130, 131, 131, 131, 132, 132, 133, 133, 134, 135, 135, 135

Step 2. Compute the position of the p^th percentile (index i):

i = (p / 100) * n), where p = 85 and n = 20

i = (85 / 100) * 20 = 17

Step 3. The index i is an integer ⇒ the 85^th percentile is the average of the values in the 16^th and 17^th positions (134 and 135 respectively)

Answer: the 85^th percentile is (134 + 135) / 2 = 134.5

126   126   127   128   128   129   129   130   130   131   131   131   132   132   133   133   134   135   135   135   

mediam=131+131/2=131