4630 Problem Set 1: SOLUTIONS

1. a. There should be 6 classes.

b.  Since the largest observation is 3.2 and the smallest is 1.5, the class interval should be approximately (3.2 – 1.5)/6, or 0.28. For ease of calculation, let us use 0.3 as the class interval.

c. 

Class ranges / midpoint / f / f/n / cum. f/n
1.45-1.749 / 1.6 / 11 / 0.275 / 0.275
1.75-2.049 / 1.9 / 11 / 0.275 / 0.550
2.05-2.349 / 2.2 / 7 / 0.175 / 0.725
2.35-2.649 / 2.5 / 6 / 0.150 / 0.875
2.65-2.949 / 2.8 / 4 / 0.100 / 0.975
2.95-3.249 / 3.1 / 1 / 0.025 / 1.000

d.

IQR = 2.4 – 1.7227 = 0.6773

e.

The actual mean (based on raw data) is 2.1.

So S = 0.4286. The actual variance and standard deviation (based on raw data) are 0.2113 and 0.4597, respectively.

2.  The Citizens Banking Company is studying the number of times a particular ATM is used each day. The following table lists the number of times the machine was used over each of the last 30 days:

83 / 64 / 84 / 76 / 84 / 54 / 75 / 59 / 70 / 61
63 / 80 / 84 / 73 / 68 / 52 / 65 / 90 / 52 / 77
95 / 36 / 78 / 61 / 59 / 84 / 95 / 47 / 87 / 60

a.  What is the mean number of times the machine was used each day?

b.  What is the modal number of times the machine was used each day?

The most commonly occurring value is 84.

c.  What is variance and standard deviation?

S = 14.8248

3.

a. Tabulate and graph the relative frequency distribution.

b. On the average, how many calls are there per minute?

c. Calculate the variance and standard deviation.

X / f / f/n
1 / 19 / 0.38
2 / 20 / 0.40
3 / 7 / 0.14
4 / 2 / 0.04
5 / 2 / 0.04

S = 1.0294

4.  The following data are the monthly rental prices for a sample of 10 unfurnished 1-bedroom apartments in Denton and a sample of 10 unfurnished apartments in Lewisville:

Denton / Lewisville
$680 / $550
$460 / $425
$369 / $699
$499 / $835
$539 / $700
$489 / $670
$559 / $610
$664 / $499
$644 / $608
$479 / $655

SOURCE: Apartments.com, 8/10/05

a.  For each set of data compute the mean, median, range, interquartile range, and standard deviation. NOTE: do not group these data.

statistic / Denton / Lewisville
mean / $538.20 / $625.10
median / $519.00 / $632.50
upper quartile / $649.00 / $699.25
lower quartile / $474.25 / $537.25
range / $311.00 / $410.00
st. dev / $99.87 / $115.54

b.  What can be said about unfurnished 1-bedroom apartments renting in Denton versus Lewisville?

The sample average rent is higher in Lewisville, but the variation is also greater. This makes it not totally clear that the true population average rent is higher in Lewisville.

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