Download Quiz 1 Answer Key STA 13 Fall 09 1. The mean price of 50

Quiz 1 Answer Key
STA 13 Fall 09
1. The mean price of 50 computers in a store is $1000 and the standard deviation is $200.
a. (2) What are the units of measurement and what is the variable of interest?
The variable of interest is the price of the computers. Depending on the interpretation of “units of
measurement”, an acceptable answer for the first part of this question is either “dollars” or “computers”.
b. (1) Is the data qualitative or quantitative?
The data is quantitative because the price of the computers is measured on a natural numerical scale.
c. (3) What can be said about the number of computers that cost between $600 and $1400?
Since nothing is known about the distribution of the price of the computers, we may not use the empirical
rule, which can only be applied if the distribution is mound-shaped and symmetric. Instead, we use
Chebyshev’s rule, which applies to any data set regardless of the shape of the frequency distribution of
the data.
First, we notice that $600 is $400 less than the mean price ($1000), and that $1400 is $400 more than
the mean price. Since the standard deviation is $200, we can deduce that $600 is 2 standard deviations
less than the mean and that $1400 is 2 standard deviations more than the mean. Chebyshev’s rule tells
us that at least 1 − k12 of the prices will fall within k standard deviations of the mean. In this case, k =
2, so at least 43 of the computer prices will fall between $600 and $1400, or at least 37.5 of the 50 prices.
d. (2) If we know further that the distribution of prices is mound-shaped and symmetric, how does the number in
part c change?
The distribution is now mound-shaped and symmetric, so we are allowed to apply the empirical rule. As
in part c, the range from $600 to $1400 is 2 standard deviations from the mean in either direction. By
the empirical rule, we know that approximately 95% of the prices will fall within 2 standard deviations
of the mean (the interval from $600 to $1400). We may also say that approximately 47.5 of the 50 prices
will fall in that interval.
2. (2) The US Environment Protection Agency (EPA) sets limits on the amount of lead permitted in drinking
water. The EPA action level for lead is 0.015 milligrams per liter (mg/L) of water. Under EPA guidelines, if 90%
of a water system’s study samples have lead concentration less than 0.015 mg/L, the water is considered safe for
drinking. In a particular subdivision, the 90th percentile of the study sample had lead concentration of 0.00372
mg/L. Are the customers in this subdivision at risk of drinking water with unhealthy lead levels? Explain.
By the definition of percentiles, we know that 90% of the measurements in the study sample had lead
concentration below 0.00372 mg/L. Since 0.00372 mg/L < 0.015 mg/L, all of the measurements that are
less than 0.00372 mg/L will also be less than 0.015 mg/L. Therefore, at least 90% of the measurements
in the study sample will have lead concentration below 0.015 mg/L. Therefore, we can conclude that
the water is safe for drinking, under EPA guidelines.
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