Question 1172252: The relative frequencies of blood lead concentrations for two groups of workers in Canada-one
examined in 1979 and the other in 1987-are displayed below.
Blood Lead
(µg/dl)
1979
(%)
1987
(%)
<20 11.5 37.8
20-29 12.1 14.7
30-39 13.9 13.1
40-49 15.4 15.3
50-59 16.5 10.5
60-69 12.8 6.8
70-79 8.4 1.4
80 9.4 0.4
a) In which of the two years do the workers tend to have lower blood lead levels?
b) Compute the cumulative relative frequencies for each group of workers. Use these data
to construct a pair of cumulative frequency polygons.
c) For which group of workers is the distribution of blood lead levels stochastically larger?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's analyze the blood lead concentration data.
**a) Lower Blood Lead Levels**
* By observing the first row of data, we can see that in 1987, 37.8% of workers had blood lead levels below 20 µg/dl, while in 1979, only 11.5% had levels that low.
* Also, as we go down the table, the percentages for higher lead levels are much lower in 1987.
* Therefore, the workers in **1987** tend to have lower blood lead levels.
**b) Cumulative Relative Frequencies and Cumulative Frequency Polygons**
* **Cumulative Relative Frequencies:**
* **1979:**
* <20: 11.5%
* <30: 11.5 + 12.1 = 23.6%
* <40: 23.6 + 13.9 = 37.5%
* <50: 37.5 + 15.4 = 52.9%
* <60: 52.9 + 16.5 = 69.4%
* <70: 69.4 + 12.8 = 82.2%
* <80: 82.2 + 8.4 = 90.6%
* ≥ 80: 90.6 + 9.4 = 100%
* **1987:**
* <20: 37.8%
* <30: 37.8 + 14.7 = 52.5%
* <40: 52.5 + 13.1 = 65.6%
* <50: 65.6 + 15.3 = 80.9%
* <60: 80.9 + 10.5 = 91.4%
* <70: 91.4 + 6.8 = 98.2%
* <80: 98.2 + 1.4 = 99.6%
* ≥ 80: 99.6 + 0.4 = 100%
* **Cumulative Frequency Polygons:**
* To construct the polygons, plot the cumulative frequencies against the upper limit of each blood lead concentration interval.
* The 1987 polygon will be shifted to the left of the 1979 polygon, visually indicating lower blood lead levels.
**c) Stochastic Dominance**
* A distribution is stochastically larger if its cumulative distribution function is always below or equal to the other distribution's cumulative distribution function.
* In this case, the cumulative relative frequencies for 1987 are consistently higher than those for 1979.
* Therefore, the **1987** group of workers has a stochastically larger distribution of blood lead levels. This indicates that the 1987 group has a greater probability of having lower blood lead levels.
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