Download Normal Distribution Teacher Notes

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Normal Distribution
Pages 739-743
A normal distribution shows data that vary randomly from the mean in the
pattern of a bell shaped curve.
Examples of data that tend to have a normal distribution about their mean
include weight, height, and blood pressure.
A normal distribution has a symmetric bell shape centered on the mean:
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In a normal distribution:
68 % of data fall within one standard deviation of the mean
95 % of data fall within two standard deviations of the mean
99.7 % of data fall within three standard deviations of the mean
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Not all data is normally distributed. When a data set is skewed, the data do
not vary predictably from the mean. In such cases it is inappropriate to use
mean and standard deviation to estimate percentages for skewed data.
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Examples:
I.
For a population of female European eels, the mean body length is 21.1 inches
and the standard deviation is 4.7 inches. Sketch a normal curve showing eel
lengths one, two, and three standard deviations from the mean:
7
11.7
16.4
21.1
25.8 30.5
35.2
Use the normal curve you sketched above to help you answer the following:
1. Approximately what percent of female eels are between16.4 and 30.5 inches?
34% + 34% + 13.5% = 81.5%
2. What is the probability that a randomly selected eel is less than 11.7 inches?
The left tail end, so 0.15% + 2.35% = 2.5%
II.
The amount of mustard dispensed from a machine at The Hotdog Emporium is
normally distributed with a mean of 0.9 ounce and a standard deviation of 0.1
ounce. If the machine is used 500 times, approximately how many times will it
be expected to dispense 1 or more ounces of mustard? 0.9 + 0.1 is 1 ounce,
which is mean + one σ. This means the right tail from one σ over is the area of
interest. Adding up, this is 13.5% + 2.35% + 0.15% = 16%. 500 times 16%
means we can expect it will dispense more than 1 ounce about 80 times.
III.
The scores on the Algebra 2 final are normally distributed with a mean of 150
and a standard deviation of 15
a. What percentage of students who took the test scored above 180? This is
mean + two σ which means the area of interest is 0.15% + 2.35% = 2.5%
b. If 250 students took the final exam, approximately how many score
above 135? 135 is mean minus one σ. This means we need to add 50% +
34% which is 84%. 250 times 84% means about 210 students scored
above 135.
c. If 13.6% of students received a B on the final, how can you describe their
scores? Explain. The students who received a B fall between one and
two σ of the mean. When test scores are normalized, we expect the mean
to be a C, so it seems reasonable that the B students would fall within
this range.