Distribution Discussion, math homework help
The normal distribution is a symmetric, bell-shaped distribution with a single peak. Its peak corresponds to the mean, median, and mode of the distribution. Its variation is characterized by the standard deviation of the distribution.
A data set satisfying the following criteria is likely to have a nearly normal distribution.
1. Most data values are clustered near the mean giving the distribution a well-defined single peak.
2. Data values are spread evenly around the mean making the distribution symmetric.
3. Larger deviations from the mean are increasingly rare, producing the tapering tails of the distribution.
4. Individual data values result from a combination of many different factors
Which of the following variables would you expect to have a normal or nearly normal distribution?
a. Scores on a very easy test.
b. Show sizes of a random sample of adult women.
Solution
a. Tests have a maximum score of 100% that limits the size of the data values. If a test is very easy, the mean will be high and many scores will be near the maximum. The few lower scores can be spread out well below the mean. We therefore expect the distribution of scores to be left-skewed and not normal.
b. Shoe size depends on foot length which is a human trait determined by many genetic and environmental factors. We therefore expect women’s shoe size to cluster near a mean and become less common farther from the mean, giving the distribution the bell shape of a normal distribution.
About 68% (more precisely, 68.3%), or just over two-thirds, of the data points fall within 1 standard deviation of the mean.
About 95% (more precisely, 95.4%) of the data points fall within 2 standard deviations of the mean.
About 99.7% of the data points fall within 3 standard deviations of the mean.
The 68-95-99.7 rule applies to data values that are exactly 1, 2 , or 3 standard deviations from the mean.
Example
Vending machines can be adjusted to reject coins above and below certain weights. The weights of legal U.S. quarters are normally distributed with a mean of 5.67 grams and a standard deviation of 0.0700 gram. If a vending machine is adjusted to reject quarters that weigh more than 5.81 grams and less than 5.53 grams, what percentage of legal quarters will be rejected by the machine?
Solution
A weight of 5.81 is 0.14 gram, or 2 standard deviations above the mean. A weight of 5.53 is 0.14 gram, or 2 standard deviations below the mean. Therefore, by accepting only quarters within the weight range 5.53 to 5.81 grams, the machine accepts quarters that are within 2 standard deviations of the mean and rejects those that are more than 2 standard deviations from the mean. By the 68-95-99.7 rule, about 95% of legal quarters will be accepted and about 5% of legal quarters will be rejected.
Give of an example for a normal distribution in real life!
