6C - The Normal Distribution
The amount of Jen's monthly phone bill is normally distributed with a mean of $56 and a standard deviation of $12. 68% of her phone bills are between $_____ and $______.
$44 and $68
Scores on a test are normally distributed with a mean of 70 and a standard deviation of 5. What is the percentile for an exam score of 61? Round to the nearest tenth.
3.6
The Empirical Rule for a Normal Distribution
68.3% of the data values fall between -1 and +1 standard deviations of the mean. 95.4% of the data values fall between -2 and +2 standard deviations of the mean. 99.7% of the data values fall within -3 and +3 standard deviations of the mean.
Find the standard score for the given data value. A data value in the 8th percentile.
z = -1.4
Standard Score ( z-score )
Is a measure of how many standard deviations a data value is above or below the mean.
Suppose that the mean salary in a particular profession is $45,000 with a standard deviation of $2,000. To what percentile does a salary of $48,000 correspond?
Find the z-score, then associate that with the percentile from Table 6.4: 93rd
Properties of a Normal Distribution
Symmetric, bell-shaped distribution with a single peak. Its peak corresponds to the mean, median, and mode of the distribution.
Percentiles
The percentage of all data values in a data set that are less than or equal to a given data value.
Find the standard score for the given data value. A data value in the 31st percentile.
z = -0.5
For the data value, find the standard score and the percentile. A data value 2.3 standard deviations below the mean.
z = -2.3, percentile = 1.07
For the data value, find the standard score and the percentile. A data value 0.6 standard deviations above the mean.
z = 0.6, percentile = 72.57
For the data value, find the standard score and the percentile. A data value 1.3 standard deviations above the mean.
z = 1.3, percentile = 90.32