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