The amount of Jen's monthly phone bill is normally distributed with a mean of $55 and a standard deviation of $9. Within what range are 68% of her phone bills?
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Jaqueline
To determine the range within which 68% of Jen's phone bills fall, we will use the concept of the standard deviation and the properties of a normal distribution.
Step 1: Understand normal distribution
The normal distribution, also known as the bell curve, is a statistical distribution characterized by its symmetry. In a normal distribution, the mean (average) is located at the center, and the data points are symmetrically distributed around the mean. The standard deviation measures the spread or dispersion of the data points.
Step 2: Use the empirical rule
The empirical rule, also known as the 68-95-99.7 rule, is a guideline used to estimate the percentage of data that falls within a certain number of standard deviations from the mean in a normal distribution. According to the empirical rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Step 3: Apply the empirical rule to Jen's phone bills
In this case, we know that the mean of Jen's phone bills is $55, and the standard deviation is $9. Using the empirical rule, we can determine the range within which 68% of her phone bills fall.
One standard deviation above the mean: $55 + $9 = $64
One standard deviation below the mean: $55 - $9 = $46
Therefore, within one standard deviation of the mean ($46 to $64), approximately 68% of Jen's phone bills will be found.