Use the 68-95-99.7 rule to solve: The amount of Jen's monthly phone bill is normally distributed with a mean of $48 and a standard deviation of $6. Fill in the blanks. 95% of her phone bills are between $_____ and $_____.
To use the 68-95-99.7 rule to solve this problem, we need to understand the concept of standard deviation and how it relates to a normal distribution.
The 68-95-99.7 rule states that for a normal distribution:
- 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.
Given that Jen's monthly phone bill is normally distributed with a mean of $48 and a standard deviation of $6, we can apply the 68-95-99.7 rule to find the range within which 95% of her phone bills fall.
Step 1: Calculate one standard deviation.
One standard deviation is equal to $6, which represents the amount the data typically varies from the mean.
Step 2: Calculate two standard deviations.
Two standard deviations would be equal to 2 * $6 = $12.
Step 3: Determine the range of Jen's phone bills.
Since we are looking for the range within which 95% of her phone bills fall, we need to consider two standard deviations above and below the mean.
Lower limit: Mean - 2 * standard deviation = $48 - $12 = $36
Upper limit: Mean + 2 * standard deviation = $48 + $12 = $60
Therefore, 95% of her phone bills are between $36 and $60.
Note: It's important to keep in mind that the 68-95-99.7 rule is an approximation and works well for a normally distributed data set.