Use the 68-95-99.7 rule to solve the problem. - The amount of Jen's monthly phone bill is normally distributed with a mean of $53 and a standard deviation of $11. What percentage of her phone bills are between $20 and $86?
Z = (score-mean)/SD
Z = (20-53)/11 = -33/11 = -3
Calculate the other Z score and use the rule to calculate percentage between the two. Remember that the rules apply in both directions.
Or you could use a table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
99.7
To solve this problem using the 68-95-99.7 rule, also known as the empirical rule, we need to calculate the z-scores for $20 and $86, and then find the corresponding percentages.
Step 1: Calculate the z-scores.
The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the individual value, μ is the mean, and σ is the standard deviation.
z-score for $20:
z1 = ($20 - $53) / $11
z-score for $86:
z2 = ($86 - $53) / $11
Step 2: Look up the corresponding percentages.
Using the z-score values, we can look up the corresponding percentages from a standard normal distribution table or by using a calculator or software.
The 68-95-99.7 rule states that:
- Approximately 68% of the data falls within one standard deviation of the mean,
- Approximately 95% falls within two standard deviations of the mean,
- Approximately 99.7% falls within three standard deviations of the mean.
Therefore, to find the percentage between $20 and $86, we need to find the percentage of data within two standard deviations.
Step 3: Calculate the percentage.
To calculate the percentage between $20 and $86, we subtract the cumulative percentage of data below $20 from the cumulative percentage of data below $86.
P($20 < x < $86) = P(z1 < z < z2)
Now, you can use a standard normal distribution table or a calculator/software to find the percentages corresponding to the z-scores z1 and z2, and then subtract the cumulative percentages.
Keep in mind that the calculated percentages will be an approximation using the 68-95-99.7 rule.
To solve this problem using the 68-95-99.7 rule, we will calculate the z-scores for the given values and then use those z-scores to find the corresponding percentages.
Step 1: Calculate the z-scores for the given values.
The formula to calculate the z-score is: z = (x - mean) / standard deviation
For the lower value, x = $20:
z1 = (20 - 53) / 11 = -33 / 11 = -3
For the upper value, x = $86:
z2 = (86 - 53) / 11 = 33 / 11 = 3
Step 2: Use the z-scores to find the corresponding percentages.
According to the 68-95-99.7 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.
Since our z-scores are -3 and 3, which are within three standard deviations, we can conclude that the percentage of Jen's phone bills between $20 and $86 is approximately 99.7%.
Therefore, approximately 99.7% of Jen's phone bills are between $20 and $86.