Question
When Marques commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 37 minutes and a standard deviation of 4.5 minutes. Using the empirical rule, what percentage of his commutes will be between 23.5 and 50.5 minutes?
Answers
Answered by
GPT 3.5
To find the percentage of his commutes that will be between 23.5 and 50.5 minutes, we first need to calculate the z-scores for each of these times using the formula:
z = (X - μ) / σ
Where:
X = the specific time (23.5 minutes or 50.5 minutes)
μ = the mean (37 minutes)
σ = the standard deviation (4.5 minutes)
For X = 23.5 minutes:
z = (23.5 - 37) / 4.5
z = -13.5 / 4.5
z = -3
For X = 50.5 minutes:
z = (50.5 - 37) / 4.5
z = 13.5 / 4.5
z = 3
Now, we can look up the z-scores in a standard normal table or use a calculator to find the percentage of data that falls between -3 and 3 standard deviations from the mean. The empirical rule states that approximately 99.7% of the data falls within 3 standard deviations of the mean in a normal distribution.
Since -3 to 3 standard deviations cover almost all of the data, we can estimate that approximately 99.7% of Marques' commutes will be between 23.5 and 50.5 minutes.
z = (X - μ) / σ
Where:
X = the specific time (23.5 minutes or 50.5 minutes)
μ = the mean (37 minutes)
σ = the standard deviation (4.5 minutes)
For X = 23.5 minutes:
z = (23.5 - 37) / 4.5
z = -13.5 / 4.5
z = -3
For X = 50.5 minutes:
z = (50.5 - 37) / 4.5
z = 13.5 / 4.5
z = 3
Now, we can look up the z-scores in a standard normal table or use a calculator to find the percentage of data that falls between -3 and 3 standard deviations from the mean. The empirical rule states that approximately 99.7% of the data falls within 3 standard deviations of the mean in a normal distribution.
Since -3 to 3 standard deviations cover almost all of the data, we can estimate that approximately 99.7% of Marques' commutes will be between 23.5 and 50.5 minutes.