Quick Check Solutions
The average monthly high temperatures in Chicago for \(2016\) are normally distributed with a mean of \(53^{\circ}\) F and a standard deviation of \(22^{\circ}\) F. Draw and label the normal distribution and answer the following questions.
The average monthly high temperatures in Chicago for \(2016\) are normally distributed with a mean of \(53^{\circ}\) F and a standard deviation of \(22^{\circ}\) F. Draw and label the normal distribution and answer the following questions.
a) What is the probability the average high temperature will be between \(31\) degrees and \(75\) degrees?
Since this is within \(1\) standard deviation, it is \(68\%\)
b) What is the probability the average high temperature will be between \(9\) degrees and \(97\) degrees?
Since this is within \(2\) standard deviations, it is \(95\%\)
c) What is the probability the average high temperature will be greater than \(97\) degrees?
Since \(97^{\circ}\) is \(2\) standard deviations, it is \(2.5\%\)
d) What is the probability the average high temperature will be less than \(75\) degrees?
\(50\%\) are less than \(53^{\circ}\) and \(34\%\) is between \(53^{\circ}\) and \(75^{\circ}\) (half of \(68\%\)). So \(84\%\).
e) What is the probability the average high temperature will be between \(75\) degrees and \(97\) degrees?
\(50\%\) are more than \(53^{\circ}\) and \(2.5\%\) are more than \(97^{\circ}\). So \(50\% - 2.5\% = 47.5\%\).
f) What is the range for the middle \(99.7\%\) of average high temperatures?
The middle \(99.7\%\) of temperatures are between \(-13^{\circ}\) and \(119^{\circ}\).
g) The top \(16\%\) of average high temperatures are greater than what amount?
\(16\%\) are greater than \(75^{\circ}\)
h) Give two high temperatures that would be considered outliers.
Any temperature below \(-13^{\circ}\) or over \(119^{\circ}\) would be considered outliers because they are more than \(3\) standard deviations above or below the mean. For example, \(-15^{\circ}\) or \(120^{\circ}\) would be considered outliers.