Question

the ages in years are recorded for a sample of 697 visitors to a museum exhibit. summary statistics were computed based on this sample of data, and they are included in the table below. a boxplot of the data set is also given below. what can we conclude based on this information?

mean standard deviation minimum q1 median q3 maximum
30.3 13.1 10 20 26 39 77

its possible that none of the individuals in the sample were exactly 26 years old.
on average, the individuals in the sample have ages that are 13.1 years away from the median age.
the range of the middle 50% of the distribution is equal to approximately 13 years.
the percentage of exhibit attendees who are older than 26 years of age is larger than the percentage who are younger than 26 years of age.
none of the above interpretations are correct.

Explanation:

Step1: Recall median concept

The median is the middle - value when data is ordered. It's possible that no data point has the exact value of the median in a sample. Here, median = 26, and it's possible that none of the 697 visitors are exactly 26 years old.

Step2: Analyze standard - deviation misinterpretation

The standard deviation (13.1) measures the average distance of data points from the mean, not from the median. So, the statement about the average distance from the median is incorrect.

Step3: Calculate inter - quartile range

The inter - quartile range (IQR), which represents the range of the middle 50% of the data, is calculated as \(IQR = Q_3 - Q_1\). Given \(Q_1 = 20\) and \(Q_3=39\), \(IQR=39 - 20=19
eq13\).

Step4: Analyze median property

Since 26 is the median, 50% of the data is less than or equal to 26 and 50% is greater than or equal to 26. So, the percentage of those older than 26 is not larger than those younger than 26.

Answer:

It's possible that none of the individuals in the sample were exactly 26 years old.