Question

1. you are interested in how much time students spend on the internet each day. here are data on the time spent on the internet (in minutes) for a particular day reported by a random sample of 30 students at a large high school.

7 20 24 25 25 28 28 30 32 35
42 43 44 45 46 47 48 48 50 51
72 75 77 78 79 83 87 88 135 151
heres a histogram and summary statistics for these data:
summary statistics
n mean sd min q₁ med q₃ max
30 54.767 32.647 7 30 46.5 77 151
a. are there any outliers? justify your answer.
b. would it be better to use the mean and standard deviation or the median and iqr to describe the center and variability of this distribution? why?

Explanation:

Step1: Calculate the inter - quartile range (IQR)

The formula for \(IQR = Q_{3}-Q_{1}\). Given \(Q_{1} = 30\) and \(Q_{3}=77\), so \(IQR=77 - 30=47\).

Step2: Calculate the lower and upper fences

The lower fence \(=Q_{1}-1.5\times IQR\) and the upper fence \(=Q_{3}+1.5\times IQR\).
Lower fence: \(30-1.5\times47=30 - 70.5=- 40.5\)
Upper fence: \(77+1.5\times47=77 + 70.5 = 147.5\)

Step3: Check for outliers

Values less than the lower fence or greater than the upper fence are outliers. The value \(151>147.5\) is an outlier.

Step4: Analyze the distribution shape for part (b)

The histogram of the data is skewed (right - skewed as there are larger values pulling the tail to the right). The mean is affected by outliers (the value \(151\) in this case), while the median is a more robust measure of central tendency for skewed distributions. The standard deviation is also affected by outliers, and the \(IQR\) is a more robust measure of variability for skewed distributions.

Answer:

a. Yes, there is an outlier. Using the \(1.5\times IQR\) rule, \(IQR = 77-30 = 47\). Lower fence \(=30-1.5\times47=-40.5\), upper fence \(=77 + 1.5\times47=147.5\). Since \(151>147.5\), \(151\) is an outlier.
b. It is better to use the median and \(IQR\). The distribution is skewed (right - skewed as seen from the histogram and the presence of the outlier \(151\)). The mean and standard deviation are sensitive to outliers, while the median and \(IQR\) are more robust measures for skewed distributions.