Suppose internet users rated their internet service providers on value, reliability, and speed. These ratings
were combined to get an overall satisfaction score. The scores for 26 different providers (arranged in order
from high to low) are given below.
86 82 72 71 71 71 71 69 68 68 65 65 64
64 64 63 63 62 62 61 59 59 58 57 52 47
(a) Calculate the quartiles and the interquartile range.
(b) Are there outliers in this data set? If so, which observations are outliers? (Enter your answers as a
comma-separated list. If there are no outliers, enter NONE.)
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(a) To calculate the quartiles, we need to determine the median first.
Median: The median is the middle value in the data set. Since there are 26 values, the median is the average of the 13th and 14th values, which are 65 and 64. Therefore, the median is (65 + 64) / 2 = 64.5.
Quartiles:
- Lower Quartile (Q1): The lower quartile is the median of the lower half of the data set. In this case, it is the median of the values from the 1st to the 13th (inclusive). There are 13 values in this range, so the lower quartile is the average of the 6th and 7th values, which are 71 and 71. Therefore, Q1 = (71 + 71) / 2 = 71.
- Upper Quartile (Q3): The upper quartile is the median of the upper half of the data set. In this case, it is the median of the values from the 14th to the 26th (inclusive). There are 13 values in this range, so the upper quartile is the average of the 6th and 7th values, which are 64 and 65. Therefore, Q3 = (64 + 65) / 2 = 64.5.
Interquartile Range (IQR):
The interquartile range is the range of values between the lower and upper quartiles. IQR = Q3 - Q1 = 64.5 - 71 = -6.5.
(b) To identify outliers, we can use the 1.5 * IQR rule. Any value below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR is considered an outlier.
In this case, Q1 - 1.5 * IQR = 71 - 1.5 * (-6.5) = 71 + 9.75 = 80.75.
Q3 + 1.5 * IQR = 64.5 + 1.5 * (-6.5) = 64.5 - 9.75 = 54.75.
There are no observations below 80.75 or above 54.75. Therefore, there are no outliers in this data set.
Answer:
(a) Quartiles: Q1 = 71, Median = 64.5, Q3 = 64.5
Interquartile Range: IQR = -6.5
(b) None
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