Over a nine-month period, OutdoorGearLab tested hardshell jackets designed for ice climbing, mountaineering, and backpacking. Based on the breathability, durability, ver-satility, features, mobility, and weight of each jacket, an overall rating ranging from 0 (lowest) to 100 (highest) was assigned to each jacket tested. The following data show the results for 20 top-of-the line jackets (OutdoorGearLab website, February 27, 2013).
42 66 67 71 78 62 61 76 71 67
61 64 61 54 83 63 68 69 81 53
a. Compute the mean, median, and mode.
b. Compute the first and third quartiles.
c. Compute and interpret the 90th percentile.

Answer:

(a) The mean is 65.9, the median is, 66.5 and the mode is. 61.

(b) The first and third quartiles are 61 and 71 respectively.

(c) The value of the 90th percentile is, 78.3.

Step-by-step explanation:

The ratings of 20 top-of-the line jackets, arranged in ascending order are as follows:

S = {42 , 53 , 54 , 61 , 61 , 61 , 62 , 63 , 64 , 66 , 67 , 67 , 68 , 69 , 71 , 71 , 76 , 78 , 81 , 83}

(a)

Compute the mean as follows:

[tex]\bar x=\frac{1}{n}\sum\limits^{n}_{i=1}{x_{i}}=\frac{1}{20}\times 1318=65.9[/tex]

Compute the median as follows:

For an even number of values the median is the average of the middle two values.

[tex]M=\frac{10^{th}obs.+11^{th}obs.}{2}=\frac{66+67}{2}=66.5[/tex]

Compute the mode as follows:

The mode of a dataset is the term that occurs most of the time.

[tex]m=61[/tex]

Thus, the mean is 65.9, the median is, 66.5 and the mode is. 61.

(b)

The first quartile is the median of the first half of the data set.

S₁ = {42 , 53 , 54 , 61 , 61 , 61 , 62 , 63 , 64 , 66}

[tex]Q_{1}=\text{Median of }S_{1}=\frac{5^{th}obs.+6^{th}obs.}{2}=\frac{61+61}{2}=61[/tex]

The third quartile is the median of the second half of the data set.

S₂ = {67 , 67 , 68 , 69 , 71 , 71 , 76 , 78 , 81 , 83}

[tex]Q_{3}=\text{Median of }S_{2}=\frac{15^{th}obs.+16^{th}obs.}{2}=\frac{71+71}{2}=71[/tex]

Thus, the first and third quartiles are 61 and 71 respectively.

(c)

The pth percentile is a data value such that at least p% of the data set is less-than or equal to this data value and at least (100 - p)% of the data-set are more-than or equal to this data value.

Use the Excel formula "=PERCENTILE.INC(array,0.9)" to compute the 90th percentile.

The value of the 90th percentile is, 78.3.

This value implies that 90% of the ratings are less than 78.3.

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