Question

a researcher determined the iq of 42 subjects and obtained the following list
68 70 73 76 77 77 79 83 83 84
85 86 87 89 89 91 91 94 94 95
96 97 99 100 100 102 105 106 108 108
109 110 110 111 112 115 117 117 121 125
127 150
here is the histogram of these values.
determine the quartiles, the iqr, and then use the 1.5 x iqr rule to check for outliers.
q1 =
q3 =
iqr =
a data value must be lower than
to be considered a low outlier (the lower limit)
a data value must be higher than
to be considered a high outlier (the upper limit)
list any data values flagged as outliers. list your answers, separated with commas.
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Explanation:

Step1: Calculate position of Q1

The position of $Q_1$ for $n = 42$ data - points is $L_{Q1}=\frac{1}{4}(n + 1)=\frac{1}{4}(42+1)=10.75$. So, $Q_1$ is the value at the 10 - th position plus 0.75 times the difference between the 11 - th and 10 - th values. The 10 - th value is 84 and the 11 - th value is 85. Then $Q_1=84 + 0.75\times(85 - 84)=84.75$.

Step2: Calculate position of Q3

The position of $Q_3$ is $L_{Q3}=\frac{3}{4}(n + 1)=\frac{3}{4}(42 + 1)=32.25$. So, $Q_3$ is the value at the 32 - nd position plus 0.25 times the difference between the 33 - rd and 32 - nd values. The 32 - nd value is 110 and the 33 - rd value is 111. Then $Q_3=110+0.25\times(111 - 110)=110.25$.

Step3: Calculate IQR

The inter - quartile range $IQR = Q_3−Q_1=110.25−84.75 = 25.5$.

Step4: Calculate lower limit for outliers

The lower limit for outliers is $Q_1-1.5\times IQR=84.75-1.5\times25.5=84.75 - 38.25=46.5$.

Step5: Calculate upper limit for outliers

The upper limit for outliers is $Q_3 + 1.5\times IQR=110.25+1.5\times25.5=110.25+38.25 = 148.5$.

Step6: Identify outliers

Since all the data values in the list are greater than 46.5 and 150 is greater than 148.5, the outlier is 150.

Answer:

$Q_1 = 84.75$
$Q_3 = 110.25$
$IQR = 25.5$
Lower limit for low outliers: 46.5
Upper limit for high outliers: 148.5
Outliers: 150