Question

find the interquartile range and sample standard deviation for the set of data. single family home prices 339,400 345,000 332,600 341,500 325,200 318,500 358,000 321,100 323,700 333,800 351,000 337,700 355,100 339,400 352,000 341,500 337,700 iqr = 11,000 and s = 8,850.79 iqr = 18,300 and s = 12,299.9 iqr = 22,900 and s = 12,466.55 iqr = 19,100 and s = 11,760.5

Explanation:

Step1: Sort the data

First, sort the data set: \(318500, 321100, 323700, 325200, 332600, 333800, 337700, 337700, 339400, 339400, 341500, 341500, 345000, 351000, 352000, 355100, 358000\)

Step2: Find quartiles

The number of data points \(n = 17\).
The median (\(Q_2\)) is the \(\frac{n + 1}{2}=9^{th}\) value. So \(Q_2=339400\)
The lower half of the data is \(318500, 321100, 323700, 325200, 332600, 333800, 337700, 337700\). The median of the lower half (\(Q_1\)) is the \(\frac{8}{2}=4^{th}\) value. So \(Q_1 = 325200\)
The upper half of the data is \(341500, 341500, 345000, 351000, 352000, 355100, 358000\). The median of the upper half (\(Q_3\)) is the \(\frac{7+1}{2}=4^{th}\) value. So \(Q_3=351000\)

Step3: Calculate IQR

\(IQR=Q_3 - Q_1=351000 - 325200 = 25800\) (Wait, let's check the formula again. For \(n = 17\), \(Q_1\) is the \(\frac{n+1}{4}=4.5^{th}\) value (using linear - interpolation). The \(4^{th}\) value is \(325200\) and \(5^{th}\) value is \(332600\). \(Q_1=325200+\frac{332600 - 325200}{2}=328900\). \(Q_3\) is the \(3\times\frac{n + 1}{4}=13.5^{th}\) value. The \(13^{th}\) value is \(345000\) and \(14^{th}\) value is \(351000\). \(Q_3=345000+\frac{351000 - 345000}{2}=348000\). \(IQR = Q_3-Q_1=348000 - 328900=19100\)

Step4: Calculate sample mean \(\bar{x}\)

\(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{318500+321100+\cdots+358000}{17}=\frac{5777000}{17}\approx339823.53\)

Step5: Calculate sample variance \(s^{2}\)

\(s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^{2}}{n - 1}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^{2}=(318500 - 339823.53)^{2}+(321100 - 339823.53)^{2}+\cdots+(358000 - 339823.53)^{2}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^{2}=2307979411.76\)
\(s^{2}=\frac{2307979411.76}{16}\approx144248713.235\)

Step6: Calculate sample standard deviation \(s\)

\(s=\sqrt{s^{2}}=\sqrt{144248713.235}\approx11760.5\)

Answer:

\(IQR = 19100\) and \(s = 11760.5\) (corresponding to the last option \(IQR = 19,100\) and \(s = 11,760.5\))