Questions: Find the 74th percentile, P74, from the following data. 1100 1200 1300 1500 2200 2800 3500 3800 4100 4200 4500 4700 4800 5100 5200 5300 5400 5700 5800 6400 6500 6600 6700 6900 7000 7100 7200 7300 7500 7800 8200 8300 8400 8500 8600 8700 9000 P74=

Solution Steps

The given data set is: \[ \{1100, 1200, 1300, 1500, 2200, 2800, 3500, 3800, 4100, 4200, 4500, 4700, 4800, 5100, 5200, 5300, 5400, 5700, 5800, 6400, 6500, 6600, 6700, 6900, 7000, 7100, 7200, 7300, 7500, 7800, 8200, 8300, 8400, 8500, 8600, 8700, 9000\} \]

To find the \(74^{\text{th}}\) percentile, we use the formula for rank: \[ \text{Rank} = Q \times (N + 1) \] where \(Q = 0.74\) and \(N = 37\) (the number of data points). Thus, \[ \text{Rank} = 0.74 \times (37 + 1) = 0.74 \times 38 = 28.12 \]

The rank \(28.12\) indicates that the \(74^{\text{th}}\) percentile lies between the \(28^{\text{th}}\) and \(29^{\text{th}}\) values in the sorted data set. These values are: \[ X_{\text{lower}} = 7300 \quad \text{and} \quad X_{\text{upper}} = 7500 \]

Using the averaging formula: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{7300 + 7500}{2} = \frac{14800}{2} = 7400.0 \]

Final Answer