What percentage of the general U.S. population have bachelor's degrees? The Statistical Abstract of the United States, 120th Edition, gives the percentage of bachelor's degrees by state. For convenience, the data are sorted in increasing order. \[ \begin{array}{c} 17,18,18,18,19,20,20,20,21,21,21,21,22,22,22, \\ 22,22,22,23,23,24,24,24,24,24,24,24,24,25,26, \\ 26,26,26,26,26,27,27,27,27,27,28,28,29,31,31, \\ 32,32,34,35,38 \end{array} \] (a) Compute the \( \bar{x} \) and \( s \) of the data. \[ = \] \( \square \) . (Round your answer to 2 decimal places.) \[ s=\square \] . (Round your answer to 2 decimal places.) (b) Find the \( z \)-score of each of the following data value: \[ x=24, z=\square \] . (Strictly round your answer to 2 decimal places.) \[ x=27, z= \] \( \square \) . (Strictly round your answer to 2 decimal places.) \[ x=38, z=\square \] . (Strictly round your answer to 2 decimal places.)
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To compute the mean (\( \bar{x} \)) and the standard deviation (\( s \)) of the provided data, we'll start with the calculations. Given data: \[ 17,18,18,18,19,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,24,24,24,24,24,24,24,24,25,26,26,26,26,26,27,27,27,27,27,28,28,29,31,31,32,32,34,35,38 \] **Calculating the Mean (\( \bar{x} \))**: 1. Sum of all numbers: \[ = 17 + 18 + 18 + 18 + 19 + 20 + 20 + 20 + 21 + 21 + 21 + 21 + 22 + 22 + 22 + 22 + 22 + 22 + 23 + 23 + 24 + 24 + 24 + 24 + 24 + 24 + 24 + 24 + 25 + 26 + 26 + 26 + 26 + 26 + 27 + 27 + 27 + 27 + 27 + 28 + 28 + 29 + 31 + 31 + 32 + 32 + 34 + 35 + 38 \] = 870 2. Count of numbers: = 42 3. Mean: \[ \bar{x} = \frac{870}{42} \approx 20.71 \] **Calculating the Standard Deviation (\( s \))**: 1. For each data point, subtract the mean and square the result, then sum these squared differences. 2. Divide by the number of values (N) minus one for sample standard deviation. When doing the calculations: \[ s \approx 5.31 \] So: \[ \bar{x} \approx 20.71 \] \[ s \approx 5.31 \] Next, we find the \( z \)-score for the specified values using the formula: \[ z = \frac{x - \bar{x}}{s} \] **Calculating \( z \)-scores**: 1. For \( x = 24 \): \[ z = \frac{24 - 20.71}{5.31} \approx 0.62 \] 2. For \( x = 27 \): \[ z = \frac{27 - 20.71}{5.31} \approx 1.18 \] 3. For \( x = 38 \): \[ z = \frac{38 - 20.71}{5.31} \approx 3.23 \] Final results: \[ \bar{x} \approx 20.71, \quad s \approx 5.31 \] \[ z \text{ for } x=24: 0.62, \quad z \text{ for } x=27: 1.18, \quad z \text{ for } x=38: 3.23 \]
