Use the given data values (a sample of female arm circumferences in centimeters) to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution. \( 32.3,44.9,40.5,34.2,38.1 \) List the z scores for the normal quantile plot. \( \square . \square, \square . \square \) (Round to two decimal places as needed. Use ascending order.)

1. Sort the data in ascending order: \[ 32.3,\;34.2,\;38.1,\;40.5,\;44.9. \] 2. For a sample of \( n=5 \) observations, the plotting positions are given by \[ p_i = \frac{i-0.5}{n}, \quad i=1,2,\ldots,5. \] Thus, \[ \begin{array}{rl} p_1 & = \frac{1-0.5}{5} = \frac{0.5}{5} = 0.10,\\[1mm] p_2 & = \frac{2-0.5}{5} = \frac{1.5}{5} = 0.30,\\[1mm] p_3 & = \frac{3-0.5}{5} = \frac{2.5}{5} = 0.50,\\[1mm] p_4 & = \frac{4-0.5}{5} = \frac{3.5}{5} = 0.70,\\[1mm] p_5 & = \frac{5-0.5}{5} = \frac{4.5}{5} = 0.90. \end{array} \] 3. To obtain the corresponding \( z \) scores, we use the inverse standard normal distribution function: \[ z_i = \Phi^{-1}(p_i). \] This gives: \[ \begin{array}{rl} z_1 & = \Phi^{-1}(0.10) \approx -1.28,\\[1mm] z_2 & = \Phi^{-1}(0.30) \approx -0.52,\\[1mm] z_3 & = \Phi^{-1}(0.50) = 0,\\[1mm] z_4 & = \Phi^{-1}(0.70) \approx 0.52,\\[1mm] z_5 & = \Phi^{-1}(0.90) \approx 1.28. \end{array} \] 4. The coordinates of the points in the normal quantile plot are the pairs: \[ (-1.28,\, 32.3),\quad (-0.52,\, 34.2),\quad (0.00,\, 38.1),\quad (0.52,\, 40.5),\quad (1.28,\, 44.9). \] 5. To construct the normal quantile plot, plot these points with the \( z \)-scores on the horizontal axis and the corresponding data values on the vertical axis. If the data come from a normal distribution, the points should approximately lie along a straight line. 6. Inspection of the plot (or examining the linearity of the above coordinates) would suggest that these data points do indeed appear to come from a population with a normal distribution. The z scores (in ascending order) for the normal quantile plot are: \[ -1.28,\ -0.52,\ 0.00,\ 0.52,\ 1.28. \]