Question: (\[ z = \frac{x - \mu}{\sigma} \] \[ x = \mu + z\sigma \] A normal distribution has a…

\(\[ z = \frac{x - \mu}{\sigma} \]

\[ x = \mu + z\sigma \]

A normal distribution has a mean of 80 and a standard deviation of 5. Convert each raw data score to a z-score.

28. 82

29. 92

30. 77

31. 80

32. 75

33. 71\)

Solution

To convert each raw data score to a z-score, use the formula: \[ z = \frac{x - \mu}{\sigma} \] where \( x \) is the raw score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Given: - Mean (\(\mu\)) = 80 - Standard Deviation (\(\sigma\)) = 5 Calculate the z-scores: 28) \( x = 82 \) \[ z = \frac{82 - 80}{5} \] \[ z = \frac{2}{5} = 0.4 \] 29) \( x = 92 \) \[ z = \frac{92 - 80}{5} \] \[ z = \frac{12}{5} = 2.4 \] 30) \( x = 77 \) \[ z = \frac{77 - 80}{5} \] \[ z = \frac{-3}{5} = -0.6 \] 31) \( x = 80 \) \[ z = \frac{80 - 80}{5} \] \[ z = \frac{0}{5} = 0 \] 32) \( x = 75 \) \[ z = \frac{75 - 80}{5} \] \[ z = \frac{-5}{5} = -1 \] 33) \( x = 71 \) \[ z = \frac{71 - 80}{5} \] \[ z = \frac{-9}{5} = -1.8 \]