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Math  /  Data & Statistics

QuestionSuppose the random variable xx is best described by a normal distribution with μ=20\mu=20 and σ=6.8\sigma=6.8. Find the zz-score that corresponds to each of the following xx values.  (a) x=29z=1.32\begin{array}{l} \text { (a) } x=29 \\ z=1.32 \end{array} (b) x=26x=26 z=□z=\square  (c) x=15z=□\begin{array}{l} \text { (c) } x=15 \\ z=\square \end{array} (d) x=10x=10 z=−1.47z=-1.47 (e) x=12x=12 z=□z=\square (f) x=14x=14 z=□z=\square

Studdy Solution

STEP 1

1. The random variable x x follows a normal distribution with mean μ=20 \mu = 20 and standard deviation σ=6.8 \sigma = 6.8 .
2. The z z -score is calculated using the formula: $ z = \frac{x - \mu}{\sigma} \]

STEP 2

1. Calculate the z z -score for each given x x value using the formula.

STEP 3

Calculate the z z -score for x=26 x = 26 .
z=26−206.8=66.8≈0.88z = \frac{26 - 20}{6.8} = \frac{6}{6.8} \approx 0.88

STEP 4

Calculate the z z -score for x=15 x = 15 .
z=15−206.8=−56.8≈−0.74z = \frac{15 - 20}{6.8} = \frac{-5}{6.8} \approx -0.74

STEP 5

Calculate the z z -score for x=12 x = 12 .
z=12−206.8=−86.8≈−1.18z = \frac{12 - 20}{6.8} = \frac{-8}{6.8} \approx -1.18

STEP 6

Calculate the z z -score for x=14 x = 14 .
z=14−206.8=−66.8≈−0.88z = \frac{14 - 20}{6.8} = \frac{-6}{6.8} \approx -0.88
The calculated z z -scores are: (b) x=26, z≈0.88(c) x=15, z≈−0.74(e) x=12, z≈−1.18(f) x=14, z≈−0.88\begin{array}{l} \text{(b) } x=26, \, z \approx 0.88 \\ \text{(c) } x=15, \, z \approx -0.74 \\ \text{(e) } x=12, \, z \approx -1.18 \\ \text{(f) } x=14, \, z \approx -0.88 \\ \end{array}

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