Z Score Sample 7E7E59

1. **State the problem:** We have a sample of $n=7$ scores: $5, 0, 4, 5, 1, 2, 4$. We need to find the z score for each score in the sample. 2. **Recall the formula for z score:** $$z = \frac{X - \mu}{\sigma}$$ where $X$ is the score, $\mu$ is the mean of the sample, and $\sigma$ is the standard deviation of the sample. 3. **Calculate the mean $\mu$:** $$\mu = \frac{5 + 0 + 4 + 5 + 1 + 2 + 4}{7} = \frac{21}{7} = 3$$ 4. **Calculate the sample standard deviation $\sigma$:** First, find each squared deviation from the mean: $$ (5-3)^2 = 2^2 = 4 $$ $$ (0-3)^2 = (-3)^2 = 9 $$ $$ (4-3)^2 = 1^2 = 1 $$ $$ (5-3)^2 = 2^2 = 4 $$ $$ (1-3)^2 = (-2)^2 = 4 $$ $$ (2-3)^2 = (-1)^2 = 1 $$ $$ (4-3)^2 = 1^2 = 1 $$ Sum of squared deviations: $$4 + 9 + 1 + 4 + 4 + 1 + 1 = 24$$ Since this is a sample, use $n-1=6$ for variance: $$s^2 = \frac{24}{6} = 4$$ Standard deviation: $$\sigma = s = \sqrt{4} = 2$$ 5. **Calculate each z score:** - For $X=5$: $$z = \frac{5 - 3}{2} = \frac{2}{2} = 1$$ - For $X=0$: $$z = \frac{0 - 3}{2} = \frac{-3}{2} = -1.5$$ - For $X=4$: $$z = \frac{4 - 3}{2} = \frac{1}{2} = 0.5$$ - For $X=1$: $$z = \frac{1 - 3}{2} = \frac{-2}{2} = -1$$ - For $X=2$: $$z = \frac{2 - 3}{2} = \frac{-1}{2} = -0.5$$ **Final answers:** - $z$ score for $X=5$ is $1$ - $z$ score for $X=0$ is $-1.5$ - $z$ score for $X=4$ is $0.5$ - $z$ score for $X=1$ is $-1$ - $z$ score for $X=2$ is $-0.5$