The Z Score
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Question 1 of 4
1. Question
If a set of scores have a mean of `11` and a standard deviation of `0.9`, sketch the normal curve then find the z-score for `12`.Hint
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`Z` score
`Z=(x-barx)/S`Given Values
Mean `barX= 11`Standard Deviation`=0.9`First, complete the labels of the bell curve by using the mean and standard deviation.For example, start with the mean, `11`. Then add and subtract `0.9` to get the values `1` standard deviation above and below the mean.
Keep adding and subtracting the standard deviation until the labels are completed.
Add the z-score values.
Input the values to the z-score formula to solve for the z-score.`Z` `=` `(x-barx)/S` `=` `(12-11)/0.9` Substitute values `=` `1.11` Simplify `1.11` -
Question 2 of 4
2. Question
Given that the mean is `78.4` and the standard deviation is `3.2`, find the z-scores for `68.2` and `82.4`.-
`68.2` z-score: (-3.19, -3.1875)`82.4` z-score: (1.25)
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`Z` score
`Z=(x-barx)/S`Given Values
Mean `barX= 78.4`Standard Deviation`=3.2`First, complete the labels of the bell curve by using the mean and standard deviation.For example, start with the mean, `78.4`. Then add and subtract `3.2` to get the values `1` standard deviation above and below the mean.
Keep adding and subtracting the standard deviation until the labels are completed.
Add the z-score values.
Input the values to the z-score formula to solve for the z-score of `68.2`.`Z` `=` `(x-barx)/S` `=` `(68.2-78.4)/3.2` Substitute values `=` `(-10.2)/3.2` Simplify `=` `-3.19` Do the same to solve for the z-score of `82.4`.`Z` `=` `(x-barx)/S` `=` `(82.4-78.4)/3.2` Substitute values `=` `4/3.2` Simplify `=` `1.25` `68.2` z-score: `-3.19``82.4` z-score: `1.25` -
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Question 3 of 4
3. Question
Jack had a test for both Math and English. Given the following information, in which of the two subjects did he perform better?
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`Z` score
`Z=(x-barx)/S`First, find the z-score for Math.`Z` `=` `(x-barx)/S` `=` `(86-62)/13` Substitute values `=` `(24)/13` `=` `1.85` First, find the z-score for English.`Z` `=` `(x-barx)/S` `=` `(86-73)/6` Substitute values `=` `13/6` `=` `2.17` In this problem, the better score is determined by how further away it is from the right of the mean.
Visualizing the z-scores, we have:
Therefore, Jack performed better in EnglishEnglish -
Question 4 of 4
4. Question
Women’s heights in a survey had a mean of `166.8`cm, a standard deviation of `3.1`cm, and is normally distributed.(a) Where do most heights almost certainly lie?(b) Find the z-score for the height of `173.5`cm.(c) Find the height for the z-score of `-1.8`.-
(a) (157.5)cm to (176.1)cm(b) z-score: (2.16)(c) height: (161.22)cm
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`Z` score
`Z=(x-barx)/S`Given Values
Mean `barX= 166.8` `cm`Standard Deviation`=3.1` `cm`First, complete the labels of the bell curve by using the mean and standard deviation.For example, start with the mean, `166.8` `cm`. Then add and subtract `3.1` `cm` to get the values `1` standard deviation above and below the mean.
Keep adding and subtracting the standard deviation until the labels are completed.
Add the z-score values.
Solution for (a)`99.7%` of scores lie within `3` standard deviations away from the mean.
Hence, almost all heights measure `157.5` `cm` to `176.1` `cm`For part (b), input the values to the z-score formula to solve for the z-score of the height, `173.5` `cm`.`Z` `=` `(x-barx)/S` `=` `(173.5-166.8)/3.1` Substitute values `=` `6.7/3.1` Simplify `=` `2.16` The actual height of `173.5`cm is equivalent to a z-score of `2.16`.
For part (c), input the known values into the z-score formula to solve for the corresponding height of the z-score, `-1.8`.`Z` `=` `(x-barx)/S` `-1.8` `=` `(x-166.8)/3.1` Substitute values `-1.8``xx3.1` `=` `(x-166.8)/3.1``xx3.1` Multiply both sides by `3.1` `-5.58` `=` `x-166.8` Simplify `-5.58``+166.8` `=` `x-166.8``+166.8` Add `166.8` to both sides `161.22` `=` `x` `x` `=` `161.22` `cm` The z-score of `-1.8` is equivalent to a height of `161.22`cm.
(a) `157.5`cm to `176.1`cm(b) z-score: `2.16`(c) height: `161.22`cm -