QuestionCheck if $X$ is an outlier for these sets: a. $\mu=12.3, \sigma=2.1, X=8.0$ b. $\mu=21.75, \sigma=7.4, X=15.13$ c. $\mu=51.13, \sigma=5.41, X=41.75$ d. $\mu=14.13, \sigma=1.3, X=10.1$

Studdy Solution

STEP 1

Assumptions1. The data are normally distributed. An outlier is typically defined as a value that is more than3 standard deviations away from the mean3. The mean (μ) and standard deviation (σ) are given for each set of scores4. The value of X is the score we are testing to see if it is an outlier

STEP 2

First, we need to calculate the range within which a value would not be considered an outlier. We can do this by calculating the values of μ -σ and μ +σ.
$Lower\, limit = \mu -\sigma$ $Upper\, limit = \mu +\sigma$

STEP 3

For the first set of scores (a), plug in the given values for μ and σ to calculate the lower and upper limits.
$Lower\, limit =12.3 -3(2.1)$ $Upper\, limit =12.3 +3(2.1)$

STEP 4

Calculate the lower and upper limits for the first set of scores.
$Lower\, limit =12.3 -3(2.1) =.9$ $Upper\, limit =12.3 +3(2.1) =18.7$

STEP 5

Now, compare the value of X to the lower and upper limits. If X is less than the lower limit or greater than the upper limit, it is considered an outlier.
For the first set of scores, X =8.0. Since8.0 is greater than the lower limit (5.9) and less than the upper limit (18.7), X is not an outlier.

STEP 6

Repeat steps3 to5 for the remaining sets of scores.
For the second set of scores (b), calculate the lower and upper limits.
$Lower\, limit =21.75 -3(.4)$ $Upper\, limit =21.75 +3(.4)$

STEP 7

Calculate the lower and upper limits for the second set of scores.
$Lower\, limit =21.75 -3(7.4) = -0.45$ $Upper\, limit =21.75 +3(7.4) =43.95$

STEP 8

For the second set of scores, X =15.13. Since15.13 is greater than the lower limit (-0.45) and less than the upper limit (43.95), X is not an outlier.

STEP 9

For the third set of scores (c), calculate the lower and upper limits.
$Lower\, limit =51.13 -3(5.41)$ $Upper\, limit =51.13 +3(5.41)$

STEP 10

Calculate the lower and upper limits for the third set of scores.
$Lower\, limit =51.13 -3(5.41) =34.9$ $Upper\, limit =51.13 +3(5.41) =67.36$

STEP 11

For the third set of scores, X =41.75. Since41.75 is greater than the lower limit (34.9) and less than the upper limit (67.36), X is not an outlier.

STEP 12

For the fourth set of scores (d), calculate the lower and upper limits.
$Lower\, limit =14. -(.)$ $Upper\, limit =14. +(.)$

STEP 13

Calculate the lower and upper limits for the fourth set of scores.
$Lower\, limit =.13 -3(.3) =10.23$ $Upper\, limit =.13 +3(.3) =18.03$

STEP 14

For the fourth set of scores, X =10.. Since10. is less than the lower limit (10.23), X is considered an outlier.
In conclusion, for the given sets of scores, X is not an outlier for sets a, b, and c, but it is an outlier for set d.

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