Math Problem Statement

A sample consists of the following n = 7 scores: 5, 0, 4, 5, 1, 2, 4. Compute the mean and standard deviation, find the z-scores, and transform the sample with a new mean and standard deviation.

Solution

Solution to Problem 5.26:


Part (a): Compute the mean and standard deviation

Given sample: n=7n = 7, scores = 5,0,4,5,1,2,45, 0, 4, 5, 1, 2, 4

Step 1: Calculate the Mean (MM) M=ΣXnM = \frac{\Sigma X}{n} M=5+0+4+5+1+2+47=217=3M = \frac{5 + 0 + 4 + 5 + 1 + 2 + 4}{7} = \frac{21}{7} = 3

Step 2: Compute the Sum of Squares (SS) The computational formula for SS is: SS=ΣX2(ΣX)2nSS = \Sigma X^2 - \frac{(\Sigma X)^2}{n} ΣX=21,ΣX2=52+02+42+52+12+22+42=25+0+16+25+1+4+16=87\Sigma X = 21, \quad \Sigma X^2 = 5^2 + 0^2 + 4^2 + 5^2 + 1^2 + 2^2 + 4^2 = 25 + 0 + 16 + 25 + 1 + 4 + 16 = 87 SS=872127=874417=8763=24SS = 87 - \frac{21^2}{7} = 87 - \frac{441}{7} = 87 - 63 = 24

Step 3: Compute the Variance (s2s^2) s2=SSn1s^2 = \frac{SS}{n - 1} s2=2471=246=4s^2 = \frac{24}{7 - 1} = \frac{24}{6} = 4

**Step 4: Compute the Standard Deviation (\

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Standard Deviation
Z-scores
Data Transformation

Formulas

Mean formula: M = ΣX / n
Sum of squares formula: SS = ΣX^2 - (ΣX)^2 / n
Variance formula: s^2 = SS / (n - 1)
Standard deviation formula: s = √s^2
Z-score formula: z = (X - M) / s
Transformation formula: New X = (z * New s) + New M

Theorems

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Suitable Grade Level

Grades 10-12