Math Problem Statement

answer the exercises with solution

Solution

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Exercise 1

Ratings given: $4, 5, 8, 7, 9, 8, 7, 3, 6, 7$

(a) Range:
Range = Max - Min
$\text{Range} = 9 - 3 = 6$

(b) Variance:

Find the mean:
$\text{Mean} = \frac{\sum x_i}{n} = \frac{4 + 5 + 8 + 7 + 9 + 8 + 7 + 3 + 6 + 7}{10} = \frac{64}{10} = 6.4$
Variance formula:
$\text{Variance} = \frac{\sum (x_i - \text{mean})^2}{n}$ $\text{Variance} = \frac{(4-6.4)^2 + (5-6.4)^2 + \dots + (7-6.4)^2}{10}$

Calculating: $\text{Variance} = \frac{5.76 + 1.96 + 2.56 + 0.36 + 6.76 + 2.56 + 0.36 + 11.56 + 0.16 + 0.36}{10} = \frac{34.4}{10} = 3.44$

Exercise 2

Frequency distribution:

\hline X & f \\ \hline 10 & 1 \\ 9 & 0 \\ 8 & 2 \\ 7 & 4 \\ 6 & 7 \\ 5 & 11 \\ 4 & 5 \\ 3 & 3 \\ 2 & 4 \\ 1 & 2 \\ 0 & 1 \\ \hline N & 40 \\ \hline \end{array}$$ 1. Calculate the mean: $$\text{Mean} = \frac{\sum f \cdot X}{N}$$ $$\text{Mean} = \frac{(10 \cdot 1) + (9 \cdot 0) + (8 \cdot 2) + \dots + (0 \cdot 1)}{40} = \frac{146}{40} = 3.65$$ 2. Variance formula: $$\text{Variance} = \frac{\sum f \cdot (X - \text{Mean})^2}{N}$$ Calculate $$(X - \text{Mean})^2$$, multiply by $$f$$, sum up, and divide by $$N$$. 3. Standard Deviation: $$\text{SD} = \sqrt{\text{Variance}}$$ --- ### Exercise 3 Grouped frequency distribution: $$\begin{array}{|c|c|} \hline \text{Class Limit} & f \\ \hline 46-48 & 1 \\ 43-45 & 1 \\ 40-42 & 2 \\ 37-39 & 3 \\ 34-36 & 3 \\ 31-33 & 4 \\ 28-30 & 7 \\ 25-27 & 5 \\ 22-24 & 3 \\ 19-21 & 2 \\ 16-18 & 2 \\ 13-15 & 1 \\ 10-12 & 1 \\ \hline N & 35 \\ \hline \end{array}$$ 1. Compute the midpoints ($$m$$) for each class: Example for $$46-48$$: midpoint $$= \frac{46+48}{2} = 47$$. Repeat for all. 2. Calculate the mean: $$\text{Mean} = \frac{\sum f \cdot m}{N}$$ 3. Variance formula: $$\text{Variance} = \frac{\sum f \cdot (m - \text{Mean})^2}{N}$$ 4. Standard Deviation: $$\text{SD} = \sqrt{\text{Variance}}$$ Would you like a detailed table and step-by-step calculations for any of these exercises? Let me know!

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Frequency Distribution
Grouped Data
Measures of Dispersion

Formulas

Range = Max - Min
Variance (Ungrouped Data): Variance = (∑(x_i - mean)^2) / n
Standard Deviation: SD = √Variance
Variance (Frequency Distribution): Variance = (∑f * (x_i - mean)^2) / N
Mean (Grouped Data): Mean = (∑f * m) / N

Theorems

Properties of Variance and Standard Deviation
Mean and its Applications in Data Distribution

Suitable Grade Level

Undergraduate or High School (Grades 11-12)

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