Math Problem Statement
That's incorrect.
First determine the sample mean and then use the mean to find the sample standard deviation. The sample mean and sample standard deviation are given by the formulas
x overbar equals StartFraction Upper Sigma x Subscript i Baseline f Subscript i Over Upper Sigma f Subscript i EndFraction equals StartFraction x 1 f 1 plus x 2 f 2 plus ... plus x Subscript n Baseline f Subscript n Baseline Over f 1 plus f 2 plus ... plus f Subscript n EndFractionx=ΣxifiΣfi=x1f1+x2f2+...+xnfn f1+f2+...+fn and s equals StartRoot StartFraction Upper Sigma left parenthesis x Subscript i Baseline minus x overbar right parenthesis squared f Subscript i Over left parenthesis Upper Sigma f Subscript i Baseline right parenthesis minus 1 EndFraction EndRoots=Σxi−x2fiΣfi−1
where
x Subscript ixi
is the midpoint or value of the ith class,
f Subscript ifi
is the frequency of the ith class, and n is the number of classes.
Solution
You are correct. Let's follow the precise steps to calculate the sample mean and then the sample standard deviation using the given formulas.
Step 1: Calculate the Sample Mean
The formula for the sample mean is:
where is the midpoint of each class and is the frequency of each class.
Step 1.1: Midpoints and Frequency Products
\text{Class Interval} & \text{Midpoint} (x_i) & \text{Frequency} (f_i) & f_i \cdot x_i \\ \hline 10.00 - 19.99 & 15.00 & 5 & 75 \\ 20.00 - 29.99 & 25.00 & 14 & 350 \\ 30.00 - 39.99 & 35.00 & 27 & 945 \\ 40.00 - 49.99 & 45.00 & 20 & 900 \\ 50.00 - 59.99 & 55.00 & 10 & 550 \\ \end{array}$$ ### Step 1.2: Sum of Frequency Products and Total Frequency $$\sum f_i x_i = 75 + 350 + 945 + 900 + 550 = 2820$$ $$\sum f_i = 5 + 14 + 27 + 20 + 10 = 76$$ ### Step 1.3: Calculate the Sample Mean $$\bar{x} = \frac{2820}{76} \approx 37.11$$ ### Step 2: Calculate the Sample Standard Deviation The formula for the sample standard deviation $$s$$ is: $$s = \sqrt{\frac{\sum (x_i - \bar{x})^2 f_i}{\sum f_i - 1}}$$ ### Step 2.1: Calculate $$(x_i - \bar{x})^2$$ and $$(x_i - \bar{x})^2 f_i$$ $$\begin{array}{c|c|c|c|c} \text{Class Interval} & \text{Midpoint} (x_i) & \text{Frequency} (f_i) & (x_i - \bar{x})^2 & (x_i - \bar{x})^2 f_i \\ \hline 10.00 - 19.99 & 15.00 & 5 & (15.00 - 37.11)^2 = 488.3521 & 5 \cdot 488.3521 = 2441.7605 \\ 20.00 - 29.99 & 25.00 & 14 & (25.00 - 37.11)^2 = 146.6521 & 14 \cdot 146.6521 = 2053.1294 \\ 30.00 - 39.99 & 35.00 & 27 & (35.00 - 37.11)^2 = 4.4521 & 27 \cdot 4.4521 = 120.207 \\ 40.00 - 49.99 & 45.00 & 20 & (45.00 - 37.11)^2 = 62.3521 & 20 \cdot 62.3521 = 1247.042 \\ 50.00 - 59.99 & 55.00 & 10 & (55.00 - 37.11)^2 = 321.3521 & 10 \cdot 321.3521 = 3213.521 \\ \end{array}$$ ### Step 2.2: Sum of $$(x_i - \bar{x})^2 f_i$$ $$\sum (x_i - \bar{x})^2 f_i = 2441.7605 + 2053.1294 + 120.207 + 1247.042 + 3213.521 = 9075.66$$ ### Step 2.3: Calculate the Sample Standard Deviation $$s = \sqrt{\frac{9075.66}{76 - 1}} = \sqrt{\frac{9075.66}{75}} = \sqrt{121.0088} \approx 11.00$$ So, the sample standard deviation for the cost is approximately $11.00. --- Do you want more details or have any questions? Here are some related questions for further practice: 1. How would you calculate the variance from this data? 2. What is the range of the monthly pizza costs? 3. How would the sample standard deviation change if the highest class interval had a higher frequency? 4. How do you interpret the standard deviation in the context of this problem? 5. What is the difference between sample and population standard deviation? 6. How would you visualize this data using a histogram? 7. Can you calculate the median monthly pizza cost from this data? 8. What assumptions are we making about the distribution of costs within each class interval? **Tip:** When working with grouped data, ensure accuracy in calculating midpoints and consider using a calculator or software for precise computation of statistical measures.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Statistics
Descriptive Statistics
Sample Mean
Sample Standard Deviation
Formulas
Sample Mean formula
Sample Standard Deviation formula
Theorems
-
Suitable Grade Level
Advanced High School or College
Related Recommendation
Calculate Mean and Standard Deviation for Grouped Data: Step-by-Step Guide
Grouped Data: Calculating Mean, Median, Mode, and Standard Deviation for Central Tendency
Standard Deviation of Grouped Data: Step-by-Step Guide and Example
Calculating Sample Mean and Sample Standard Deviation for Data Set
How to Calculate Sample Standard Deviation and Round to Two Decimals