Math Problem Statement
Solution
To find the mean and standard deviation of the weights, let's use the provided data points:
Weights: 29.3, 28.2, 29.4, 28.6, 28.6, 28.6 grams
Step 1: Calculate the Mean ()
The formula for the mean is: where are the individual weights and is the number of data points.
Calculating the sum of weights:
Now, divide by the number of weights (6):
Step 2: Calculate the Standard Deviation (s)
The formula for the sample standard deviation is:
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Calculate each deviation from the mean, square it, and sum them up:
Sum of squared deviations:
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Divide by :
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Take the square root:
Answers:
- Mean () = 28.78 grams
- Standard deviation () = 0.47 grams
Would you like further details or explanations?
Here are some additional related questions you might find interesting:
- How would the mean change if an additional bag with a weight of 29.0 grams was added?
- What is the variance of these weights, and how does it relate to the standard deviation?
- How would you interpret the standard deviation in the context of this data?
- If the standard deviation was much higher, what could that indicate about the packaging process?
- How does sample size affect the accuracy of the mean and standard deviation?
Tip: When working with small sample sizes, the sample standard deviation () provides a better estimate of population variability than using the population formula.
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Math Problem Analysis
Mathematical Concepts
Statistics
Mean
Standard Deviation
Formulas
Mean formula: \( \bar{y} = \frac{\sum y_i}{n} \)
Sample Standard Deviation formula: \( s = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n - 1}} \)
Theorems
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Suitable Grade Level
Grades 10-12
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