Math Problem Statement

Groups of six people are chosen at random and the number, x , of people in each group who normally wear glasses is recorded. The table gives the results from 200 groups. Number wearing glasses, x 0 1 2 3 4 5 6 Number of occurrences 17 53 65 45 18 2 0 a. Calculate, from the above data, the mean value of x

Solution

To calculate the mean value of xx from the given data, we follow these steps:

Step 1: Multiply each value of xx by its corresponding number of occurrences.

This gives us the total contribution of each group to the overall sum.

Sum of x=(0×17)+(1×53)+(2×65)+(3×45)+(4×18)+(5×2)+(6×0)\text{Sum of } x = (0 \times 17) + (1 \times 53) + (2 \times 65) + (3 \times 45) + (4 \times 18) + (5 \times 2) + (6 \times 0)

=0+53+130+135+72+10+0= 0 + 53 + 130 + 135 + 72 + 10 + 0

=400= 400

Step 2: Divide the total sum by the total number of groups (200) to get the mean value of xx.

Mean value of x=400200=2\text{Mean value of } x = \frac{400}{200} = 2

So, the mean number of people wearing glasses in each group is 2\boxed{2}.

Would you like more details, or do you have any questions?

Here are some related questions to deepen your understanding:

  1. How would you calculate the variance and standard deviation for this data?
  2. What is the mode of the number of people wearing glasses based on this data?
  3. What is the median number of people wearing glasses in these groups?
  4. How would you graph this data as a probability distribution?
  5. How could this data be modeled using a binomial distribution?

Tip: The mean of a data set gives a measure of the central tendency, but other statistics like variance or standard deviation provide insights into the spread of the data.

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Math Problem Analysis

Mathematical Concepts

Statistics
Mean
Data Distribution

Formulas

Mean = (Sum of x * occurrences) / Total number of occurrences

Theorems

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Suitable Grade Level

Grades 8-10