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3 parts
a
7
b
6.9
c
7

Step-by-step solution

8 steps

Here's an explanation of the columns in your frequency table:

  • xix_i (Variable Value): These are the distinct data points or categories.
  • fif_i (Frequency): This shows how many times each xix_i value appears.
  • CF (Cumulative Frequency): This is the running total of frequencies, indicating the count of observations less than or equal to the current xix_i.
  • xifix_i f_i (Product): This is the product of each xix_i and its fif_i, used for calculating the mean.

Let's solve for the mode, mean, and median using the provided table.

The data from the table is:

xix_ifif_iCFxifix_i f_i
2224
4134
65830
771549
931827
1222024
Total20138

Part a· 2 steps

What is the mode?

  1. Identify the highest frequency

    The highest value in the fif_i column is 7.

  2. Find the corresponding xix_i value

    The xix_i value corresponding to the frequency of 7 is 7.
    The mode is 7\boxed{7}.

Part b6.9

Mean

  1. Use the formula for the mean

    The mean (xˉ\bar{x}) is given by the sum of xifix_i f_i divided by the sum of fif_i.

    xˉ=∑xifi∑fi\bar{x} = \frac{\sum x_i f_i}{\sum f_i}
  2. Substitute the sums from the table

    From the table, ∑xifi=138\sum x_i f_i = 138 and ∑fi=20\sum f_i = 20.

    xˉ=13820\bar{x} = \frac{138}{20}
  3. Calculate the mean

    xˉ=6.9\bar{x} = 6.9

    The mean is 6.9\boxed{6.9}.

Part c7

Median

  1. Determine the position of the median

    The total number of observations is N=∑fi=20N = \sum f_i = 20. Since NN is even, the median is the average of the N2\frac{N}{2}-th and (N2+1)(\frac{N}{2}+1)-th values.

    N2=202=10\frac{N}{2} = \frac{20}{2} = 10

    We need the 10th and 11th values.

  2. Find the values using cumulative frequency (CF)

    Looking at the CF column:

    • The 10th value falls within the range where CF is 15, which corresponds to xi=7x_i = 7.
    • The 11th value also falls within this range, corresponding to xi=7x_i = 7. The 10th value is 7. The 11th value is 7.
  3. Calculate the median

    Median=7+72=7\text{Median} = \frac{7+7}{2} = 7

    The median is 7\boxed{7}.

Part d

Draw a frequency table to show the distribution

Here is the frequency table:

xix_ifif_iCFxifix_i f_i
2224
4134
65830
771549
931827
1222024
Total20138
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