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3 parts
a
7
b
6.9
c
7

Step-by-step solution

7 steps

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

  • xix_i (Variable Value): This column lists the distinct values or categories observed in your dataset.
  • fif_i (Frequency): This column shows how many times each specific xix_i value appears in the dataset.
  • CF (Cumulative Frequency): This is the running total of the frequencies. For each row, it represents the total number of observations that are less than or equal to the current xix_i value.
  • xifix_i f_i (Product): This column is the result of multiplying each variable value (xix_i) by its corresponding frequency (fif_i). The sum of this column is used to calculate the mean.

Now, let's calculate the mode, mean, and median, and then present the frequency table.

The data from the table is:

xix_ifif_iCF (calculated)xifix_i f_i
2224
4134
65830
771549
931827
1222024
Total20138

Part a· 2 steps

What is the mode?

  1. Identify the highest frequency

    Looking at the fif_i column, the highest frequency is 7.

  2. Determine the corresponding xix_i value

    The xix_i value that corresponds 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 calculated by dividing the sum of the products xifix_i f_i by the sum of the frequencies 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
    ∑fi=20\sum f_i = 20

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

    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 an even number, 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

    So, we need to find the 10th and 11th values.

  2. Use cumulative frequencies to find the corresponding values

    Using the calculated cumulative frequency (CF) column:

    • The first 2 values are 2.
    • The values from the 3rd to the 3rd are 4.
    • The values from the 4th to the 8th are 6.
    • The values from the 9th to the 15th are 7. Therefore, the 10th value is 7. And the 11th value is also 7. The median is the average of these two values:
    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 showing the distribution, with the correct cumulative frequencies:

xix_ifif_iCFxifix_i f_i
2224
4134
65830
771549
931827
1222024
Total20138
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