High School

For the following sample:

139.9, 133.8, 81, 111, 136.4, 135.2, 53.2, 103.2, 117.2, 81.8, 119.1, 137.7, 122.7, 95.7, 52.6, 100.5, 82.1, 112.6, 83.3

Use default R settings to calculate the 10% quantile.

If the median of a random variable is [tex]$Md=23$[/tex], then what is the value?

Answer :

To calculate the 10% quantile of a given dataset using R's default settings, you need to follow these steps.

  1. Sort the Data: First, organize the data from smallest to largest. For the given sample:

    [tex]52.6, 53.2, 81.0, 81.8, 82.1, 83.3, 95.7, 100.5, 103.2, 111.0, 112.6, 117.2, 119.1, 122.7, 133.8, 135.2, 136.4, 137.7, 139.9[/tex]

  2. Determine the Position: The 10% quantile is the value below which 10% of the data falls. To find this position in the sorted array, use the formula:

    [tex]P = (n + 1) \times 0.10[/tex]

    where [tex]n[/tex] is the number of data points. For this dataset, [tex]n = 19[/tex].

    [tex]P = (19 + 1) \times 0.10 = 2.0[/tex]

  3. Locate the Value: The position 2.0 suggests that the 10% quantile is the value at the second position in the sorted list.

    Therefore, the 10% quantile is [tex]53.2[/tex].

  4. Relation to Median: The median is the 50% quantile of a distribution. It's given that [tex]Md = 23[/tex], but this value is unrelated to the calculation of the 10% quantile in this context unless a separate data set or distribution is being referred. Typically, the 10% quantile is independent of the median value provided for interpreting random variables.

In conclusion, the 10% quantile of the provided sample data, when calculated using default R settings, is [tex]53.2[/tex]. This value indicates that 10% of the data are less than or equal to [tex]53.2[/tex].

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