Find the values of the 30th and 90th percentiles of the data. Please show your work.
129, 113, 200, 100, 105, 132, 100, 176, 146, 152
There are 10 scores. The 90th percentile is the score with 9 below it.
Similarly, the 30th percentile is the score with 3 below it.
Did you check the related questions below?
Since the 3rd score is 105 and the 4th score is 113, I guess any score x where
105 <= x < 113
would work as the 30th %ile.
Not sure how the percentiles are demarcated in such discrete distributions. Maybe your text discusses the topic...
To find the values of the 30th and 90th percentiles of the given data, we need to arrange the data in ascending order first:
100, 100, 105, 113, 129, 132, 146, 152, 176, 200
To find the 30th percentile, we need to find the value that is larger than 30% (0.30) of the data and smaller than the remaining 70% (1 - 0.30) of the data.
First, calculate the number of observations (n):
n = 10
To find the position of the 30th percentile, multiply the total number of observations by the percentile (0.30) (n * 0.30):
position = 10 * 0.30 = 3
The position is 3, indicating that the value of the 30th percentile lies between the 3rd and the 4th observations. Since there is no exact value, we interpolate to find the value.
Value at the 3rd position = 105
Value at the 4th position = 113
To interpolate, we can use the formula:
Value of the 30th percentile = (Value at the 3rd position) + (Position - Lower position) * (Value at the 4th position - Value at the 3rd position)
Value of the 30th percentile = 105 + (3 - 3) * (113 - 105)
Value of the 30th percentile = 105
Therefore, the value of the 30th percentile is 105.
To find the 90th percentile, follow a similar process. Multiply the total number of observations by the percentile (0.90) (n * 0.90):
position = 10 * 0.90 = 9
The position is 9, indicating that the value of the 90th percentile lies between the 9th and the 10th observations.
Value at the 9th position = 152
Value at the 10th position = 176
Using the interpolation formula:
Value of the 90th percentile = (Value at the 9th position) + (Position - Lower position) * (Value at the 10th position - Value at the 9th position)
Value of the 90th percentile = 152 + (9 - 9) * (176 - 152)
Value of the 90th percentile = 152
Therefore, the value of the 90th percentile is 152.