May is concerned that her "screen time' (the amount of time she spends on her phone) istoo high. Suppose that the amounts of time (in minutes) May used her phone on each ofthe previous 7 days were the following:100,129,63,57, 116,93,103Based on the formula given in lectures, calculate the difference between the 83.8th and21.7th percentiles. i.e.. the 83.8th percentile minus the 21.7th percentile. Do not includeany units in your answer.
Question
May is concerned that her "screen time' (the amount of time she spends on her phone) istoo high. Suppose that the amounts of time (in minutes) May used her phone on each ofthe previous 7 days were the following:100,129,63,57, 116,93,103Based on the formula given in lectures, calculate the difference between the 83.8th and21.7th percentiles. i.e.. the 83.8th percentile minus the 21.7th percentile. Do not includeany units in your answer.
Solution
First, we need to sort the data in ascending order: 57, 63, 93, 100, 103, 116, 129.
Next, we calculate the positions of the 21.7th and 83.8th percentiles. The position of a percentile is given by the formula:
Position = percentile/100 * (n + 1)
where n is the number of data points. For the 21.7th percentile:
Position = 21.7/100 * (7 + 1) = 1.736
For the 83.8th percentile:
Position = 83.8/100 * (7 + 1) = 6.704
Since these positions are not whole numbers, we need to interpolate between the data points to find the percentiles.
For the 21.7th percentile, the position is between the first and second data points (57 and 63). We interpolate as follows:
57 + (63 - 57) * (1.736 - 1) = 57 + 6 * 0.736 = 61.416
For the 83.8th percentile, the position is between the sixth and seventh data points (116 and 129). We interpolate as follows:
116 + (129 - 116) * (6.704 - 6) = 116 + 13 * 0.704 = 125.152
Finally, we subtract the 21.7th percentile from the 83.8th percentile:
125.152 - 61.416 = 63.736
Therefore, the difference between the 83.8th and 21.7th percentiles is approximately 63.736.
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