Question

using a sleep - tracking app, two new parents track the length of time, in minutes, that their newborn baby sleeps uninterrupted from the time they put the baby to sleep at night until the baby wakes. the results for the first 16 days are shown in the dotplot.

90 100 110 120 130 140 150 160 170 180
time (minutes)

a. the value of 110 is at what percentile?

b. which value is at the 75th percentile?

1. sienna says that the maximum value in any data set is always at the 100th percentile. do you agree? explain.

Explanation:

Step1: Count data values less than 119

There are 7 data - values less than 119 in the dot - plot.

Step2: Calculate the percentile for 119

Percentile of 119=$\frac{7}{16}\times100 = 43.75$

Step3: Find the position for 75th percentile

$n = 16$, position of 75th percentile$=0.75\times16=12$

Step4: Locate the value at 75th percentile

Counting from the smallest value in the dot - plot, the 12th value is 150.

Step5: Analyze the statement about 100th percentile

The 100th percentile represents a value such that 100% of the data is less than or equal to it. In a data set, if there are no ties for the maximum value, the maximum value is at the 100th percentile. But if there are multiple maximum values, the maximum value is not at the 100th percentile in the strict sense of the percentile definition. For example, if half of the data points are the maximum value, the maximum value is at the 50th percentile. So, we do not agree with Sienna's statement.

Answer:

a. 43.75th percentile
b. 150

1. Disagree. The 100th percentile means all data is less than or equal to it. In case of multiple maximum values, the maximum is not at the 100th percentile.