91Ó°ÊÓ

The following data give the numbers of text messages sent by a high school student on 40 randomly selected days during 2012: \(\begin{array}{llllllllll}32 & 33 & 33 & 34 & 35 & 36 & 37 & 37 & 37 & 37 \\\ 38 & 39 & 40 & 41 & 41 & 42 & 42 & 42 & 43 & 44 \\ 44 & 45 & 45 & 45 & 47 & 47 & 47 & 47 & 47 & 48 \\ 48 & 49 & 50 & 50 & 51 & 52 & 53 & 54 & 59 & 61\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value 49 fall in relation to these quartiles? b. Determine the approximate value of the 91 st percentile. Give a brief interpretation of this percentile. c. For what percentage of the days was the number of text messages sent 40 or higher? Answer by finding the percentile rank of 40 .

Step by step solution

01

Organize the data

First, the provided data need to be added to a list and sorted in ascending order.

02

Determine quartiles

To determine the quartiles, the sorted data is then divided into four equal parts.The first quartile (Q1) is the median of the first half of the data, the second quartile (Q2) is the median of the whole data set (which is also known as median), and the third quartile (Q3) is the median of the second half of the data set.

03

Calculate interquartile range

The Interquartile Range (IQR) is calculated by subtracting the first quartile from the third quartile.

04

Position of 49

To find out where the value 49 falls in relation to the quartiles, we need to see if it lies within Q1 and Q2, Q2 and Q3, or is greater than Q3.

05

Calculate 91st percentile

The 91st percentile is the value below which 91% of the data fall. We calculate it by taking 0.91 times the total number of data, rounded up to the nearest whole number.

06

Interpret 91st percentile

91% percentile indicates that 91% of the days, the adolescent sent lesser or equal number of messages than this percentile value.

07

Calculate percentile rank of 40

To compute the percentile rank of 40, we have to figure out what percentage of the data values were below 40.

08

Interpret percentile rank of 40

This percentile rank indicates that on this percentage of days, the student sent 40 or fewer messages.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Interquartile Range

The Interquartile Range (IQR) is a measure of statistical dispersion, or how spread out the data is in a set. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). By focusing on the middle 50% of the data, the IQR gives a sense of the distribution's central tendency, excluding outliers. If Q1 is 37 and Q3 is 47, then the IQR is 47 - 37 = 10.
This value helps to understand the range in which the majority of the data points lie, offering insights into variability and anomalies within the data set.

Calculating Percentiles

Percentiles divide a data set into 100 equal parts, indicating the value below which a given percentage of observations fall. The 91st percentile, for instance, represents the value below which 91% of data points can be found. To calculate it, multiply the percentile rank (0.91) by the total number of data points (40), which results in a ranking position that identifies this value.
Understanding percentiles is crucial in comparing individual scores to a larger group, as higher percentiles typically indicate higher performance or frequency.

Understanding Percentile Rank

Percentile rank is used to determine the percentage of scores in a data set that a particular value is equal to or greater than. For example, if you want to find the percentile rank of 40, you need to count how many data points are below 40 and divide that by the total number of data points, then multiply by 100.
This tells you the relative standing of 40 compared to the rest of the data, showing how this score ranks relative to all other scores.

Statistical Analysis Made Simple

Statistical analysis involves collecting and scrutinizing every data sample in a set of items from which samples can be drawn. It's about discovering patterns or trends. Basic statistical tools like mean, median, and mode are used to summarize data. Advanced analysis includes quartile calculations and understanding the distribution with tools like the IQR and percentiles.
Such analyses help in making informed decisions, finding relationships between variables, and predicting future trends based on historical data.