Question

1. unit 1 part ii 1e - 1fb

name

1. an ogive shows the cumulative relative frequency of student heights. there are 30 total

students. use the ogive to answer the following questions.
heights of students

a. how many students have a height less than 160 cm?
b. how many students are at least 170 cm tall?
c. what percentage of students are under 180 cm?
d. estimate the median height from the ogive.
e. between which height intervals do the greatest number of students fall?
f. approximately how many students are between 160 cm and 180 cm tall?
g. explain how you could use the ogive to estimate the interquartile range (iqr) of
student heights.

Explanation:

Step1: Find cumulative relative frequency for height < 160 cm

From the ogive, the cumulative relative frequency for height less than 160 cm is approximately 0.3.

Step2: Calculate number of students

Number of students = Cumulative relative frequency × Total number of students. So, \(0.3\times30 = 9\).

Step3: Find cumulative relative frequency for height < 170 cm

From the ogive, the cumulative relative frequency for height less than 170 cm is approximately 0.2.

Step4: Calculate number of students at least 170 cm

Number of students at least 170 cm = Total students - Students with height < 170 cm. So, \(30-(0.2\times30)=30 - 6=24\).

Step5: Find cumulative relative frequency for height < 180 cm

From the ogive, the cumulative relative frequency for height less than 180 cm is approximately 0.8. So the percentage is \(80\%\).

Step6: Estimate median height

The median is the value where cumulative relative frequency is 0.5. From the ogive, this occurs at approximately 165 cm.

Step7: Determine interval with greatest number of students

The greatest number of students fall where the slope of the ogive is steepest. This is between 160 - 170 cm.

Step8: Calculate students between 160 - 180 cm

Cumulative relative frequency at 180 cm is 0.8 and at 160 cm is 0.3. Number of students = \((0.8 - 0.3)\times30=15\).

Step9: Explain IQR estimation

The IQR is \(Q3 - Q1\). \(Q1\) (25th percentile) is the height where cumulative relative frequency is 0.25 and \(Q3\) (75th percentile) is the height where cumulative relative frequency is 0.75. Subtract the two heights.

Answer:

a. 9 students
b. 24 students
c. \(80\%\)
d. 165 cm
e. 160 - 170 cm
f. 15 students
g. Find \(Q1\) (height at 0.25 cumulative relative frequency) and \(Q3\) (height at 0.75 cumulative relative frequency), then \(IQR=Q3 - Q1\)