A local community center surveyed a group of 20 students about the number of hours they vo… — ACT

A local community center surveyed a group of $20$ students about the number of hours they volunteered last month. The table below displays the results of the survey, where $a$ and $b$ represent the number of students in their respective categories.

Hours Volunteered	Number of Students
$2$	$a$
$5$	$b$
$8$	$4$
$12$	$3$
$15$	$2$

If the mean number of hours volunteered per student for this group is $6.6$ , what is the median number of hours volunteered per student for these $20$ students?

A
2.0
B
3.5
5.0Cevap
D
6.6
E
8.0

Cevap

The median number of hours volunteered per student is 5.0.

The correct answer is 5.0. To find this, we first establish two equations from the given information: the total count of students (

a + b + 9 = 20 \implies a + b = 11

) and the mean of the data (

\frac{2a + 5b + 32 + 36 + 30}{20} = 6.6 \implies 2a + 5b = 34

). Solving this system of equations gives

a = 7

and

b = 4

. To find the median of the 20 values, we look at the 10th and 11th sorted values. Since there are seven 2s (positions 1–7) and four 5s (positions 8–11), both the 10th and 11th values are 5, making the median 5.0.

Adım Adım Çözüm

Set up an equation for the total number of students.

a + b = 11

The total number of students in the survey is 20. Summing the frequencies gives

a + b + 4 + 3 + 2 = 20

, which simplifies to

a + b = 11

Set up an equation for the mean of the dataset.

2a + 5b = 34

The mean of the data is 6.6. Using the formula for the weighted mean:

\frac{2a + 5b + 8(4) + 12(3) + 15(2)}{20} = 6.6

. Multiplying both sides by 20 gives

2a + 5b + 98 = 132

, which simplifies to

2a + 5b = 34

Solve the system of linear equations for

a

and

b

a = 7

and

b = 4

Multiply the first equation by 2 to get

2a + 2b = 22

. Subtract this from

2a + 5b = 34

to find

3b = 12 \implies b = 4

. Substitute

b = 4

back into the first equation to find

a = 7

Find the median of the 20 sorted values.

5.0

With 20 data points, the median is the average of the 10th and 11th data values in ascending order. Since there are seven 2s followed by four 5s, the 10th and 11th values are both 5. The average of 5 and 5 is 5.0.

Anahtar Kavram

Determining the median of a grouped frequency distribution by solving for missing frequencies using the total count and the mean.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru