A bookstore recorded the number of books purchased by a sample of 25 customers. The freque… — ACT

A bookstore recorded the number of books purchased by a sample of $25$ customers. The frequency table below summarizes the purchases made by $20$ of these customers:

Number of Books Purchased	Frequency
1	6
2	5
3	4
4	3
5	2

The remaining $5$ customers each purchased the exact same number of books, $n$ . If the mean number of books purchased by all $25$ customers is $2.8$ , what is the median number of books purchased by the entire group of $25$ customers?

A
2
B
2.5
3Cevap
D
3.5
E
4

Cevap

The median number of books purchased by the entire group of 25 customers is 3.

The total number of books purchased by all 25 customers is

25 \times 2.8 = 70

. The sum of books for the first 20 customers is

(1 \times 6) + (2 \times 5) + (3 \times 4) + (4 \times 3) + (5 \times 2) = 50

. The remaining 5 customers bought a total of

70 - 50 = 20

books, meaning each bought

20 \div 5 = 4

books (

n = 4

). Updating the frequency table, the number of customers buying 4 books becomes

3 + 5 = 8

. For 25 ordered values, the median is the 13th value. Calculating cumulative frequencies: 6 customers bought 1 book, 11 bought

\le 2

books, and 15 bought

\le 3

books. Therefore, the 13th value is 3.

Adım Adım Çözüm

Calculate the total number of books purchased by all 25 customers.

\text{Total Books} = 25 \times 2.8 = 70

Mean is equal to total sum divided by total count, so total sum equals mean multiplied by count.

Calculate the sum of books purchased by the first 20 customers listed in the table.

(1 \times 6) + (2 \times 5) + (3 \times 4) + (4 \times 3) + (5 \times 2) = 6 + 10 + 12 + 12 + 10 = 50

Multiply each book amount by its corresponding frequency and sum the results.

Determine the value of

n

for the remaining 5 customers.

\text{Remaining Books} = 70 - 50 = 20

, so

5n = 20 \implies n = 4

Subtracting the sum of the 20 customers from the grand total leaves the books purchased by the 5 remaining identical customers.

Update the frequency distribution and locate the median position.

New frequencies: 1 book (6), 2 books (5), 3 books (4), 4 books (3 + 5 = 8), 5 books (2). The median of 25 values is the

\frac{25 + 1}{2} = 13^{\text{th}}

ordered value.

For an odd number of data points

N

, the median position is

\frac{N+1}{2}

Find the 13th value using cumulative frequencies.

1 book: 6 customers (positions 1-6)
2 books: 5 customers (positions 7-11)
3 books: 4 customers (positions 12-15)
Since position 13 falls within the 3-book group, the median is 3.

The cumulative total reaches 11 at 2 books and 15 at 3 books, so the 13th customer is in the 3-book category.

Anahtar Kavram

Descriptive Statistics: Finding Missing Values via Mean and Computing Median from Frequency Tables

Tahmini Süre:2m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru