﻿ Sampling Distribution of Proportion Worksheet | Problems & Solutions Sampling Distribution of Proportion Worksheet

Sampling Distribution of Proportion Worksheet
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1.
A fashion apparel store gives choice to its clients to order T-shirts as per their personal tastes & preferences in terms of
Size: small, medium, large, extra large
Color: white, ivory, pastel brown, pastel yellow, gray
Texture: pure wool, synthetics, silk, cotton,
Style: plain (for men) & embroidered (for ladies), printed (for children)
How many types of orders can be placed if a client chooses one texture, one size, one color and one style? a. 192 b. 16 c. 420 d. 240

Solution:

There are 4 sizes, 5 colors, 4 textures, and 3 styles to choose from, if a customer chooses one size, one color, one style and one texture

Types of orders possible for a client = Number of possible combinations

Types of orders possible for a client if he chooses one texture, one size, one colour and one style = 4 × 5 × 4 × 3 = 240.

2.
How many 5-digit numbers (not starting with 0) can be made out of digits 0 - 9, if the numbers can be used only once? a. 252 b. 30240 c. 27216 d. 100000

Solution:

There are 5 spaces to fill up and 10 numbers to choose from.

The first digit cannot be a zero, so the first place can be filled by any of the 9 numbers from 1 to 9.

Since the numbers are used only once, the second place can be filled in 9 ways.
[0 to 9 excluding the digit used in first place.]

The third place can be filled in 8 ways.

The fourth place can be filled in 7 ways. The fifth place can be filled in 6 ways.

The total number of 5 digit numbers possible = 9 × 9 × 8 × 7 × 6 = 27216

3.
A television news director wishes to use three news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of ten stories to choose from, how many possilbe ways can the program be set up? a. 6 b. 120 c. 3 d. 720

Solution:

The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taking r objects at a time. It is written as nPr, and the formula is nPr = n!(n-r)!

We have to choose 3 stories from the given 10 stories, since the order is important, this can be done in 10P3 ways.

10P3 = 10!(10-3)!

10P3 = (10× 9× 8× 7×6×5×4×3×2×1)(7×6×5 ×4×3×2×1)

10P3 = 720
[Simplify.]

Hence, the program can be set up in 720 possible ways.

4.
How many combinations of 6 objects are there, taken 4 at a time? a. 15 b. 24 c. 360 d. 4

Solution:

A selection of distinct objects without regard to order is called a combination. The number of combinations of r objects selected from n objects is denoted by nCr, and the formula is nCr = n!(n-r)!r!

In this case order is not important, so we go for combinations.

The number of combinations of 4 objects selected from 6 objects is denoted by 6C4

6C4 = 6!(6-4)! x 4!

6C4 = 6 x 5 x 4 x 3 x 2 x 1(2 x 1) x (4 x 3 x 2 x 1) = 15
[Substitute n = 6, r = 4 in nCr & Simplify.]

Hence, 15 combinations of 6 objects are there, taken 4 at a time.

5.
A committee of 4 men and 3 women is to be chosen from 6 men and 6 women. How many different possibilities are there? a. 300 b. 43200 c. 792 d. 3

Solution:

A selection of distinct objects without regard to order is called a combination. The number of combinations of r objects selected from n objects is denoted by nCr, and the formula is nCr = n!(n-r)!r!

Here, one must select 3 women from 6 women, which can be done in 6C3 ways.

4 men can be selected from 6 men in 6C4 ways.

Using the fundamental counting rule, the total number of different ways = 6C3 × 6C4

6C3 × 6C4 = 6!(6-3)! × 3! × 6!(6-4)! × 4! = 20 × 15 = 300
[Simplify.]

So, there are 300 possible ways to select a committee of 4 men and 3 women.

6.
In a shelf there are 10 different math books, 8 different physics books, and 4 different psychology books. A student must select one book of each type. How many different ways can this be done? a. 32 b. 320 c. 22 d. 80

Solution:

Number of ways a math book can be selected = 10

Number of ways a physics book can be selected = 8

Number of ways a psychology book can be selected = 4

Selecting one book of each type = number of ways of selecting a math book × number of ways of selecting a physics book × number of ways of selecting a psychology book

Selecting one book of each type = 10 × 8 × 4 = 320
[Substitute.]

So, the number of ways in which a student can select one book of each type is 320.

7.
How many different 3-color code stripes can be made on a cloth if each code consists of the colors red, blue, green, violet and yellow? All colors are used only once. a. 243 b. 60 c. 125 d. 15

Solution:

Number of different colors = 5

The first color on the stripe can be choosen in 5 ways.

Number of ways the second color on the stripe can be choosen = 4
[Since colors cannot be repeated, we are left with 4 colors to choose.]

Number of ways the third color on the stripe can be choosen = 3
[Since colors cannot be repeated, we are left with 3 colors to choose.]

Total number of different 3-color code = number of ways of selecting the first stripe × number of ways of selecting the second stripe ×number of ways of selecting the third stripe

Total number of different 3-color code = 5 × 4 × 3 = 60

So, the total number of different 3-color code stripes is 60.

8.
There are three cities A, B, C. There are 5 major roads from city A to city B and 4 major roads from city B to city C. How many different routes can be covered while travelling from city A to city C passing through city B? a. 20 b. 9 c. 4 d. 5

Solution:

Number of ways a road can be selected between city A and city B = 5

Number of ways a road can be selected between city B and city C = 4

Number of different routes that can be covered while travelling from city A to city C passing through city B = Number of ways a road can be selected between city A and city B × Number of ways a road can be selected between city B and city C = 5 × 4 = 20

So, the total number of different routes that can be covered while travelling from city A to city C passing through city B is 20.

9.
In how many ways can a boy select 3 red balls and 4 green balls from a box containing 6 balls of each kind? a. 300 b. 12C7 c. 3 d. 12

Solution:

Number of ways a red ball can be selected = 6C3
[Selecting 3 balls from 6.]

Number of ways a green ball can be selected = 6C4
[Selecting 4 balls from 6.]

Number of ways of selecting 3 red balls and 4 green balls = Number of ways of selecting 3 red balls × Number of ways of selecting 4 green balls = 6C3 × 6C4

Number of ways of selecting 3 red balls and 4 green balls = 20 × 15 = 300
[6C3 = 20 , 6C4 = 15]

So, the number of many ways in which a boy can select 3 red balls and 4 green balls is 300.

10.
A football team manager has to select a team. He has 3 goalkeepers, 6 defenders, 7 midfielders, 6 attackers to choose from. He decides to go for 4 defenders, 3 midfielders, 3 attackers. How many ways can he choose his team? a. 22C11 b. 36 c. 8 d. 31500

Solution:

Number of ways a goalkeeper can be selected = 3C1

Number of ways 4 defenders can be selected from 6 defenders = 6C4

Number of ways 3 midfielders can be selected from 7 midfielders = 7C3

Number of ways 3 attackers can be selected from 6 attackers = 6C3

Total number of ways the team can be choosen = number of ways of selecting a goalkeeper × number of ways of selecting a defender × number of ways of selecting a midfielder × number of ways of selecting an attacker

Total number of ways the team can be choosen = 3C1 × 6C4 × 7C3 × 6C3
[Substitute.]

Total number of ways the team can be choosen = 3 × 15 × 35 × 20 = 31500
[Simplify.]

So, a football team manager can select a team in 31500 ways.