Sampling distribution involves the proportions of a data element in a given sample.
- <em>The proportion of Good TV set is 0.67</em>
- <em>The number of ways of selecting 5 from 6 TV sets is 6</em>
- <em>The number of ways of selecting 4 from 6 TV sets is 15</em>
<em />
Given
![n = 6](https://tex.z-dn.net/?f=n%20%3D%206)
Sample Space = Good, Good, Defective, Defective, Good, Good
<u>(a) Proportion that are good</u>
From the sample space, we have:
![Good = 4](https://tex.z-dn.net/?f=Good%20%3D%204)
So, the proportion (p) that are good are:
![p = \frac{Good}{n}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7BGood%7D%7Bn%7D)
![p = \frac{4}{6}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7B4%7D%7B6%7D)
![p = 0.67](https://tex.z-dn.net/?f=p%20%3D%200.67)
<u>(b) Ways to select 5 samples (without replacement)</u>
This is calculated using:
![^nC_r = \frac{n!}{(n - r)!r!}](https://tex.z-dn.net/?f=%5EnC_r%20%3D%20%5Cfrac%7Bn%21%7D%7B%28n%20-%20r%29%21r%21%7D)
Where
![r = 5](https://tex.z-dn.net/?f=r%20%3D%205)
So, we have:
![^6C_5 = \frac{6!}{(6 - 5)!5!}](https://tex.z-dn.net/?f=%5E6C_5%20%3D%20%5Cfrac%7B6%21%7D%7B%286%20-%205%29%215%21%7D)
![^6C_5 = \frac{6!}{1!5!}](https://tex.z-dn.net/?f=%5E6C_5%20%3D%20%5Cfrac%7B6%21%7D%7B1%215%21%7D)
![^6C_5 = \frac{6 \times 5!}{1 \times 5!}](https://tex.z-dn.net/?f=%5E6C_5%20%3D%20%5Cfrac%7B6%20%5Ctimes%205%21%7D%7B1%20%5Ctimes%205%21%7D)
![^6C_5 = \frac{6}{1}](https://tex.z-dn.net/?f=%5E6C_5%20%3D%20%5Cfrac%7B6%7D%7B1%7D)
![^6C_5 = 6](https://tex.z-dn.net/?f=%5E6C_5%20%3D%206)
Hence, there are 6 ways
<u>(c) All possible sample space of 4</u>
First, we calculate the number of ways to select 4.
This is calculated using:
![^nC_r = \frac{n!}{(n - r)!r!}](https://tex.z-dn.net/?f=%5EnC_r%20%3D%20%5Cfrac%7Bn%21%7D%7B%28n%20-%20r%29%21r%21%7D)
Where
![r = 4](https://tex.z-dn.net/?f=r%20%3D%204)
So, we have:
![^6C_4 = \frac{6!}{(6 - 4)!4!}](https://tex.z-dn.net/?f=%5E6C_4%20%3D%20%5Cfrac%7B6%21%7D%7B%286%20-%204%29%214%21%7D)
![^6C_4 = \frac{6!}{2!4!}](https://tex.z-dn.net/?f=%5E6C_4%20%3D%20%5Cfrac%7B6%21%7D%7B2%214%21%7D)
![^6C_4 = \frac{6 \times 5 \times 4}{2 \times 1 \times 4!}](https://tex.z-dn.net/?f=%5E6C_4%20%3D%20%5Cfrac%7B6%20%5Ctimes%205%20%5Ctimes%204%7D%7B2%20%5Ctimes%201%20%5Ctimes%204%21%7D)
![^6C_4 = \frac{30}{2}](https://tex.z-dn.net/?f=%5E6C_4%20%3D%20%5Cfrac%7B30%7D%7B2%7D)
![^6C_4 = 15](https://tex.z-dn.net/?f=%5E6C_4%20%3D%2015)
So, the table is as follows:
![\left[\begin{array}{ccc}TV&Good&Proportion\\1,2,3,4&2&0.5&2,3,4,5&2&0.5&3,4,5,6&2&0.5\\4,5,6,1&3&0.75&5,6,1,2&4&1&6,1,2,3&3&0.75\\1,2,3,5&3&0.75&3,5,6,2&3&0.75&1,3,4,5&2&0.5\\1,3,4,6&2&0.5&1,4,5,2&3&0.75&2,4,6,1&3&0.75\\2,4,6,3&2&0.5&2,4,6,5&3&0.75&3,5,6,1&3&0.75\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7DTV%26Good%26Proportion%5C%5C1%2C2%2C3%2C4%262%260.5%262%2C3%2C4%2C5%262%260.5%263%2C4%2C5%2C6%262%260.5%5C%5C4%2C5%2C6%2C1%263%260.75%265%2C6%2C1%2C2%264%261%266%2C1%2C2%2C3%263%260.75%5C%5C1%2C2%2C3%2C5%263%260.75%263%2C5%2C6%2C2%263%260.75%261%2C3%2C4%2C5%262%260.5%5C%5C1%2C3%2C4%2C6%262%260.5%261%2C4%2C5%2C2%263%260.75%262%2C4%2C6%2C1%263%260.75%5C%5C2%2C4%2C6%2C3%262%260.5%262%2C4%2C6%2C5%263%260.75%263%2C5%2C6%2C1%263%260.75%5Cend%7Barray%7D%5Cright%5D)
The proportion column is calculated by dividing the number of Good TVs by the total selected (4) i.e.
![p = \frac{Good}{n}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7BGood%7D%7Bn%7D)
<u>(d) The sampling distribution</u>
In (a), we have:
--- proportion of Good TV
The sampling error is calculated as follows:
![SE_n = |p - p_n|](https://tex.z-dn.net/?f=SE_n%20%3D%20%7Cp%20-%20p_n%7C)
So, we have:
![\left[\begin{array}{ccc}TV&Good&SE\\1,2,3,4&2&0.17&2,3,4,5&2&0.17&3,4,5,6&2&0.17\\4,5,6,1&3&0.08&5,6,1,2&4&0.33&6,1,2,3&3&0.08\\1,2,3,5&3&0.08&3,5,6,2&3&0.08&1,3,4,5&2&0.17\\1,3,4,6&2&0.17&1,4,5,2&3&0.08&2,4,6,1&3&0.08\\2,4,6,3&2&0.17&2,4,6,5&3&0.08&3,5,6,1&3&0.08\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7DTV%26Good%26SE%5C%5C1%2C2%2C3%2C4%262%260.17%262%2C3%2C4%2C5%262%260.17%263%2C4%2C5%2C6%262%260.17%5C%5C4%2C5%2C6%2C1%263%260.08%265%2C6%2C1%2C2%264%260.33%266%2C1%2C2%2C3%263%260.08%5C%5C1%2C2%2C3%2C5%263%260.08%263%2C5%2C6%2C2%263%260.08%261%2C3%2C4%2C5%262%260.17%5C%5C1%2C3%2C4%2C6%262%260.17%261%2C4%2C5%2C2%263%260.08%262%2C4%2C6%2C1%263%260.08%5C%5C2%2C4%2C6%2C3%262%260.17%262%2C4%2C6%2C5%263%260.08%263%2C5%2C6%2C1%263%260.08%5Cend%7Barray%7D%5Cright%5D)
Read more about sampling distributions at:
brainly.com/question/10554762