Answer :
Sure, let's go through the questions step-by-step:
### Question 1: Scrabble Tiles
#### Part (a): 10% Condition
To check the 10% condition, we need to ensure that our sample size is no more than 10% of the population size. For Kira's Scrabble tiles:
- Total number of tiles = 100
- Sample size (n) = 7
The 10% condition is met if [tex]\(10 \times n < \text{Total number of tiles}\)[/tex].
[tex]\[10 \times 7 = 70 < 100\][/tex]
Since 70 is less than 100, the 10% condition is met.
#### Part (b): Large Counts Condition
The Large Counts condition requires that both [tex]\(n \cdot \hat{p} \geq 10\)[/tex] and [tex]\(n \cdot (1 - \hat{p}) \geq 10\)[/tex].
- Proportion of vowels in the bag, [tex]\(p = \frac{42}{100} = 0.42\)[/tex]
- Sample size, [tex]\(n = 7\)[/tex]
Calculate [tex]\(n \cdot p\)[/tex] and [tex]\(n \cdot (1 - p)\)[/tex]:
[tex]\[
7 \cdot 0.42 = 2.94 \quad (\text{less than 10})
\][/tex]
[tex]\[
7 \cdot (1 - 0.42) = 7 \cdot 0.58 = 4.06 \quad (\text{less than 10})
\][/tex]
Both products are less than 10, so the Large Counts condition is not met.
### Question 2: Factory Workers
#### Part (a): 10% Condition
The total number of workers is 3,000, and the sample size for the survey is 15.
The 10% condition is met if [tex]\(10 \times n < \text{Total number of workers}\)[/tex].
[tex]\[10 \times 15 = 150 < 3000\][/tex]
Since 150 is less than 3000, the 10% condition is met.
#### Part (b): Large Counts Condition
The Large Counts condition requires that both [tex]\(n \cdot \hat{p} \geq 10\)[/tex] and [tex]\(n \cdot (1 - \hat{p}) \geq 10\)[/tex].
- Proportion of males in the workforce, [tex]\(p = 0.9\)[/tex]
- Sample size, [tex]\(n = 15\)[/tex]
Calculate [tex]\(n \cdot p\)[/tex] and [tex]\(n \cdot (1 - p)\)[/tex]:
[tex]\[
15 \cdot 0.9 = 13.5 \quad (\text{greater than 10})
\][/tex]
[tex]\[
15 \cdot (1 - 0.9) = 15 \cdot 0.1 = 1.5 \quad (\text{less than 10})
\][/tex]
Only the first product is greater than 10, so the Large Counts condition is not met.
### Summary
- For the Scrabble tiles: The 10% condition is met, but the Large Counts condition is not.
- For the factory workers: The 10% condition is met, but the Large Counts condition is not.
### Question 1: Scrabble Tiles
#### Part (a): 10% Condition
To check the 10% condition, we need to ensure that our sample size is no more than 10% of the population size. For Kira's Scrabble tiles:
- Total number of tiles = 100
- Sample size (n) = 7
The 10% condition is met if [tex]\(10 \times n < \text{Total number of tiles}\)[/tex].
[tex]\[10 \times 7 = 70 < 100\][/tex]
Since 70 is less than 100, the 10% condition is met.
#### Part (b): Large Counts Condition
The Large Counts condition requires that both [tex]\(n \cdot \hat{p} \geq 10\)[/tex] and [tex]\(n \cdot (1 - \hat{p}) \geq 10\)[/tex].
- Proportion of vowels in the bag, [tex]\(p = \frac{42}{100} = 0.42\)[/tex]
- Sample size, [tex]\(n = 7\)[/tex]
Calculate [tex]\(n \cdot p\)[/tex] and [tex]\(n \cdot (1 - p)\)[/tex]:
[tex]\[
7 \cdot 0.42 = 2.94 \quad (\text{less than 10})
\][/tex]
[tex]\[
7 \cdot (1 - 0.42) = 7 \cdot 0.58 = 4.06 \quad (\text{less than 10})
\][/tex]
Both products are less than 10, so the Large Counts condition is not met.
### Question 2: Factory Workers
#### Part (a): 10% Condition
The total number of workers is 3,000, and the sample size for the survey is 15.
The 10% condition is met if [tex]\(10 \times n < \text{Total number of workers}\)[/tex].
[tex]\[10 \times 15 = 150 < 3000\][/tex]
Since 150 is less than 3000, the 10% condition is met.
#### Part (b): Large Counts Condition
The Large Counts condition requires that both [tex]\(n \cdot \hat{p} \geq 10\)[/tex] and [tex]\(n \cdot (1 - \hat{p}) \geq 10\)[/tex].
- Proportion of males in the workforce, [tex]\(p = 0.9\)[/tex]
- Sample size, [tex]\(n = 15\)[/tex]
Calculate [tex]\(n \cdot p\)[/tex] and [tex]\(n \cdot (1 - p)\)[/tex]:
[tex]\[
15 \cdot 0.9 = 13.5 \quad (\text{greater than 10})
\][/tex]
[tex]\[
15 \cdot (1 - 0.9) = 15 \cdot 0.1 = 1.5 \quad (\text{less than 10})
\][/tex]
Only the first product is greater than 10, so the Large Counts condition is not met.
### Summary
- For the Scrabble tiles: The 10% condition is met, but the Large Counts condition is not.
- For the factory workers: The 10% condition is met, but the Large Counts condition is not.