1. In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Kira chooses a simple random sample (SRS) of 7 tiles. Let [tex]\hat{p}[/tex] be the proportion of vowels in her sample.

(a) Is the 10% condition met in this case? Show your work to justify your answer.

[tex]
\begin{array}{lc}
10 \times n < N & \sigma_{\hat{p}} = \frac{\sqrt{p(1-p)}}{n} \\
10 \times 7 < 100 & \\
70 < 100 & \\
\end{array}
[/tex]

The 10% condition is met.

(b) Is the Large Counts condition met in this case? Show your work to justify your answer.

2. A factory employs 3000 unionized workers, 90% of whom are male. A random sample of 15 workers is selected for a survey about worker satisfaction. Let [tex]\hat{p}[/tex] be the proportion of males in the sample.

(a) Is the 10% condition met in this case? Show your work to justify your answer.

(b) Is the Large Counts condition met in this case? Show your work to justify your 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.