High School

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.

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.