Q. 32 - The Practice of Statistics for AP

Question

Scrabble 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. Cait chooses an SRS of 7 tiles. Let $\hat{p}$ be the proportion of vowels in her sample.

(a) Is the $10\%$ condition met in this case? Justify your answer.

(b) Is the Normal condition met in this case? Justify your answer.

Step-by-Step Solution

Answer

a). Yes, is the $10\%$ condition met in this case.

b). No, the Normal condition is met in this case.

Part (a) Step 1: Given Information

Number of tiles $=100$

Vowels$=42$,

Consonants $=56$,

Number of blank tiles $=2$

Number of tiles for SRS$=7$

Part (a) Step 2: Explanation

Given a population of $100$ people, a sample size of $7$ represents less than ten percent of the total. This indicates that the ten percent criteria has been met. As a result, the condition is satisfied.

Part (b) Step 1: Given Information

Number of tiles $=100$

Vowels $=42$

Consonants $=56$

Number of blank tiles $=2$

Number of tiles for SRS $=7$

Part (b) Step 2: Explanation

Given for the population:

$x=42$

$n=100$

The population proportion is calculated by dividing the sample size by the number of successes:

$$\begin{gathered} p=\frac{x}{n} \\ =\frac{42}{100} \end{gathered}$$

Sample given:  $n=7$

As long as $np$ and $n(1-p)$ are both at least $10$ , the condition is met.

$$\begin{gathered} np=7\times \frac{42}{100} \\ \approx 2.94 \end{gathered}$$

$$\begin{gathered} n(1-p)=7\times \left(1-\frac{42}{100}\right) \\ \approx 4.06 \end{gathered}$$

Neither of them exceeds $10$, so the Normal condition has not been met.