Chapter 7: Q. 32 (page 440)

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.

Short Answer

Expert verified

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

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

Step by step solution

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:

localid="1650011266324" $p = \frac{x}{n} = \frac{42}{100}$

Sample given: $n = 7$

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

localid="1650011270157" $n p = 7 \times \frac{42}{100} \approx 2.94$

localid="1650011273619" $n (1 - p) = 7 \times (1 - \frac{42}{100}) \approx 4.06$

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

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Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

Part (b) Step 1: Given Information

Part (b) Step 2: Explanation

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