Question

the average number of words in a romance novel is 64,182 and the standard deviation is 17,154.
a. find the proportion of all novels that are between 50,000 and 60,000 words.
select

b. on a shelf with 350 novels, how many would be estimated to have more than 70,000 words? select

Explanation:

Step1: Define given values

$\mu = 64182$, $\sigma = 17154$

Step2: Calculate z-scores for Part A

For $x_1=50000$: $z_1=\frac{50000-64182}{17154}\approx-0.827$
For $x_2=60000$: $z_2=\frac{60000-64182}{17154}\approx-0.244$

Step3: Find proportion for Part A

Use z-table: $P(-0.83$P(Z<-0.24)=0.4052$, $P(Z<-0.83)=0.2033$
Proportion: $0.4052-0.2033=0.2019$

Step4: Calculate z-score for Part B

For $x=70000$: $z=\frac{70000-64182}{17154}\approx0.339$

Step5: Find proportion for Part B

Use z-table: $P(Z>0.34)=1-P(Z<0.34)$
$P(Z<0.34)=0.6331$, so $P(Z>0.34)=1-0.6331=0.3669$

Step6: Calculate count for Part B

Number of novels: $350\times0.3669\approx128.42$

Answer:

A. $\approx 0.202$ (or 20.2%)
B. $\approx 128$ novels