Question

16 The table contains Flesch-Kincaid Grade Level data for 12 pages from 3 books. Higher numbers indicate more difficult reading levels for a page. Tom Clancy: The Bear and J. K. Rowling: Harry Potter Leo Tolstoy (translated): 5.2 3.7 6.1 4.9 4.4 3.2 5.6 6.9 5.7 4.1 6.7 4.4 and the Sorcerers Stone War and Peace the Dragon 8.8 5.4 5.0 7.6 9.0 3.2 12.0 4.9 4.6 5.0 6.4 6.1 8.6 9.8 6.1 5.9 7.7 10.9 8.2 7.2 11.0 11.5 8.4 5.9 a. Construct a 95% confidence interval estimate of the mean Flesch- Kincaid Grand Level rating for the population of all pages from Harry Potter and the Sorcerer's Stone by J. K. Rowling. b. Construct a 95% confidence interval estimate of the mean Flesch- Kincaid Grande Level rating for the population of all pages from War and Peace by Leo Tolstoy c. Compare the preceding confidence intervals. What do you conclude about the grade level readability ratings? d. If you want to estimate the mean Flesch-Kincaid Grade Level rating for pages from J. R. R. Tolkien's Lord of the Rings, how many pages mu you randomly select if you want to be 90% confident that the sample mean is within 0.5 of the population mean? Because the samples of pages from Clancy, Rowling and Tolstoy have Flesch-Kincaid Grade Level ratings with standard deviations of 2.45, 1.17 and 2.01, you may assume that o 2.45 for Lord of the Rings.

Answer 1

a.
From the data,
mean of level rating for Harry Potter = 5.075
standard deviation of level rating for Harry Potter = 1.167846
Standard error of sample mean = sd/ $\sqrt{n}$ = 1.167846 / $\sqrt{12}$ = 0.3371281

Degree of freedom = n - 1 = 12 - 1 = 11

Critical value of t for 95% confidence interval and df = 11 is 2.201

95% confidence interval of mean rating is,

(5.075 - 2.201 * 0.3371281, 5.075 - 2.201 * 0.3371281)

= (4.332981, 5.817019)

b.

From the data,
mean of level rating for War and Peace = 8.433333
standard deviation of level rating for War and Peace = 2.008316
Standard error of sample mean = sd/ $\sqrt{n}$ = 2.008316 / $\sqrt{12}$ = 0.5797509

95% confidence interval of mean rating is,

(8.433333 - 2.201 * 0.5797509, 8.433333 - 2.201 * 0.5797509)

= (7.157301, 9.709365)

c.

As, the lower limit of 95% confidence interval of mean rating of War and Peace is greater than the the upper limit of 95% confidence interval of mean rating of Harry Potter, there is statistically significant difference in Flesch-Kincaid rating for Harry Potter and War and Peace.

d.

Critical value of z for 90% confidence interval is 1.645

Margin of error = 0.5

Sample size, n = ( $\sigma$ * t / E)²

= (2.45 * 1.645 / 0.5)²

= 64.97166

$\approx$ 65 (Ronding to next integer)