Question

4. In written English, the letter .e, is 10% of all individual written letters. (a) A linguist selects a random sample of 298 letters in written English. Let p denote the sample proportion that are the letter ‘e." What are the center, standard deviation, and (approximate) shape of the sampling distribution of p? Show any calculations you perform. Also, provide numerical justification for the theorem that guarantees the shape. Hints: See lecture 20, 21.] (b) Use the sampling distribution of p to compute the probability the probability that more than 12% of the letters in the sample are the letter ‘e." Make a sketch of the distribution with the center labeled and the probability in question shaded; convert to Standard Normal [i.e., standardize, and show work], and then use the Normal table to get the final answer (this is how you will do it on the exams). [Hint: See lecture 21.]

Answer 1

A.

n = 298

$\hat{p}$ = 0.1

center = 0.1

Std. Deviation = 0.0174 (

VP(1-p) /n

When np.(i.е., X) 5 and n(1-p) (i.e., n-X) 25, sampling distribution of p can be approximated bya normal distribution with mean and standard deviation

Since,

np = 29.8> 5 and

n(1-p) = 268.2 > 5

Thus assumptions to consider sampling distribution of $\hat{p}$ approximately normal are fulfilled.

2.

P($\hat{p}$>0.12) = 0.1252

0.048 0.065 0.083 0.1 0117 0.135 0.152

Let z be the standard normal variable. so,

Z = ($\hat{p}$-p)/$\sigma _{\hat{p}}$

P($\hat{p}$>0.12) = p(Z>(0.12-0.1)/0.0174)

= P(Z>1.1494)

= 0.1252

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