Question

In written English, the letter ‘e’ is 11% of all individual written letters.

(a) A linguist selects a random sample of 382 letters in written English. Let pˆ denote the sample proportion that are the letter ‘e.’ What are the center, standard deviation, and (approximate) shape of the sampling distribution of pˆ ? Show any calculations you perform. Also, provide numerical justification for the theorem that guarantees the shape.

(b) Use the sampling distribution of pˆ to compute the probability the probability that less than 7% 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.

Answer 1

(a) Center = 0.11

Standard deviation = 0.11*(1 - 0.11)/382 = 0.016

n*p = 382*0.11 = 42.02 10

n*(1 - p) = 382*(1 - 0.11) = 339.98 10

The data is normally distributed.

(b) The test statistic, z = (p̂ - p)/√p(1-p)/n

z = (0.07 - 0.11)/√0.11(1-0.11)/382

z = -2.50

The p-value is 0.0062.

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