Question

Alice told Bob that, in any English text it is equally likely to have letters from the set V and from the set C that are defined below: V = {A, a,E, e,I, i, O, o, U, u}, C = {B, b, C, c, . . . , Y, y, Z, z}.
Note that the entries of C are all the letters of the English alphabet (upper-case and lower-case) which are not included in V.
Bob was not convinced that the claim is true and, because of that, he decided to investigate it by analyzing Shakespeare’s Sonnet 130. More precisely, he took the text of the sonnet and replaced each letter from V by symbol ‘V’. Then he replaced any letter from C by symbol ‘C’. He also removed all the punctuation marks, but he maintained the spaces between the original words.

The text produced after these alterations is:
CC CVCCCVCC VCVC VCV CVCCVCC CVCV CCV CVC CVCVC VC CVC CVCV CVC CCVC CVC CVCC CVC VC CCVC CV CCVCV CCC CCVC CVC CCVVCCC VCV CVC VC CVVCC CV CVCVC CCVCC CVCVC CCVC VC CVC CVVC V CVCV CVVC CVCVC CVCVCCVC CVC VCC CCVCV CVC CV CVCC CVCVC CVV V VC CVC CCVVCC VCC VC CVCV CVCCVCVC VC CCVCV CVCV CVCVCCC CCVC VC CCV CCVVCC CCVC CCVC CC CVCCCVCC CVVCC V CVCV CV CVVC CVC CCVVC CVC CVCC V CCVC CCVC CVCVC CVCC V CVC CVCV CCVVCVCC CVVCC V CCVCC V CVCVC CVC V CVCCVCC CV CC CVCCCVCC CCVC CCV CVCCC CCVVCC VC CCV CCVVCC VCC CVC CC CVVCVC V CCVCC CC CVCV VC CVCV VC VCC CCV CVCVVC CVCC CVCCV CVCCVCV
After counting carefully, Bob and Alice came to the conclusion that, out of the 464 letters of the text, only 171 are from V. Answer the following questions:

a) Out of the 464 letters of the text, let X be the number of the letters from V. Under the assumptions that (i) the symbols in the altered text have no influence on each other and (ii) it is equally likely to have in the altered text the symbols ‘V’ and ‘C’, state the distribution of X, with parameters.

b) We wish to test the hypothesis that it is equally likely to have letters from the sets V and C in the English texts. Formulate the null and alternative hypotheses in terms of the distribution of X and its parameters. Specify the full distribution of X and use a two-sided alternative hypothesis.

c) Sketch as a curve the shape of the probability function of X under the null hypothesis. Your sketch should have axes labelled x and P(X = x). Mark on the sketch the upper and lower limits of x, and the approximate value of x where the curve peaks under the null hypothesis. Also mark the observed value x = 171 so that you can see the tail probabilities required for the p-value, and shade under the curve the area represented by the p-value.

Answer 1

This problem is the test for single proportion by using normal approximation.

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