Question

t was determined that the percentage of​ a's in the English language in the 1800s was 12% A random sample of 600 letters from a current newspaper contained 80 a's. Test the hypothesis that the proportion of​ a's has changed in modern​ times, using the 0.05 level of significance. Treat the newspaper as a random sample of all letters used.

Determine the​ z-test statistic. z= Answer _____ ​(Round to two decimal places as​ needed.)

What is the ​p-value = Answer ____ ​(Round to three decimal places as​ needed.)

Answer 1

Let p be the true proportion of a's in the English language in modern​ times

Null Hypothesis H0: p = 0.12

Alternative Hypothesis H0: p 0.12

np(1-p) = 600 * 0.12 * (1 - 0.12) = 63.36

Since, np(1-p) > 10, the sample size is large enough to assume that the sampling distribution of proportion is normal and we can use one sample z test.

Significance level = 0.05

Decision - Reject null hypothesis H0 if p-value is less then 0.05.

Standard error of mean, SE =   = 0.0132665

Sample proportion, = 80/600 = 0.1333333

Test statistic, z = ( - p) / SE = (0.1333333 - 0.12) / 0.0132665 = 1.01

z = 1.01

For two-tail test, P-value = 2 * P(z > 1.01) = 0.312

Since, p-value is greater than 0.05 significance level, we fail to reject null hypothesis H0 and conclude that there is no strong evidence that the true proportion of a's in the English language in modern​ times has changed from 0.12 (12%)