Question

To what extent do syntax textbooks, which analyze the structure of sentences, illustrate gender bias? A study of this question sampled sentences from 10 texts. One part of the study examined the use of the words "girl," "boy," "man," and "woman." We will call the first two words juvenile and the last two adult. Is the proportion of female references that are juvenile (girl) equal to the proportion of male references that are juvenile (boy)? Here are data from one of the texts:

Gender	n	X(juvenile)
Female	62	47
Male	133	50

(a) Find the proportion of juvenile references for females and its standard error. Do the same for the males. (Round your answers to three decimal places.)

p̂F	= _______
SEF	= _______
p̂M	= _______
SEM	= _______

(b) Give a 90% confidence interval for the difference. (Do not use rounded values. Round your final answers to three decimal places.)

_______, _______

(c) Use a test of significance to examine whether the two proportions are equal. (Use p̂_F − p̂_M. Round your value for z to two decimal places and round your P-value to four decimal places.)

z	=	_______
P-value	=	_______

State your conclusion.

There is sufficient evidence to conclude that the two proportions are different.

There is not sufficient evidence to conclude that the two proportions are different.    

Answer 1

Solution:-

a) The proportion of juvenile references for females and its standard error are

p̂F	= 0.758
SEF	= 0.054
p̂M	= 0.376
SEM	= 0.042

(b) 90% confidence interval for the difference is C.I = (0.256, 0.509).

0.256, 0.509

C.I = (0.758 - 0.376) + 1.645 × 0.0769

C.I = 0.382 + 0.1265

C.I = (0.256, 0.509)
(c)

z	=	4.97
P-value	=	0.0000

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P₁ = P₂
Alternative hypothesis: P₁ P₂

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂) =   
p = 0.4974
SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.0769
z = (p₁ - p₂) / SE

z = 4.97

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than -2.13 or greater than 2.13.

Thus, the P-value = 0.00

Interpret results. Since the P-value (0.00) is less than the significance level (0.05), we have to reject the null hypothesis.

There is sufficient evidence to conclude that the two proportions are different.