1) Examine the data set age-at-menarche which describes the age when a female experiences her first menstrual cycle. The study compared women who were born in 1990-1994 with women who were 80-84 when these data were collected). This problem uses the age-at-menarche dataset.

a) Researchers wanted to test whether age-at-menarche was getting younger over time. One paragraph in the reading is taken verbatim here:

White girls in the United States now menstruate at an average age of 12.6 years and black girls at 12.1 years, while Mexican American girls menstruate at an average age of 12.2 years (though it is not clear how far these differences in menarche can be traced back, since most of the historical documentation focuses on white American and European girls).

All the women in the data set were white. Test whether our data on younger women could have come from a population of young black women. Be sure to state the null hypothesis, whether you reject it, and at what p-value. Give a careful definition of the p-value. Also interpret the 95%CI. (1=younger, 0=older)

b) Test whether our sample of older white women is consistent in age-at-menarche with a sample of younger Mexican-American women. Be sure to state the null hypothesis, whether you reject it, and at what p-value. Give a careful definition of the p-value. Also interpret the 95%CI.

c) In another article on Korean girls born between 1988 and 2003 (roughly when the younger women in our sample were born), a sample of 351,006 were quizzed on sexual maturation events. It's stated that the proportion of Korean girls who experienced "early age-at-menarche" (defined as 10.5 years old by the researchers) in 2015 was 3.2%. Test whether that value is consistent with the proportion of young women in our data set who experienced "early age-at-menarche" defined as 10.5 years old. Be sure to state the null hypothesis, whether you reject it, and at what p-value. Give a careful definition of the p-value. Also interpret the relevant 95%CI.

d) Test the main hypothesis in our dataset which was that the two groups have different ages-at-menarche. State the null hypothesis, whether you reject it, and at what t-score and p-value. Give a careful definition of the p-value. Also interpret the 95%CI.