Independent Random Samples

Question

1. The Connecticut Board of Education is concerned that first year female high school teachers are receiving lower salaries than their male counterparts. Two independent random samples have been selected 750 observations from population 1 (female high school teachers) and 710 from population 2 (male high school teachers). The sample means obtained are X1(bar)=$45 k and X2(bar)=$47 k.  It is known from previous studies that the population variances are 3.6 and 5.0 respectively. Using a level of significance of .05, is there evidence that the first year female high school teachers are receiving lower salaries? Fully explain your answer.    

Question

Stating in terms of the model parameters the hypotheses that you are testing write down a test statistic to investigate whether the mean level of the histamine for product (A) is less than for Product (B). Find the corresponding p-value and draw conclusions. 

Product (A) : 16.61,15.38,15.70,17.58,16.66,17.13.
Product (B) : 18.66,19.52,16.98,18.19,17.20.

Question

Suppose that k is chosen so that the three numbers 6k+1, 12k+1, 18k+1 are all prime numbers.
a) Prove that their product n= (6k+1)(12k+1)(18k+1) is a Carmichael number.
b) Find the five first values of k for which this method works and give the Carmichael number produced by the method