Question

An education researcher claims that stipends for PhD students have increased from one academic year to the next. The table shows the PhD stipends (in dollars per year) for seven randomly chosen, different fields of study at various institutions in two consecutive years. Assume the populations are normally distributed. At α=0.05 , is there enough evidence to support the education researcher’s claim? Place commas between the 7 answers for parts a. and b. Field of Study 1 2 3 4 5 6 7 Stipend (First Year) 30,600 29,658 15,200 26,233 11,900 33,000 33,590 Stipend (Second Year) 32,850 30,770 19,800 27,100 11,000 33,000 30,312 d 〖b.) d〗^2 c.) Verify that the samples taken are random _____ and dependent._____ and that the populations are known to be normally distributed or n > 30. _____ Identify H_(0∶ _______________) and H_a: ____________ Find the mean of the differences, d ̅ and ∑▒d^2 . Place commas between the three answers. ∑▒d= __________________〖_〗d ̅ = (∑▒d)/n = _______________ ∑▒d^2 = _____________ Find the standard deviation of the diffe rences: s_d= √((∑▒〖〖d 〗^2- (∑▒d)^2/n〗)/n) Find the standardized test statistic, t. t = (d ̅- μ_d)/(s_d⁄√n) Find the number of degrees of freedom: d.f. = n - 1 For α=0.05, use Table 5 to identify t and find and shade the rejection region. Determine whether to reject or fail to reject H_(0 ). Interpret this decision in the context of the original claim.

Answer 1

First we make a table for the better understanding of the data.

Filed of Study	1	2	3	4	5	6	7
Stipend (First Year)	30600	29658	15200	26233	11900	33000	33590
Stipend (Second Year)	32850	30770	19800	27100	11000	33000	30312

2c

Given that the scholars are randomly chosen and hence the sample is random and independent.

Also given that the population is normally distributed.

Let 1 be the population mean of the stipend(first year) and 2 be the population mean of the stipend(second year). Now the null and alternative hypotheses are given by

Now the columns d and d2 are given as

d	2250	1112	4600	867	-900	0	-3278	Total
d2	5062500	1236544	21160000	751689	810000	0	10745284	39766017

The test statistic is given by

df=499

The critical value is -1.645

As the test statistic is greateer than the critical value, we fail to reject the null hypothesis at 5% level of significance. Hence we cannot conclude that there was significant rise in their stipends.

Hopefully this will help you. In case of any query, do comment. If you are satisfied with the answer, give it a like. Thanks.