Are population means score in statistics different between 2006 and 2016 students? ( Help ) (Stats)

Part A

We want to determine whether there is a significant difference between the population mean scores of the students. We will test the hypotheses:

#H_0: mu_"y" = mu_"x"#
#H_a: mu_"y" < mu_"x"#

where "y" represents data from 2016, and "x" represents data from 2006.

We can conduct a two-sample t-test for the difference between two means using a significance level #alpha = .05# if the following conditions for inference are met.

• Random: both samples (2006 and 2016) are random samples
• 10% condition / Independence: We must assume that the population of test takers ...
• in 2006 is greater than 140 #N_x >= 10*14#
• in 2016 is greater than 200 #N_y >= 10*20#
• Large/Normal samples: We must assume that the distributions for both populations are each approximately Normal

The formula for the test statistic t is:

#t = frac{barx - bary}{sqrt(frac{s_x^2}{n_x}+frac{s_y^2}{n_y})}#

with degrees of freedom (using the lower n) #"df" = n_x-1#

Substitute values:
#t = frac{73.0-70.8}{sqrt(frac{3.2^2}{14}+frac{4.6^2}{20})}#

#"df" = 14-1=13#

Now, using the table of t critical values or a calculator, we can find that our #p# value lies between #.05# and #.10#.
(Using a calculator):

#p = .0549#

Since our #p# value #p = .0549# is greater than our significance level #alpha = .05#, we fail to reject our null hypothesis. There is not convincing evidence of a significant difference between 2006 and 2016 tests.

Part B

We want to construct a 95% confidence interval for the difference between two means.

We can use a two-sample t-interval . The conditions for inference were verified in part (a).

The formula for the two-sample t-interval with 95% confidence is:

#(barx - bary) +- t^("*")sqrt(frac{s_x^2}{n_x}+frac{s_y^2}{n_y})#

Substitute values:
Find t using the table of critical t values (linked above). This is where we specify the 95% confidence.

#(73.0-70.8) +- (2.160)sqrt(frac{3.2^2}{14}+frac{4.6^2}{20})#

#2.2 +- 2.889#

We are 95% confident that the interval from #-0.689# to #5.089# captures the true difference #barx - bary# between the population mean test scores in 2006 and 2016.