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

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1 Answer
May 24, 2018

Part a) No, there is not a significant difference between 2006 and 2016 students' test scores.

Part b) Interval: (0.689,5.089)

Explanation:

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:

H0:μy=μx
Ha:μy<μx

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 α=.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 Nx1014
  • in 2016 is greater than 200 Ny1020
  • 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=¯x¯ys2xnx+s2yny

with degrees of freedom (using the lower n) df=nx1

Substitute values:
t=73.070.83.2214+4.6220

df=141=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 α=.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:

(¯x¯y)±t*s2xnx+s2yny

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

(73.070.8)±(2.160)3.2214+4.6220

2.2±2.889

We are 95% confident that the interval from 0.689 to 5.089 captures the true difference ¯x¯y between the population mean test scores in 2006 and 2016.