Hypothesis Testing using Means
For the first hypothesis test we have two data sets that initially show SAT scores that were taken by nine students and then another set of SATs results, for the same students, after they had completed a preparation course. Using this information we draw the null hypothesis (H o ) stating that the preparation course had no effect on their second SAT scores while the alternative hypothesis (H a ) would be that the preparation course did indeed had an effect of on their results.
The metrics provided for both SAT I and SAT II are tabulated in the below table, additionally to provide context to the numbers shown herein, we have proceeded to determine the mean, standard deviation, and the standard error for the difference between scores.
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|
SAT I |
SAT II |
d = SATI-SATII |
|
720 |
740 |
-20 |
|
860 |
860 |
0 |
|
850 |
840 |
10 |
|
880 |
920 |
-40 |
|
860 |
890 |
-30 |
|
710 |
720 |
-10 |
|
850 |
840 |
10 |
|
1200 |
1240 |
-40 |
|
950 |
970 |
-20 |
|
Mean |
-15.56 |
|
|
Standard Deviation |
19.44 |
|
|
Standard Error |
6.48 |
|
Degrees of Freedom |
8.00 |
|
|
Critical T-Score |
2.306 |
|
|
Margin for Error 'E' |
14.94 |
|
From the above table we can categorically state that the upper limit is -0.62 while the lower limit is -30.05 for a 95% confidence interval. The also implies that that its a two tailed test and that we can accept the null hypothesis as how it is stated in the assumption made (H o ) i.e. the preparation test had no effect on the average scores of the nine individuals who recorded the observations because the P value is greater than the stated alpha.
US Government Data
The second query of the assignment gives us 580 out of 1066 adults which believe that the US government isn’t doing much for the environment while ten years prior to this sample, 514 out of 1038 individuals were of the same opinion, the alpha in this case is 0.10. The confidence coefficient α / 2 is equal to 0.05 which implies that the test in question is a two tailed test and with the help of the z distribution table we can identify that the P value of 1.73 is greater than the alpha.
The preceding metric also further proves that the null hypothesis is correct and that there hasn’t been much of a change in mindset among American adults when it comes to considering the government’s stance on the environment.
References
Anderson, T. W., Anderson, T. W., Anderson, T. W., Anderson, T. W., & Mathématicien, E. U. (1958). An introduction to multivariate statistical analysis (Vol. 2, pp. 5-3). New York: Wiley.
Lehmann, E. L., & Romano, J. P. (2006). Testing statistical hypotheses . Springer Science & Business Media.
