Statistics: GPA Distribution, Significance Level, and Confidence Interval

This article discusses the distribution of GPAs in private colleges, testing claims of grade inflation, and calculating confidence intervals. It also explains the difference between 5% and 1% significance levels and the preference for 95% confidence intervals. The article cites relevant sources to support its claims.

Statistics
Name:
Institution:
21st February 2019

Question 1:
Past experience indicates that the distribution of GPAs in private college is normal with μ=2.5
and σ²=0.25 (compactly: x≃n (2.5, 0.25), where x = student’s GPA). Some people have claimed
however, that due to grade inflation the GPA has increased. Use a sample (random) of N=2.5 and
x̄= 2.68 to test the indicated claim at the 5% significance level. Would your answer be different
if you had tested at the 1% significance level? Which test would you prefer? Explain your
answer.
Answer
STEP 1:
We choose the significance level which in this case is α =0.05
The critical z value is 1.645. Reject H0 if Z > 1.645.
STEP 2:
We calculate the Z score
Z= ¯x−μ
σ /√ N = 2.68−2.5
0.25 / √2.5 = 0.18
0.1581 =1.1384
STEP 3:
From the computed Z score, we fail to reject the null hypothesis (H0) since Z < 1.645
STEP 4:
We conclude that there is no sufficient significant evidence to conclude that GPA has increased.
Yes the answer would be different if I had tested at the 1% significance level. I would prefer 1%
significance level instead of 5% significance level because 1% significance level is more
accurate than 5% significance level (Faherty, 2008).

Question 2:
Use the data in assignment 1 to calculate a 95 percent confidence of the GPA distribution. Would
you prefer a 90 percent confidence interval? Why or Why not? In the case of the 95 percent
confidence interval would you say that there is a 95 percent probability that the specific interval
contains the true μ? Why or Why not?
Answer
95% confidence interval computation
x ± Zα /2∗σ
√ N → 2.68 ± 1.96∗0.25
√2.5
2.68 ±1.96∗0.1581
2.68 ± 0.3099
Lower limit: 2.68−0.3099=2.3701
Upper limit: 2.68+0.3099=2.9899
We are 95% confident that the average GPA for the sample is between 2.3701 and 2.9899.
I would not prefer a 90 percent confidence interval but rather I would prefer 95 percent
confidence interval. 95% confidence interval is wider than 90% confidence interval hence 95%
confidence interval is more accurate and precise as compared to the 90% confidence interval
(Morey, et al., 2016).
No; for the case of 95 percent confidence interval it would not be proper to say that there is a 95
percent probability that the specific interval contains the true μ (Steiger, 2014). This is because,
the correct interpretation is supposed to be if repeated samples were taken and the 95%
confidence interval was computed for each sample, 95% of the intervals would contain the
population mean μ (Andrea , 2010).

References
Andrea , K., 2010. Overlapping Confidence Intervals and Statistical Significance. Cornell
Statistical Consulting Unit, 5(2), pp. 45-61.
Faherty, V., 2008. Probability and statistical significance. Applied Quantitative Analysis for
Social Services, 12(5), p. 127–138.
Morey, R. D. et al., 2016. The fallacy of placing confidence in confidence intervals.
Psychonomic Bulletin & Review, 23(1), p. 103–123.
Steiger, J. H., 2014. Beyond the F test: Effect size confidence intervals and tests of close fit in
the analysis of variance and contrast analysis. Psychological Methods, 9(2), p. 164–182.