Using confidence interval versus using Hypothesis testing for inference

Question

When is it more appropriate to use a confidence interval and more appropriate to use hypothesis testing when one wants to make inferences from a sample to a population?
I think that is what confuses me about the two ways of inference making.

Please, help me out. Thanks..

Answer 1

Suppose you have $n = 150$ normally distributed test scores from your school, summarized (in R) as follows:

summary(x); length(x); sd(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  54.15   89.54  101.44  100.64  110.66  148.02 
[1] 150        # sample size
[1] 17.32858   # sample SD

The nationwide mean is 97.3. Is the average at our school significantly larger? A t test rejects the null hypothesis that our scores are the same, in favor of the alternative that ours are higher, at significance level 1% because the P-value is smaller than $0.01.$

t.test(x, mu = 97.3, alt="g")

        One Sample t-test

data:  x
t = 2.3617, df = 149, p-value = 0.009743
alternative hypothesis: 
  true mean is greater than 97.3
95 percent confidence interval:
 98.29965      Inf
sample estimates:
mean of x 
 100.6415 

Then the issue arises whether the difference between nationwide 97.3 and our 100.6 is of practical importance. This may be more a question for educators than for statisticians.

However, the one-sided 95% confidence interval $(98.3, \infty),$ shows that, given the variability in our scores, our sample mean might be considered as consistent with an actual performance as low as 98.3. And this is only one point above the national average.

So maybe it is best to interpret our higher scores as a hint of progress, rather than a fantastic breakthrough.

Note: Fake data for this illustration simulated in R as shown below. If you use the same seed for the sampling, you will get exactly the same data.

set.seed(2020)
x = rnorm(150, 100, 15)