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EDIT: My apologies, I had a coding error. I accidentally used the same standard deviation for both samples. Now that I fixed that, both the normal and Student's confidence intervals are stupidly similar (they don't contain 0), and the pvalues are both identical (0.0358). I suppose this makes the question pointless, but I'm unsure if the mods prefer to trash it, leave it with this edit, or something else.

The correct confidence intervals are:

  • Normal: (91.9098, 2682.0902)
  • Student's: (90.7354, 2683.2646)

Original question

I'm doing a statistics exercise where I have a dataset with the health insurance costs of a certain sample, $676$ males and $662$ females. What I'm trying to determine is whether the average costs at a population level are different for males and females.

My null hypothesis is that they aren't different.

The sample difference is $1387$ (male costs are larger). As I understand it, while not huge, my sample sizes are large enough to assume that the sampling distribution is approximately normal (because the samples have both far more than $30$ independent observations), so I decided to do hypothesis testing both using the Normal distribution and Student's t-distribution.

I got the following results.

Normal distribution

  • $p$-value: $0.0505$
  • $95\%$ confidence interval: $(-3.1067, 2777.1067)$

According to the above, the probability of observing a sample difference of $1387$ is higher than $5\%$, so at a $95\%$ confidence level we can't reject the null hypothesis.

Student's t:

  • $p$-value: $0.0358$
  • $95\%$ confidence interval: $(-5.6567, 2779.6567)$

According to this one, the probability of observing a sample difference of $1387$ is lower than $5\%$, so at a $95\%$ confidence level we can reject the null hypothesis.

So they contradict each other, even though both intervals contain $0$, which means in both cases there's a chance that the two means are the same at a population level (and to be fair, in both cases it is much more likely that the male costs are higher, based on the confidence intervals.)

I'm not sure whether I should conclude that the null hypothesis can be rejected or not, because I'm not sure on what ground I could choose either methods.

As far as I've read, the criteria to safely assume normality vary a great deal, from "you can always assume it when the sample size is larger than $30$" to "z-tests are sloppy and you should NEVER use them!", so I don't really know how to proceed.

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  • $\begingroup$ Your $p$-value for the $t$-test looks implausible given $0$ is inside the $95\%$ confidence interval and the $p$-value for the normal-test is so much higher. How did you find it? $\endgroup$
    – Henry
    Commented May 18, 2022 at 10:13
  • $\begingroup$ @Henry. SciPy. stats.t.interval gave me the interval, and stats.ttest_ind calculated the p-value. I only fed it the average, the standard error, the desired confidence level, and the degrees of freedom. $\endgroup$
    – Nicola
    Commented May 18, 2022 at 11:03
  • $\begingroup$ Does that give you a one-tailed probability? $\endgroup$
    – 1Rock
    Commented May 18, 2022 at 11:10
  • $\begingroup$ @1Rock, no, it's two-tailed by default. $\endgroup$
    – Nicola
    Commented May 18, 2022 at 11:15
  • $\begingroup$ You are welcome to delete or retain the question. It, together with the comments and answer, make the point that for large sample sizes, the normal and $t$-test give similar answers. If you do retain it, it would be helpful if your edit at the top had the new confidence intervals $\endgroup$
    – Henry
    Commented May 18, 2022 at 12:17

1 Answer 1

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There isn't really a perfect answer to which test you should use. The 95% confidence interval is just a convention, and while the t distribution better represents the distribution (and failing a t test is a slightly higher standard), the normal distribution is arguably more conventional, especially for a sample size of hundreds. A good case can be made for either.

As @Henry notes in the comments, there appears to be an error in your $p$-value for the t test. If your 95% confidence interval contains 0, then by definition your $p$-value should be greater than 0.05.

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