I am currently making my way through a textbook called "Modern Statistics: A Graphical Introduction". (I'm not currently taking a related course through university or school.) I came across a question that I had no idea how to do. I tried googling to find an answer, but the results asked for information that the question didn't really seem to have. I re-read the chapter that the question was next to but didn't find anything useful.

Here is the question: Assume that IQ scores are normally distributed in the population with standard deviation 14. A class of fourteen nine-year-olds has the following IQ scores: 115, 123, ... 121 (a) Graph a 95% confidence interval for the mean of this population. (b) Is the normality assumption reasonable?

Part (a) was straightforward since confidence intervals were actually covered in the chapter. The answer for (b) mentioned using a "p-value for test of normality" but nothing like this has been mentioned yet in the textbook at all. Any help appreciated!

Edit: I looked at the entries in the index under 'normality assumption' and 'p-value'.

  • There were a few sections that said something like "to assess the normality assumption use a normal scores plot and check if it's linear", which seems like a very different type of test.
  • There was a section about testing the normality assumption for proportions: check that $n \times \pi_0$ and $n \times (1 - \pi_0)$ are greater than 5.
  • One section mentioned using Student's t-test but I'm not sure how to apply that to this question since I don't have a hypothesized mean.
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    $\begingroup$ This is presumably an error in the organization of the book. Is there an index in which you could look for "normality tests" or similar? $\endgroup$ – Stephan Kolassa Mar 15 '18 at 12:04
  • $\begingroup$ When the answer mentions a p-value, it's because they expect you to make some kind of formal statistical test. Is there anything along the lines of "normality test", "Jarque-Bera test" somewhere in the book? If the book introduces Pearsons Chi squared test you can use this one as well. $\endgroup$ – Duffau Mar 15 '18 at 12:07
  • $\begingroup$ I wouldn't recommend a Chi squared test with only 14 data points. $\endgroup$ – Stephan Kolassa Mar 15 '18 at 12:15

Sounds like a bad answer in the book. It happens (I briefly edited text books, there's tons of problems). The question "Is normality reasonable?" is a good one, but not an easy one. And the assumption of an sd of 14 is odd, since you can find the sd of the sample.

There are several statistical tests of normality (e.g. Komogorov Smirnov) but with such a small sample I wouldn't use them (I might not use them even with a large sample).

Since they say they expect an sd of 14, I'd first check that to see if it is close. Then I'd look at a quantile normal plot. This can be done in your statistics program (certainly in R and SAS, probably in most other programs).

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