The Right Questions about Statistics: How confidence intervals work Maths Learning Centre

The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed to give a RANGE of possible answers for a “WHAT’S THE NUMBER?” question, using DATA from a sample. You calculate which numbers could have been likely to produce your data.

What isthe median number of chapters in a novel? Randomly choose 12 books

Which possible medians might be likely to produce this data? For each particular median, we could do a hypothesis test to answer the question (in this case it’s called the sign test) … 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Is the median for all books 10? SIGN TEST P-value = 0.0005 Under 0.05, so NO, the median is not 10. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ?

Is the median for all books 15? SIGN TEST P-value = 0.23 Over 0.05, so YES, the median could be 15. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ?

This is the (95%)confidence interval It shows all the medians that could be true based on hypothesis tests at the 5% level of significance. That is, all the medians that would give a p-value of over 0.05. You could say it’s the medians I am “happy to believe” based on my data. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 “95% confidence interval”

Another example... What isthe mean weightin a novel? Randomly choose 12 books

Another example... Which possible means might be likely to produce this data? For each particular mean, we could do a hypothesis test to answer the question (in this case it’s called the t-test) … 200 250 300 350 400

Is the mean for all books 200? T-TEST P-value = 0.00007 Under 0.05, so NO, the mean is not 200. ? 200 250 300 350 400

And so on... 200 250 300 350 400

This is the (95%)confidence interval It shows all the means that could be true based on hypothesis tests at the 5% level of significance. That is, all the means that would give a p-value of over 0.05. You could say it’s the means I am “happy to believe” based on my data. “95% confidence interval” 200 250 300 350 400

BUT you don’t have to go through all this to find a confidence interval! You just have to figure out where the ends are. These ends would have given exactly 0.05 in the hypothesis test. So you find them by doing the hypothesis test calculation backwards... “95% confidence interval” 200 250 300 350 400

HYPOTHESIS TEST 95% confidence interval is between these. 95% CI is from 269 to 347 Assume mean is 300 Use assumed mean and data to calculate a test statistic Use critical values to get two means. CI ends = mean ± t* × std. Error = 269 & 347 t = mean – 300 std. Error = 0.46 Use test statistic to get p-value. Use p-value to get “critical” test statistics. t* = ± 2.20 p = 0.65 More than 0.05, so the mean could be 300. Assume P-value is 0.05 Compare p-value to 0.05 to decide. CONFIDENCE INTERVAL 200 250 300 350 400 -2.0 -1.0 0.0 1.0 2.0

Let’s go over that again… What isthe mean weight in a novel? Randomly choose 12 books Choose a hypothesis test, and a significance level. T-Test, 0.05 t* = ± 2.20 Find the critical values of the test statistic. CI ends = mean ± t* × std. Error = 269 & 347 Use critical values and your data to calculate the two ends. The 95% CI is from 269g to 347g. State the answer. -2.0 -1.0 0.0 1.0 2.0

So this is how to find a confidence interval: • Have a “what’s the number?” question. • Collect data. • Choose a matching hypothesis test. • Work backwards to calculate two ends. • The confidence interval is between these two values.

And this is what aconfidence interval means: • The values in the CI would be retained with a matching hypothesis test. • The values in the CI have a high chance of producing data like yours. • The values in the CI are those you are “happy to believe” based on your data.

The Right Questions about Statistics: How confidence intervals work Maths Learning Centre