## Confidence Interval: Definition

### Alphas, P-Values, and Confidence Intervals, Oh My! | …

The , LR^{+} and LR^{-}, can be easily computed from the sensitivity and specificity as described above. Since they can also be seen as nonlinear functions (ratios) of model parameters, they can be computed using the , which provides a confidence interval for each. PROC GENMOD is used to fit this linear probability model with TEST as the response and RESPONSE as a categorical predictor:

### Understanding Hypothesis Tests: Confidence Intervals …

The point estimates of LR^{+} and LR^{-} agree with the computations above (2.1154 and 0.2564 respectively). The 95% confidence interval for LR^{+} is (0.3339, 3.8968) and for LR^{-} is (-0.1168, 0.6296).

The results show that a little over two subjects (2.0690) need to be treated, on average, to obtain one more positive response. A 95% large sample confidence interval for the NNT is (0.4666, 3.6713).

## a 95% confidence interval for the mean of a population

The accuracy is again found to be 0.7391 with a confidence interval of (0.56, 0.92). Asymptotic and exact tests of the null hypothesis that accuracy = 0.5 are similar and significant.

## Confidence Interval & Hypothesis Testing

One of the main goals of statistical hypothesis testing is to estimate the *P* value, which is the probability of obtaining the observed results, or something more extreme, if the null hypothesis were true. If the observed results are unlikely under the null hypothesis, your reject the null hypothesis. Alternatives to this "frequentist" approach to statistics include Bayesian statistics and estimation of effect sizes and confidence intervals.

## significance can be based on the 95% confidence interval:

The can be computed by creating a binary variable (ACC) indicating whether test and response agree in each observation. As above, the BINOMIAL option in the TABLES and EXACT statements can be used to obtain asymptotic and exact tests and confidence intervals.

## Statistical Hypothesis testing with confidence interval 3

Usually, the null hypothesis is boring and the alternative hypothesis is interesting. For example, let's say you feed chocolate to a bunch of chickens, then look at the sex ratio in their offspring. If you get more females than males, it would be a tremendously exciting discovery: it would be a fundamental discovery about the mechanism of sex determination, female chickens are more valuable than male chickens in egg-laying breeds, and you'd be able to publish your result in *Science* or *Nature*. Lots of people have spent a lot of time and money trying to change the sex ratio in chickens, and if you're successful, you'll be rich and famous. But if the chocolate doesn't change the sex ratio, it would be an extremely boring result, and you'd have a hard time getting it published in the *Eastern Delaware Journal of Chickenology*. It's therefore tempting to look for patterns in your data that support the exciting alternative hypothesis. For example, you might look at 48 offspring of chocolate-fed chickens and see 31 females and only 17 males. This looks promising, but before you get all happy and start buying formal wear for the Nobel Prize ceremony, you need to ask "What's the probability of getting a deviation from the null expectation that large, just by chance, if the boring null hypothesis is really true?" Only when that probability is low can you reject the null hypothesis. The goal of statistical hypothesis testing is to estimate the probability of getting your observed results under the null hypothesis.

## 95% one-sided confidence, null hypothesis - …

A related criticism is that a significant rejection of a null hypothesis might not be biologically meaningful, if the difference is too small to matter. For example, in the chicken-sex experiment, having a treatment that produced 49.9% male chicks might be significantly different from 50%, but it wouldn't be enough to make farmers want to buy your treatment. These critics say you should estimate the effect size and put a on it, not estimate a *P* value. So the goal of your chicken-sex experiment should not be to say "Chocolate gives a proportion of males that is significantly less than 50% (*P*=0.015)" but to say "Chocolate produced 36.1% males with a 95% confidence interval of 25.9 to 47.4%." For the chicken-feet experiment, you would say something like "The difference between males and females in mean foot size is 2.45 mm, with a confidence interval on the difference of ±1.98 mm."