Calculating confidence intervals for the mean

The main reason we want a model is the prospect of using it to make inferences about population parameters. Let us forget for a while that we weighed 140 beans. Suppose that only one bean had been randomly selected and that its mass was 0.1188g, the first value in Table 2.2. What does this value allow us to say about the mean population mass, p. 

If the masses of the beans follow a normal distribution, the interval [fj. - 1.96(7, p + 1.96cr] contains 95% of all the possible observations. This means that the single measurement 0.1188 g has a 95% probability of being within this interval. Of course this also implies that there is a 5% chance that it falls outside. If we assume the normal model holds, we can say that we have 95% confidence in the double inequality 

Taking the inequality on the left and adding 1.96(7 to both sides, we have 

Subtracting 1.96(7 from the inequality on the right, we also have 

Combining the two, we arrive at a 95% confidence interval for the population mean  

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