I seek to understand what confidence interval is with the aid of following example (which I know how to solve but do not understand the rationale behind it);
Suppose it is known that the weight of cement in packed bags is distributed normally with a standard deviation of 0.2 Kg. A sample of 25 bags is picked up at random and the mean weight of cement in these 25 bags is only 49.7 Kg. We want to find a 90% confidence interval for the mean weight of cement in filled bags.
The textbook from which this question comes explains confidence interval as follows;
The confidence level, therefore, may be defined as the probability that the interval estimate will contain the true value of the population parameter that is being estimated. If we say that a 95% confidence interval for the population mean is obtained by spanning 1.96 times the standard error of the mean on either side of the sample mean, we mean that we take a large number of samples of size n, say 1000, and obtain the interval estimates from each of these 1000 samples and then 95% of these interval estimates would contain the true population mean.
The textbook also states that;
It is to be noted that the true population mean is a constant and is not a variable. On the other hand, the interval that we specify is a random interval whose position depends on the sample mean.
Questions:
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While solving the example will we assume that the estimate of true population mean is the mean of 1 sample we have picked i.e. 49.7 Kg or will the estimate of true population mean be one of the values in the interval we eventually calculate (49.6342, 49.7658)?
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We say that probability that the sample picked at random from the population has mean ranging from (x – 1.645(sigma)/root(n) to (x + 1.645(sigma)/root(n) has probability of 90%. How does this translate to probability of picking up a sample which contains true population mean in the interval (x – 1.645(sigma)/root(n) to (x + 1.645(sigma)/root(n)?
If these questions don't make any sense, I will be grateful if the concept of confidence intervals could be explained with respect to the example above. There seem to be gaps in my knowledge.
Best Answer
There is not 'the' estimate but there are several possible estimates. A common estimate is the population mean, 49.7, which is also known as the maximum likelihood estimate (given the assumed population properties of a Gaussian distribution).
To indicate not just the maximum, but also that multiple other values can be equally good candidates for an estimate we use a range or interval (typically the more precise and large the sample, the smaller this range can be made).
A confidence interval is a range of potential estimates for which the confidence is high. For any of the values in the range, if it would be the true value, then our observed sample mean of 49.7 wouldn't be an unlikely outlier and would be within at least 90% of the observations.
The logic of a confidence interval is inversed. It relates to the probability of the data, given the predicted value (instead of the inverse 'the probability of the predicted value, given the data', which relates to a credibility interval).
The confidence interval can be seen as a range of population parameter values for which the data has a high probability (but as a fiducial probability based on p-values rather than a likelihood).
To construct a 90% confidence interval
For each hypothetical population parameter you compute a range within which 90% of the observed samples would fall*.
Following that you pick the hypothetical population parameters for which the observed sample is within their 90% range of observations.
Conditional on the true population parameter, in 90% of the time you will have observed a sample such that you picked that true population value as part of the confidence interval, in 10% of the time you will have observed a sample for which you compute an erroneous range.
See for instance the image from
The basic logic of constructing a confidence interval
and
Can we reject a null hypothesis with confidence intervals produced via sampling rather than the null hypothesis?
The idea behind a 90% confidence interval containing 90% of the time the true population is the conditioning on the true parameter instead of the observed experimental sample see: Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?
* The computation of such 90% range can be done differently, and there is not a single 'the' confidence interval, instead there are many different ways to compute a confidence interval with different tradeoffs. See e.g. Asymmetric confidence intervals and What is a rigorous, mathematical way to obtain the shortest confidence interval given a confidence level?