Confidence interval for the population mean

Alternatively, a confidence interval can be expressed in terms of the population s standard deviation and the value of a single member drawn from the population. Thus, equation 4.9 can be rewritten as a confidence interval for the population mean... 

The population standard deviation for the amount of aspirin in a batch of analgesic tablets is known to be 7 mg of aspirin. A single tablet is randomly selected, analyzed, and found to contain 245 mg of aspirin. What is the 95% confidence interval for the population mean ... 

Q, therefore a lower one-sided confidence interval for the population mean could be used as an acceptance limit. The criterion is that the lower bound on the confidence interval must be greater than Q. 

Now, let us take a closer look at expression (4). The value at the center, p, is the population mean, which is the unknown quantity we are estimating. The two expressions on the right-hand and the left-hand sides of (4) are variables calculated from the data. Thus, expression (4) represents a random interval containing the population mean p. Expression (3) assigns a probability 1 — y that (4) holds. The interpretation of this is that if we conduct an experiment and calculate the lower and upper limits of the interval, Ly and Uy, respectively, then the interval (Ly, Uy) will contain the true (and unknown ) population mean with probability 1 — y. The interval (4) is called a confidence interval for the population mean, and 1 — y is called the confidence level of the interval, often expressed as a percent. 

We can therefore express a two-sided (f — a)% confidence interval for the population mean as ... 

Therefore, the expression written above for the confidence interval for the population mean also applies to any continuous random variable as long as the sample size is large (as just noted, of the order of 200 or more). The other rather restrictive assumption required for this confidence interval is that the population variance be known. Such a scenario is neither common nor realistic. 

A reasonable suggestion for devising a confidence interval for the population mean would be to substitute the sample estimate, s, for the corresponding population parameter, a and proceed as described earlier in Section 6.10. However, when the sample size is small (particularly < 30) the use of the Z distribution is less appropriate. William S Gossett, writing anonymously as "Student" while employed at Guinness Brewery, proposed the following statistic as an alternative. When X is a normally distributed variable and the sample size is small, the statistic... 

As was the case with the normal distribution, the shape of the t distribution can be used to find two values that define a central area under the density curve of size (f - a). It can be shown that, once a value of t associated with an area of interest is determined, the difference between the sample mean x is within Ks/Jn of the population mean, g. This enables us to calculate a confidence interval for the population mean when the sample size is small and the population variance unknown. 

Scientists from the pharmaceutical company believe that reporting a 95% confidence interval for the population mean change in SBP may prove helpful. Following the confidence interval defined in Section 6.10, a 95% confidence interval for the population mean is ... 

Exampie. From the ISO statistical methods (1981) we learn that the one sided confidence interval for the population mean is defined as foliows... 

Roughly, then, x s/y/n is the interval in which the tme population mean jju will be found 68 times out of 100. The 95% confidence interval for the population mean /r is x 2.0s jy/n. However, because s/y/n slightly overestimates the interval, 95 out of 100 times, the true /r will be contained in the interval, x 1.96s/v , given the sample size is large enough to assure a normal distribution. If not, the Student s t distribution (Table B) is used, instead of the Z distribution (Table A). 

For a specified degree or level of confidence P (for example 90%), a one or two sided confidence interval for the population mean can be constructed from the sample data. The value of P can be altered depending on the desired level of confidence. The larger the value of P, that is, the greater the degree of confidence that is desired, the wider the corresponding interval will be. 

Confidence intervals serve another useful purpose. They can assist one in determining whether the population mean may equal a specific value. To illustrate, suppose prior to the collection of the example data, one wished to determine whether the population mean might be equal to 25. Since 25 does not fall within the confidence interval for the population mean, one could feel reasonably confident that, based on the data, the population mean did not equal 25. 

While only confidence intervals for the population mean have been discussed, confidence intervals can be computed for the population standard deviation. In addition, as described in the last example, confidence intervals can be applied to more than one sample. The construction of these intervals is discussed in most statistical textbooks including those cited at the end of this paper. 

The concentration of each THM is normally distributed in the drinking water sample. There exists a population mean concentration for each THM. We can never known this mean concentration, p, for each THM from the population we can, however, by measuring L replicates of the sample, calculate a mean concentration, x, and a standard deviation, s. We really do not know the standard deviation for the population, a. We can make L replicate measurements and use t-statistics to define a confidence interval for the population mean using chloroform as our example ... 

Flow do we form a confidence interval for the population mean from the marginal posterior density Let us say that we know the value of a so that the conditional posterior is... 

---