**11. Confidence intervals**

From property (3) above, and using tables of the Normal distribution, it can be stated that there is a 95% probability that *x* (the sample mean) will be no further than 1.96x SE(*x* ), from *μ*, the (unknown) population mean. There is also a 99% chance that it will be within 2.58 SE(*x* ) of u. The quantities 1.96 x SE(*x* ) and 2.5 x SE(*x* ) are referred to as the *maximum likely error*.

## **11.1 Known standard deviation**

The foregoing statements describe a property of the sample mean in terms of the population mean. But the sample mean is known, and the population mean unknown, so we require a statement describing the population mean in terms of the known sample mean.

This is achieved by reversing the statement above, viz., if the sample mean is within 1.96 x SE(*x )* of the population mean, then the population mean is within 1.96 x SE(*x )* of the sample mean. That is, for 95% of samples it is true that *μ* lies in the interval.

*x*̅� 1*:*96xSEð Þ *x*̅to *x*̅þ 1*:*96xSEð Þ *x*̅

This interval is called the 95% confidence interval for u and the ends of the interval are called the 95% *confidence limits*. The single value *x* is called a *point estimate* of the population mean *μ*, in contrast with the above interval *estimate*.

#### **Example 1**

The respiratory health of a sample of 25 men exposed to fumes in a dental laboratory was assessed by measuring the forced expiratory volume (FEV). The sample mean was 3.20 liters. From previous work it is known that the standard deviation of FEV is 0.5 liters.

x̅= 3.20 SE (x̅) = 0.5/ √n = 0.1 *95% confidence interval for x is* 3.20 � 1.96 x 0.1 liters =3.00 to 3.40 liters

Conclusion We are 95% confident that the interval 3.00 to 3.40 liters contains the unknown population mean *μ.*

*99% confidence interval for x* is 3.20 � 2.5 x 0.1 liters =2.94 to 3.46 litres

Conclusion We are 99% confident that the interval 3.00 to 3.46 liters contains the unknown population mean *μ.*

**Note:** The 99% confidence interval is wider because of the extra confidence that the interval contains the population mean *μ.*

#### **11.2 Unknown standard deviation**

In the above it was assumed that the standard deviation in the population was known. In many practical situations this will not be the case and the standard deviation has to be estimated from the sample data. It therefore seems natural to replace **σ** by its estimate *s* and argue exactly as above. However, there is a loss of precision because the standard deviation has its own sampling error. This extra imprecision is included by widening the confidence intervals by using larger constants than 1.96 (for 5%) or 2.58 (for 1%).

**Example 2.** In (example 1) the values of FEV were:


*Probability and Sampling in Dentistry DOI: http://dx.doi.org/10.5772/intechopen.97705*

x̅= 3.20 *s* 0.54 (24*df*).

The sample mean is 3.20 liters and the sample standard deviation 0.54 liters. The standard error of the mean is 0.11 liters. The estimated standard deviation has 24 degrees of freedom and the 5% and 1% points of the *t* distribution with 24 df are 2.06 and 2.80 respectively.

Thus the 95% confidence interval is: 3.20 2.06x0.11 = 2.97 to 3.43 liters and the 99% confidence interval is: 3.20 2.800.11 = 2.89 to 3.51 liter.
