Basic Properties of Confidence Intervals

7.1 Basic Properties of Confidence Intervals

  The basic concepts and properties of confidence intervals (CIs) are most easily intro- duced by first focusing on a simple, albeit somewhat unrealistic, problem situation. Suppose that the parameter of interest is a population mean m and that

  1. The population distribution is normal

  2. The value of the population standard deviation s is known Normality of the population distribution is often a reasonable assumption. However,

  if the value of m is unknown, it is typically implausible that the value of s would be available (knowledge of a population’s center typically precedes information con- cerning spread). We’ll develop methods based on less restrictive assumptions in Sections 7.2 and 7.3.

  ExamplE 7.1 Industrial engineers who specialize in ergonomics are concerned with designing workspace and worker-operated devices so as to achieve high productivity and com- fort. The article “Studies on Ergonomically Designed Alphanumeric Keyboards” (Human Factors, 1985: 175–187) reports on a study of preferred height for an exper- imental keyboard with large forearm–wrist support. A sample of n 5 31 trained typ- ists was selected, and the preferred keyboard height was determined for each typist. The resulting sample average preferred height was x 5 80.0 cm. Assuming that the preferred height is normally distributed with s 5 2.0 cm (a value suggested by data in the article), obtain a confidence interval (interval of plausible values) for m, the true average preferred height for the population of all experienced typists.

  n The actual sample observations x 1 ,x 2 ,…, x n are assumed to be the result of a

  random sample X 1 ,…, X n from a normal distribution with mean value m and stan-

  dard deviation s. The results described in Chapter 5 then imply that, irrespective of the sample size n, the sample mean X is normally distributed with expected value m and standard deviation s yÏn. Standardizing X by first subtracting its expected value and then dividing by its standard deviation yields the standard normal variable

  X2m

  Z5

  s yÏn

  278 ChaPter 7 Statistical Intervals Based on a Single Sample

  Because the area under the standard normal curve between 21.96 and 1.96 is .95,

  X2m

  1 s yÏn 2

  Now let’s manipulate the inequalities inside the parentheses in (7.2) so that they appear in the equivalent form l , m , u, where the endpoints l and u involve

  X and s yÏn. This is achieved through the following sequence of operations, each yielding inequalities equivalent to the original ones.

  1. Multiply through by s yÏn:

  , X2m,

  2. Subtract X from each term:

  , 2m , 2 X1 1.96 ?

  Ïn

  Ïn

  3. Multiply through by 21 to eliminate the minus sign in front of m (which reverses the direction of each inequality):

  .m. X2 1.96 ?

  Ïn

  Ïn

  that is,

  ,m, X1 1.96 ?

  Ïn

  Ïn

  The equivalence of each set of inequalities to the original set implies that

  1 Ïn

  s

  Ïn 2

  s

  P X2 1.96 ,m, X1 1.96 5 .95 (7.3)

  The event inside the parentheses in (7.3) has a somewhat unfamiliar appearance; previously, the random quantity has appeared in the middle with constants on both ends, as in a Y b. In (7.3) the random quantity appears on the two ends, whereas the unknown constant m appears in the middle. To interpret (7.3), think of a random interval having left endpoint X 2 1.96 ? s yÏn and right endpoint

  X1 1.96 ? s yÏn. In interval notation, this becomes

  1 Ïn Ïn 2

  The interval (7.4) is random because the two endpoints of the interval involve a ran dom variable. It is centered at the sample mean X and extends 1.96s yÏn to each side of X. Thus the interval’s width is 2 ? (1.96) ? s yÏn, a fixed number; only the location of the interval (its midpoint X) is random (Figure 7.2). Now (7.3) can

  be par aphrased as “the probability is .95 that the random interval (7.4) includes or covers the true value of m.” Before any data is gathered, it is quite likely that m will lie inside the interval (7.4).

  s n 1.96 s n 1.96

  2 1.96 s n

  X X 1 1.96 s n

  Figure 7.2 The random interval (7.4) centered at X

  7.1 Basic Properties of Confidence Intervals 279

  DEfinition If, after observing X 1 5 x 1 ,X 2 5 x 2 ,…, X n 5 x n , we compute the observed

  sample mean x and then substitute x into (7.4) in place of X, the resulting fixed interval is called a 95 confidence interval for m. This CI can be expressed either as

  1 Ïn

  s

  Ïn 2

  , x 1 1.96 ? is a 95 CI for m

  with 95 confidence

  Ïn

  Ïn

  A concise expression for the interval is x 6 1.96 ? s yÏn, where 2 gives the left endpoint (lower limit) and 1 gives the right endpoint (upper limit).

  ExamplE 7.2

  The quantities needed for computation of the 95 CI for true average preferred

  (Example 7.1

  height are s 5 2.0, n 5 31, and x 5 80.0. The resulting interval is

  That is, we can be highly confident, at the 95 confidence level, that

  79.3 , m , 80.7. This interval is relatively narrow, indicating that m has been rather precisely estimated.

  n