Page 313 - Applied statistics and probability for engineers

Section 8-4/Large-Sample Conidence Interval for a Population Proportion 291

normality of the population and comment on the assumptions 22.2 24.7 20.9 26.0 27.0
for the conidence interval.
8-55. An article in Technometrics (1999, Vol. 41, pp. 202– 24.8 26.5 23.8 25.6 23.9
2
211) studied the capability of a gauge by measuring the weight (a) Construct a 99% two-sided conidence interval for σ .
of paper. The data for repeated measurements of one sheet of (b) Calculate a 99% lower conidence bound for σ .
2
paper are in the following table. Construct a 95% one-sided (c) Calculate a 90% lower conidence bound for σ.
upper conidence interval for the standard deviation of these (d) Compare the intervals that you have computed.
measurements. Check the assumption of normality of the data 8-57. From the data on the pH of rain in Ingham County,
and comment on the assumptions for the conidence interval.
Michigan:
Observations 5.47 5.37 5.38 4.63 5.37 3.74 3.71 4.96 4.64 5.11 5.65
3.481 3.448 3.485 3.475 3.472 5.39 4.16 5.62 4.57 4.64 5.48 4.57 4.57 4.51 4.86 4.56
4.61 4.32 3.98 5.70 4.15 3.98 5.65 3.10 5.04 4.62 4.51
3.477 3.472 3.464 3.472 3.470
4.34 4.16 4.64 5.12 3.71 4.64
3.470 3.470 3.477 3.473 3.474
Find a two-sided 95% conidence interval for the standard
8-56. An article in the Australian Journal of Agricultural deviation of pH.
Research [“Non-Starch Polysaccharides and Broiler Perfor- 8-58. From the data on CAT scans in Exercise 8-45
mance on Diets Containing Soyabean Meal as the Sole Protein (a) Find a two-sided 95% conidence interval for the standard
Concentrate” (1993, Vol. 44(8), pp. 1483–1499)] determined deviation.
that the essential amino acid (Lysine) composition level of soy- (b) What should you do to address any reservations about the
bean meals is as shown here (g/kg): conidence interval you found in part (a)?

8-4 Large-Sample Confidence Interval
for a Population Proportion

It is often necessary to construct conidence intervals on a population proportion. For exam-
ple, suppose that a random sample of size n has been taken from a large (possibly ininite)
population and that X(≤ n) observations in this sample belong to a class of interest. Then
ˆ
P = X n is a point estimator of the proportion of the population p that belongs to this class.
Note that n and p are the parameters of a binomial distribution. Furthermore, from Chapter 4
ˆ
we know that the sampling distribution of P is approximately normal with mean p and vari-
ance p(1− p n, if p is not too close to either 0 or 1 and if n is relatively large. Typically, to
)
apply this approximation we require that np and n(1 – p) be greater than or equal to 5. We will
use the normal approximation in this section.

Normal
Approximation If n is large, the distribution of
for a Binomial
ˆ
Proportion Z = X − np = P − p
np − p) p − p)
(1
(1
n
is approximately standard normal.

To construct the conidence interval on p, note that
P − ( z / Ð Z ≤ z / ) 1≃ − ≠
α 2
α 2
so
⎛ ⎞
⎜ P − ⎟
ˆ
⎜ z / Ð p ≤ z / 2 ≃ − ≠
⎟
P − α 2 α 1
⎜ p − p) ⎟
(1
⎜ ⎝ n ⎟ ⎠