2. OBJECTIVES
1. Explain and understand the general concepts of estimating
the parameters of a population or a probability distribution.
2. Calculate and explain the important rule of the normal
distribution as a sampling distribution and the central limit
theorem.
3. INTRODUCTION
Statistical methods are used to make decisions and draw conclusions about
populations. This aspect of statistics is generally called statistical inference.
These techniques utilize the information in a sample for drawing conclusions. This
chapter covers the study of the statistical methods used in decision making.
Statistical inference has one major areas which is the parameter estimation. In
practice, the engineer will use sample data to compute a number that is in some
sense a reasonable value (a good guess) of the true population mean.
This number is called a point estimate. In this chapter, we will see that
procedures are available for developing point estimates of parameters that have
good statistical properties.
4. POINT ESTIMATION
• Point estimation is the process of using the data available to estimate the
unknown value of a parameter, when some representative statistical model has
been proposed for the variation observed in some chance phenomenon.
• A point estimate of some population parameter θ is a single numerical value of
a statistic . The statistic is called the point estimator.
6. • Let X be the height of a randomly chosen individual from a population. In order
to estimate the mean and variance of X, we observe a random sample X1, X2,
, X
⋯⋯ n. We obtain the following values (in centimeters):
• Find the values of the sample mean, the sample variance, and the sample
standard deviation for the observed sample
166.8 171.4 169.1 178.5 168.0
157.9 170.1 166.8 171.4 169.1
178.5 168.0 157.9 170.1 159.9
7. POINT ESTIMATION
Estimation problems occur frequently in engineering. We often
need to estimate
The mean μ of a single population
The variance (or standard deviation σ) of a single population
The proportion p of items in a population that belong to a class of
interest
The difference in means of two populations, μ1 − μ2
The difference in two population proportions, p1 − p2
8. SAMPLING DISTRIBUTION AND CENTRAL LIMIT
THEOREM
• SAMPLING DISTRIBUTION is the probability distribution of a
statistic.
• The sampling distribution of a statistic depends on the distribution
of the population, the size of the samples, and the method of
choosing the samples.
• The probability distribution of is called the sampling distribution of
the mean.
• The standard distribution of the sampling distribution of the sample mean is also
known as the standard error of the mean.
9. SAMPLING DISTRIBUTION AND CENTRAL LIMIT
THEOREM
• CENTRAL LIMIT THEOREM
• If we are sampling from a population that has an unknown probability distribution,
the sampling distribution of the sample mean will still be approximately normal
with mean μ and variance σ2
/n if the sample n is large.
• This is one of the most useful theorems in statistics, called the central limit
theorem.
Z=
X − μ
√σ
2
n
∨Z =
X − μ
σ
√n
10. SAMPLING DISTRIBUTION AND CENTRAL LIMIT
THEOREM
Figure illustrates how the theorem
works.
It shows how the distribution of X
becomes closer to normal as n
grows larger, beginning with the
clearly nonsymmetric distribution of
an individual observation (n = 1).
It also illustrates that the mean of X
remains μ for any sample size and
the variance of gets smaller as n
increases.
11. • An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed,
with mean equal to 800 hours and a standard deviation of 40 hours. Find the probability that a random sample of
16 bulbs will have an average life of less than 775 hours.
12. • An electronics company manufactures resistors that have a mean resistance of 100 ohms and a standard
deviation of 10 ohms. The distribution of resistance is normal. Find the probability that a random sample of n =
25 resistors will have an average resistance of greater than 95 ohms.
13. APPROXIMATE SAMPLING DISTRIBUTION OF A
DIFFERENCE IN SAMPLE MEANS
If we have two independent populations with means μ1 and μ2 and variances and
and if and are the sample means of two independent random samples of sizes
n1 and n2 from these populations, then the sampling distribution of the equation
below is approximately standard normal if the conditions of the central limit
theorem apply. If the two populations are normal, the sampling distribution of Z
is exactly standard normal.
Z=
( X1− X2)−( μ1− μ2)
√σ1
2
n1
+
σ2
2
n2
14. • Two independent experiments are run in which two different types of paint are compared. Eighteen specimens
are painted using type A, and the drying time, in hours, is recorded for each. The same is done with type B. The
population standard deviations are both known to be 1.0. If the mean drying time is equal for the two types of
paint, find , where and are average drying times for samples of size nA = nB = 18.
EXPERIMENT A EXPERIMENT B
15. The television picture tubes of manufacturer A have a mean lifetime of 6.5 years and a standard deviation of 0.9
year, while those of manufacturer B have a mean lifetime of 6.0 years and a standard deviation of 0.8 year. What
is the probability that a random sample of 36 tubes from manufacturer A will have a mean lifetime that is at least 1
year more than the mean lifetime of a sample of 49 tubes from manufacturer B?
16. A random sample of size 25 is taken from a normal population having a mean of 80 and a standard deviation of 5.
A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard
deviation of 3. Find the probability that the sample mean computed from the 25 measurements will exceed the
sample mean computed from the 36 measurements by at least 3.4 but less than 5.9.
POPULATION 1 POPULATION 2
