*ESTIMATION AND HYPOTHESIS TESTING

Statistical Inference

Estimation

-Confidence interval estimation for mean and

proportion

-Determining sample size

Hypothesis Testing

-Test for one and two means

-Test for one and two proportions

• Statistical inference is a process of drawing an inference about the data statistically.

• It concerned in making conclusion about the characteristics of a population based on information contained in a sample.

• Since populations are characterized by numerical descriptive measures called parameters, therefore, statistical inference is concerned in making inferences about population parameters.

* Statistical Inference

ESTIMATION

In estimation, there are two terms that firstly, should be understand. The two terms involved in estimation are estimator and estimate.

An estimate of a population parameter may be expressed in two ways: point estimate and interval estimate.

Point Estimate

A point estimate of a population parameter is a single value of a statistic. For example, the sample mean is a point estimate of the population mean μ. Similarly, the sample proportion is a point estimate of the population proportion p.

x

p̂

Interval estimate

An interval estimate is defined by two numbers, between which a population parameter is said to lie.

For example, a < < b is an interval estimate of the population mean μ. It indicates that the population mean is greater than a but less than b.

x

Point estimators

Choosing the right point estimators to estimate a parameter depends on the properties of the estimators it selves. There are four properties of the estimators that need to be satisfied in which it is considered as best linear unbiased estimators. The properties are: Unbiased Consistent Efficient Sufficient

Confidence Interval

• A range of values constructed from the sample data. So that the population parameter is likely to occur within that range at a specified probability.

• Specified probability is called the level of confidence.

• States how much confidence we have that this interval contains the true population parameter. The confidence level is denoted by

• Example :- 95% level of confidence would mean that if 100 confidence intervals were constructed, each based on the different sample from the same population, we would expect 95 of the intervals to contain the population mean.

To compute a confidence interval, we will consider two situations:

i. We use sample data to estimate, with and the population standard deviation is known.

ii. We use sample data to estimate, with and the population standard deviation is unknown. In this case, we substitute the sample standard deviation (s) for the population standard deviation

X

X

Example 1:

A publishing company has just published a new textbook. Before the company decides the price at which to sell this textbook, it wants to know the average price of all such textbooks in the market. The research department at the company took a sample of 36 comparable textbooks and collected the information on their prices. This information produced a mean price RM 70.50 for this sample. It is known that the standard deviation of the prices of all such textbooks is RM4.50.

(a) What is the point estimate of the mean price of all such college textbooks?

(b) Construct a 90% confidence interval for the mean price of all such college textbooks.

Solution:(a) The point estimate of the mean price of all

such college textbooks is RM70.50, that is Point estimate of μ = =

RM70.50

(b) It is known that, n = 36, μ = = RM70.50 and RM4.50

x

x

For 90% CI

From normal distribution table:

Hence, 90% CI:

Thus, we are 90% confident that the mean price of all such college textbooks is

between RM69.26 and RM 71.73.

90 1 100

1 0 90

0 1

0 052

% %

.

.

.

0 05

2

1 65.z z .

2

4 570 50 1 6449

36

70 50 1 2337

RM69 26 RM71 73

x Zn

.. .

. .

. , .

Example 2:

The brightness of a television picture tube can be evaluated by measuring the amount of current required to achieve a particular brightness level. A random sample of 10 tubes indicated a sample mean 317.2microamps and a sample standard deviation is 15.7microamps. Find (in microamps) a 99% confidence interval estimate for mean current required to achieve a particular brightness level.

Solution:

For 99% CI:

From t normal distribution table:

317 2x .15 7s . 15 7 10 30 317 2s . , n , x .

99 1 100

1 0 99

0 01

0 0052

% %

.

.

.

0 005 9

2

1 3 250. ,t ,n t .

Hence 99% CI

Thus, we are 99% confident that the mean current required to achieve a particular brightness level is between 301.0645 and 333.3355

0 005 9

15 7317 2

10

15 7317 2 3 250

10

301 0645 333 3355 microamps

. ,

.. t

.. .

. , .

Exercise 1:

Taking a random sample of 35 individuals waiting to be serviced by the teller, we find that the mean waiting time was 22.0 min and the standard deviation was 8.0 min. Using a 90% confidence level, estimate the mean waiting time for all individuals waiting in the service line.

Answer : [19.7757, 24.2243]

Example 3:According to the analysis of Women Magazine in June 2005, “Stress has become a common part of everyday life among working women in Malaysia. The demands of work, family and home place is an increasing burden on average Malaysian women”. According to this poll, 40% of working women included in the survey indicated that they had a little amount of time to relax. The poll was based on a randomly selected of 1502 working women aged 30 and above. Construct a 95% confidence interval for the corresponding population proportion.

Solution:Let p be the proportion of all working women age 30 and above, who have a limited amount of time to relax, and let pˆ be the corresponding sample proportion. From the given information,n = 1502 , pˆ = 0.40 , qˆ =1− pˆ = 1 – 0.40 = 0.60

Hence, 95% CI :

Thus, we can state with 95% confidence that the proportion of all working women aged 30 and above who have a limited amount of time to relax is between 37.5% and 42.5%.

2

0 025

0 4 0 60 40 0 4 0 01264069

1502

0 375 0 425 or 37 5% to 42%

.

ˆ ˆpqp̂ Z

n

. .. Z . .

. , . .

Exercise 2In a random sample of 70 automobiles registered in a certain state, 28 of them were found to have emission levels that exceed a state standard. Find a 95% confidence interval for the proportion of automobiles in the state whose emission levels exceed the standard.

Answer : [0.2852, 0.5148]

Exercise 3:

A paint manufacturing company claims that the mean drying time for its paint is at most 45 minutes. A random sample of 20 trials tested. It is found that the sample mean drying time is 49.50 minutes with standard deviation 3 minutes. Assume that the drying times follow a normal distribution. Construct a 99% confidence interval for the mean drying time of the paint. Explain your answer.

Answer : [47.58, 51.42]