chapter6 estimation

Excellent does not an accident, but it comes through a hard work!! 1

CHAPTER SIX

ESTIMATION

PREPARED BY: NORYANI MUHAMMAD

Excellent does not an accident, but it comes through a hard work!!

1. To identify the use of t-test or z-test procedure in solving estimation problem.

2. To construct point and interval estimate for mean and proportion.

3. To determine sample size for mean and proportion.

OBJECTIVES

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INTRODUCTION

Inference: Methodologies that allow us to draw conclusions about population parameter from sample statistic. There are two procedures for making inferences: Estimation. Hypotheses testing

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6.1 CONSEPT OF ESTIMATION

Estimator : Sample statistic that is being used in estimating population parameter. Estimation : Procedure of estimating a population parameter based on sample statistic. Estimate : The value of population parameter based on estimator.

px , p,

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6.2 POINT & INTERVAL ESTIMATE

Point Estimator: A single value or a point calculated from sample data for

which we have some expectation to the population value. Margin of Error: EXAMPLE 1. An academic performance survey has been done in a university. It is known

that the standard deviation of all students grade point average (GPA) was 0.4. A sample of nine hundred (900) graduates was randomly selected and revealed that their mean GPA was 2.7.

(a) What is the point estimate of ? (b) What is the margin of error associated with the point estimate of ?

x96.1

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Test your understanding. Given = 158 cm, = 6.5 cm, n = 100 Find the point estimate for and the margin of error for this estimate.

x

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Interval Estimator: An interval within which we would expect to find the true

value of the population parameter. It is also stated as confidence interval.

confidence level-stated how much confidence we have that this interval contains the true value of the population parameter.

Confidence level = (1 )100% Confidence coefficient = (1 ) Significance level =

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EXAMPLE

Confidence level, (1 )100%

Confidence

coefficient, (1 ) Significance

level, Critical

z-value

(z/2)

90% 0.9 0.1 1.645

93%

95%

98%

99%

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INTERVAL ESTIMATION OF A POPULATION MEAN

Confidence

Interval

Mean

Large Sample (n30)

known unknown

Small Samples (n

EXAMPLES

1. The average zinc concentration recovered from a sample of zinc measurements in 36 different locations is found to be 2.6 grams per millilitre. Find the 97% confidence interval for the mean zinc concentration in the river. Assume that the sample standard deviation is 0.3.

2. A random sample of 10 chocolate energy bars of a certain brand has, on average, 230 calories with a standard deviation of 15 calories. Construct a 99% confidence interval for the true mean calorie content of this brand of energy bar. Assume that the distribution of the calories is approximately normal.


Test your understanding. Exercise 6.3, (pg 111) In 64 randomly selected hours of production, the mean and standard deviation of the number of acceptable pieces produced by an automatic stamping machine are 1,039 and 146 respectively. (a) Find the point estimate for the average number of accepting pieces for all such pieces produced by that automatic stamping. (b) Construct a 99.2% confidence interval to estimate the average number of accepting pieces for all such pieces produced by that automatic stamping. Exercise 6.5, (pg 112) You decided to study on average, how much time engineering students spend to watch television per night. A random sample of 10 students was selected and the results (in hour) obtained: 2, 1.5, 3, 2, 3.5, 1, 0.5, 3, 2, 4. Find a 95% confidence interval on the population mean. 11


INTERVAL ESTIMATION OF A POPULATION PROPORTION

Confidence Interval

Proportion

pszpp 2

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EXAMPLES

1. A quality study was performed on the manufacturing of Application

Specific Integrated Circuits (ASIC). A sample of 1,126 units revealed 114

nonconforming units.

(a) Find the point estimate for the proportion of nonconforming units.

(b) Find the margin of error of the estimation made in (a).

(c) Construct a 97.3% confidence interval for the proportion of non

conforming units.


Test your understanding. Exercise 6.6, (pg 114) A manufacturer of compact disk players uses a set of comprehensive tests to access the electrical function of its product. All compact disk players must pass all tests prior to being sold. A random sample of 500 disk players resulted in 15 failing one or more tests. Find a 96.3% confidence interval for the proportion of compact disk players from the population that passes all tests.

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6.3 DETERMINATION OF THE SAMPLE SIZE

The sample size required to estimate the population mean, with confident level, with a specified maximum error, given by OR ; Proportion

xzE

2

2

2

E

zn

2

2

E

szn

2

2

2

E

pqz

n

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EXAMPLES

1. An efficiency expert wishes to determine the average time that it takes to drill three holes in a certain metal clamp. How large a sample will he needed to be 95% confident that his sample mean will be within 15 seconds of the true mean? Assume that it is known from previous studies that = 40 seconds.

2. Suppose that we want to estimate the true proportion of defectives in a very large shipment of adobe bricks, and we want to be at least 95% confident that the error is at most 0.04. How large a sample will we need if: (a) We have no idea what the true proportion might be. (b) We know that the true proportion does not exceed 0.12?



Test your understanding.

Exercise 6.7, (pg 115) The dean of a college wants to use the mean of a random sample to estimate the average amount of time students take to get from one class to the next, and she wants to be able to assert with 99% confidence that the error is at most 0.25 minute. If she assumes that the standard deviation is 1.40 minutes, how large a sample will she have to take? Exercise 6.8, (pg 116) A study is to be made to estimate the percentage of citizens in a town who favour having their water fluoridated. (a) How large a sample is needed if one wishes to be at least 95% confident that our estimate is within 1% of the true percentage? (b) If a random sample of 200 citizens is selected and 114 are found in favour of having their water fluoridated, how large must the sample be if we wish to be 96% confident that our sample proportion will be within 0.02 of the true fraction of the citizens?

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-End of Chapter two-

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chapter6 estimation

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