Chapter 8 : Estimation

8.1 Introduction The field of statistical inference consist of those methods used to make decisions or to draw conclusions about a population. These methods utilize the information contained in a sample from the population in drawing conclusions.

Estimator VS Estimate Estimator Estimate In statistics, the method used Estimate The value that obtained from a sample I have a sample of 5 numbers and I take the average. The estimator is taking the average of the sample. The estimator of the mean. Let say, the average = 4 the estimate.

Point Estimation Point estimate – an estimate of the parameter using a single number. E.g : is the point estimate for In choosing the point estimators, we have to depends on the properties of the estimators. Unbiased Consistent Efficient Sufficient

Unbiased If the expected value of the statistics is equal to the population parameter. Example 8.1 If a random sample size n is taken from a population with mean and variance , hence is an unbiased estimator for .

Consistent If it gets closer to the parameter value as the sample size increase Consistent if its variance decrease while the n increase. Example 8.2 If a random sample size n is taken from a population with mean, and variance , hence (mean sample) is a consistent estimator for .

Variance for sample mean, Efficient For 2 or more unbiased estimators, the one with the smallest variance is considered the most efficient estimator. Example 8.3 Sample mean is an efficient estimator compares to sample median in estimating the population mean. . This gives if Proof: Variance for sample mean, Variance for sample median, Thus, significantly, mean is more efficient in estimating

Sufficient If it used all the sample’s information

Point estimators for mean, variance, and proportion Population mean Given a sample X1, X2,X3,...,Xn of size n taken from a certain population with unknown mean, µ and variance, σ2 . The sample mean is the best estimator of µ. Population variance Given a sample X1, X2,X3,...,Xn of size n taken from a certain population with mean, µ and variance, σ2 . The sample variance is the best estimator of . Population proportion Given a sample X1, X2,X3,...,Xn of size n taken from a certain population with unknown proportion P . The sample proportion is the best estimator of P.

8.2 Interval Estimation Definition 8.1: An Interval Estimate In interval estimation, an interval is constructed around the point estimate and it is stated that this interval is likely to contain the corresponding population parameter. Definition 8.2: Confidence Level and Confidence Interval Each interval is constructed with regard to a given confidence level and is called a confidence interval. The confidence level associated with a confidence interval states how much confidence we have that this interval contains the true population parameter. The confidence level is denoted by .

8.2.1 Confidence Interval Estimates for Population Mean

Example 8.4

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Example 8. 5 : A publishing company has just published a new textbook Example 8.5 : 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. Construct a 90% confidence interval for the mean price of all such college textbooks.

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8.2.3 Confidence Interval Estimates for Population Proportion

Example 8.6 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 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.

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8.2.5 Error of Estimation and Determining the Sample size Definition 8.3:

Example 8.7: sample of size n=150 to estimate the average mechanical A team of efficiency experts intends to use the mean of a random sample of size n=150 to estimate the average mechanical aptitude of assembly-line workers in a large industry (as measured by a certain standardized test). If, based on experience, the efficiency experts can assume that for such data, what can they assert with probability 0.99 about the maximum error of their estimate?

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Definition 8.4:

Example 8.8: A study is made to determine the proportion of voters in a sizable community who favor the construction of a nuclear power plant. If 140 of 400 voters selected at random favor the project and we use as an estimate of the actual proportion of all voters in the community who favor the project, what can we say with 99% confidence about the maximum error?

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Example 8.9: How large a sample required if we want to be 95% confident that the error in using to estimate p is less than 0.05? If , find the required sample size.

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