Estimation of Parameters

Overview of Regression Analysis: || Hypothesis Testing:

Learning Objectives:

Given the learning materials and activities of this chapter, they will be able to:

 Distinguish classical methods from Bayesian methods of estimates.

 Calculate the standard error of a sample.

 Calculate the margin of error for interval estimate.

 Construct interval estimates of the population mean given a specified level of confidence.

 Construct interval estimates of the population proportion with the specified confidence level.

Introduction

The estimate of parameters and hypothesis testing are the two main domains of statistical inference. The process of determining the value of a parameter from data acquired in a sample is known as estimation. The sample size is an important consideration in estimating. The method, function, or procedure used to estimate a population parameter is known as an estimator. Estimating population parameters can be done in two ways:

a. Classical method is based strictly on information obtained from a random sample selected from the population.

b. Bayesian method utilizes prior subjective knowledge about the probability distribution of the unknown parameters in conjunction with the information provided from the sample data.

We'll use the classical technique to estimate unknown population parameters like the mean, percentage, and variance by computing statistics from a random sample and applying the theory of sampling distributions in this text. In the classical method of estimation, there are two methods: point estimate and interval estimate.

Point Estimate

A single value is used to estimate a population parameter in a point estimate. Due to sampling error, the point estimate will differ from the population mean in most cases. It's now possible to determine how close the point estimate is to the population parameter. As a result, statisticians favor a different form of estimate.

Confidence Interval (Interval Estimate)

The Confidential Interval is a range of values or an interval used to estimate a parameter. The parameter in an interval estimate is described as being between two values. Before making an interval estimate, a degree of confidence might be applied. The confidence level refers to the likelihood that the interval estimate contains the genuine population mean or proportion. 90 percent, 95 percent, and 99 percent confidence intervals are three frequent confidence levels. The values of the standard deviations and the margin of error for the most commonly used confidence level are summarized in the table below.

The probability of incorrectly concluding that a confidence interval generated will contain the parameter is denoted by the term level of significance. The greater the level of confidence, the larger the z values, the larger the margin of error, and, of course, the wider the confidence interval, as shown in the table.

A point estimator's margin of error is subtracted and added to create an interval estimate. The sample size, standard deviation, and desired confidence level all determine the length of the confidence interval.

Estimating Means

Confidence interval for means sample size greater than or equal to 30 (n>=30) and the standard deviation is known the formula is:

Example:A study of 40 bowlers showed that their average score was 186. The standard deviation of the population is 6.

a. Find the 95% confidence interval of the mean score for all bowlers.

b. Find the 99% confidence interval of the mean score of a sample of 100 bowlers instead of a sample of 40.

Thus, it can be 95% confident that the true mean score of bowlers is between 184.14 and 187.86. This means that 95% of the time, the population mean score of bowlers will be roughly between 184 and 188.

Thus, the 99% confidence interval for the population mean score is ranging from 184.542 to 187.548. This means that we can be 99% confident that the population mean score is roughly between 185 to 188.

Confidence interval for means with small sample size (n < 30) and the standard deviation is unknown

If the population standard deviation is unknown and the sample size is less than 30, the sample standard deviation can be used for the population standard deviation. The t-distribution is utilized to calculate the confidence interval in this case, and the random variable is roughly normally distributed. The formula is as follows:

To determine the value of t_(α/2) locate the critical value from the table in critical value of T-distribution: Table IV below with the corresponding degrees of freedom. The degrees of freedom are the values that are free to vary after a sample statistic has been computed. The degrees of freedom for the confidence interval for the mean is n – 1. Also note that the sample standard deviation is used instead of the population standard deviation.

Example:A sample of 20 tuna showed that they swim an average of 8.6 miles per hour. The standard deviation for the sample was 1.6. Find the 95% confidence interval of the true mean.

Solution: Given n = 20, the sample mean of 8.6 miles per hour, and the sample standard deviation s = 1.6 miles per hour. The degrees of freedom is n – 1 = 20 – 1, using the t-distribution table in Table IV yielded a critical value of t_(α/2) =2.093. Hence,

Thus, the 95% confidence interval for the population mean time is ranging from 7.851 to 9.349 miles per hour. This means that we can be 95% confident that the true mean time of tuna can swim roughly between 8 and 9 miles per hour.

When to use the z and t distribution:

- If the population standard deviation is known and sample size is large, use z-test.

- If the population standard deviation is unknown and sample size is large, use z-test.

- If the population standard deviation is unknown and sample size is small, use t-test.

Estimating Proportions

When the variable of interest is qualitative and are summarized in terms of frequencies, confidence intervals for estimating proportion may be constructed.

To construct confidence interval for estimating a population proportion based on a proportion obtain from a random sample, similar procedure used to estimate population mean.

The confidence interval for a population proportion is given by formula below:

Where;

sample proportion p=x/n, is the sample proportion, with x as the number of times that an event has occurred in n trials

√(pq/n) – standard error of the proportion

q = 1 – p

Example:A local polling organization reports that based on a local-wide survey of 500 respondents, 43% of the vote will be in favor of the administration governatorial candidate in the May X elections. Construct the 95% confidence interval for the proportion indicating preference for the administration candidate.

Solution: Given n = 500, z_(α/2) = 1.96, p = 0.43, and q = 1.0 – 0.43 = 0.57. Hence,

Thus, the 95% confidence interval is from 38.7% to 47.3%. This means that with a sample of 500, the poll has a margin of error of ±4.3% and the pollster can be 95% confident that the administration candidate will obtain roughly between 39% and 47% of the votes.

Table/Critical Value of T-distribution: Table IV

I'm sure I don't have all of the information about the Estimation of Parameters right here." I'm hoping and would love to hear your thoughts on this topic in the comment section below."

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Overview of Regression Analysis: || Hypothesis Testing:

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