Learning Objectives:
Given the learning materials and activities of this chapter, they will be able to:
Ø
Distinguish
classical methods from Bayesian methods of estimates.
Ø
Calculate
the standard error of a sample.
Ø
Calculate
the margin of error for interval estimate.
Ø
Construct
interval estimates of the population mean given a specified level of
confidence.
Ø
Construct
interval estimates of the population proportion with the specified confidence
level.
Introduction
There are two major areas in statistical inference, the estimation of parameters and hypothesis testing. Estimation is the process of estimating the value of a parameter from information obtained in a sample. An important aspect of estimation is the size of the sample. An estimator is a formula, the function or procedure used in estimating a population parameter. There are two methods of estimating population parameters, such as:
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.
In this text, we shall utilize the classical method to
estimate unknown population parameters such as the mean, proportion and the
variance by computing statistics from a random sample and applying the theory of
sampling distributions. There are two ways in the classical method of estimation,
namely: point estimate and interval estimate.
Point estimate
Point estimate consists of a single
value used to estimate a population parameter. For most parts, the point
estimate will be different from the population mean due to sampling error.
There is now way of knowing how close the point estimate is to the population
parameter. For this reason, statisticians prefer another type of estimate.
Interval estimate
Is an interval or a
range of values used to estimate the parameter. In an interval estimate, the
parameter is specified as being between two values. A degree of confidence can
be assigned before an interval estimate is made. The confidence level is the probability
that the interval estimate will contain the true population mean or population
proportion.Three common confidence level are 90%, 95% and 99% confidence intervals. The table below summarizes the values of the standard deviates and the margin of error for the most commonly used confidence level.
A
term level of significance is
defined as the probability of erroneously concluding that a confidence interval
generated will contain the parameter. As observed in the table, 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.
An interval estimate is constructed
by subtracting and adding the margin of error to a point estimator. The length
of the confidence interval is determined by the sample size, the standard deviation, and the desired confidence level.
Estimating Means
Estimation (or estimating) is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available. Typically, estimation involves "using the value of a statistic derived from a sample to estimate the value of a corresponding population parameter". The sample provides information that can be projected, through various formal or informal processes, to determine a range most likely to describe the missing information. An estimate that turns out to be incorrect will be an overestimate if the estimate exceeded the actual result, and an underestimate if the estimate fell short of the actual result.
Estimating Means Large Sample and the Standard Deviation is known
The Central Limit Theorem says that, for large samples (samples of size n ≥ 30), when viewed as a random variable the sample mean is normally distributed with mean and standard deviation. The Empirical Rule says that we must go about two standard deviations from the mean to capture 95% of the values of sample mean generated by sample after sample.
Confidence interval for means >=30 and the standard deviation is known the formula is
Confidence interval for means >=30 and the standard deviation is known the formula is
Example 1: 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 < 30 and the standard deviation is unknown (small sample)
When the population standard deviation is unknown and the sample size is less than 30, the standard deviation from the sample can be used in place of the population standard deviation. In this case, the t-distribution is used to determine the confidence interval and the random variable is approximately normally distributed. The formula is:
To
determine the value of t critical locates the critical value from the table in t distribution 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 2: 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 yielded a critical value of t =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.
Example 3: 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 2016 elections. Construct the 95% confidence interval for the proportion indicating preference for the administration candidate.
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.
Note: For Comments, Questions, and Suggestions feel free to contact at enomaratas@jrmsu.edu.ph or ednielmaratas@gmail.com. You can also post at the comment section below.
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