Estimation of Parameters

EstimationofParameters
Estimation of
Parameters
• Parameters are descriptive measures of an entire population.

However, their values are usually unknown because it is
infeasible to measure an entire population. Because of this,
you can take a random sample from the population to obtain
parameter estimates. One goal of statistical analyses is to
obtain estimates of the population parameters along with the
amount of error associated with these estimates. These
estimates are also known as sample statistics.
• Technically the mere purpose of
Estimation of Parameters is to give
us a certain interval that contains
our population proportion using our
obtained sample proportion
There are several
types of parameter
estimates:
• Point estimates are the single,

most likely value of a
parameter. For example, the
point estimate of population
mean (the parameter) is the
sample mean (the parameter
estimate).
• Confidence intervals are a
range of values likely to
contain the population
parameter.
• For an example of parameter estimates, suppose you
work for a spark plug manufacturer that is studying a
problem in their spark plug gap. It would be too costly
to measure every single spark plug that is made.
Instead, you randomly sample 100 spark plugs and
measure the gap in millimeters. The mean of the
sample is 9.2. This is the point estimate for the
population mean (μ). You also create a 95% confidence
interval for μ which is (8.8, 9.6). This means that you
can be 95% confident that the true value of the average
gap for all the spark plugs is between 8.8 and 9.6.
Margin of Errors
• in statistics, the degree of error in

results received from random sampling
surveys. A higher margin of error in
statistics indicates less likelihood of
relying on the results of a survey or
poll, i.e. the confidence on the results
will be lower to represent a population.
Confidence level
• refers to the percentage of
probability, or certainty, that the
confidence interval would contain
the true population parameter
when you draw a random sample
many times.
Think it through!
• A confidence interval for the mean is a way of estimating the true
population mean. Instead of a single number for the mean, a
confidence interval gives you a lower estimate and an upper
estimate. For example, instead of “6” as the mean you might get
{5,7}, where 5 is the lower estimate and 7 is the upper.
• let's say that the average score of students in quiz is 8, with a 95%
confidence interval range of 7-9. This means that with high certainty,
the child's true score lies between 7 and 9, even if the received score
of 8 is not 100% accurate.

𝒙
^=
𝒑
𝒏
Sample Proportion
𝒏𝒑 (𝟏 −𝒑 ) ≥𝟏𝟎
Is the formula we can use to

approximate a Binomial
Distribution with a Standard
Normal distribution
Confidence Level Z-Score
90% 1.645
95% 1.96
98% 2.33
99% 2.575
𝑬=𝒛 𝜶
𝟐 √ ^𝒒
𝒑
𝒏
^
^ −𝒛 𝜶
𝒑
𝟐 √ ^𝒒
𝒑
𝒏
^
^ +𝒛 𝜶
<𝒑< 𝒑
𝟐
^𝒒
𝒑
𝒏
^
√
Example #1
• 800 randomly selected students at university AJB were asked if

they own a laptop. 584 responded YES to survey
a. Can we approximate the Binomial Distribution with a Normal Distribution
b. Conduct a 95% confidence interval to estimate the True Population Proportion of

students at University AJB who owns a laptop and determine the margin of error
c. What sample size is needed to estimate the true proportion with 2% margin of
error at a 90% confidence interval?
a. Can we approximate the Binomial Distribution with a
Normal Distribution
a.
𝒏=𝟖𝟎𝟎 𝒔𝒂𝒎𝒑𝒍𝒆𝒔
𝒙=𝟓𝟖𝟒 𝒏𝒐. 𝒐𝒇 𝒔𝒖𝒄𝒄𝒆𝒔𝒔 , 𝒔𝒊𝒏𝒄𝒆𝒕𝒉𝒆𝒚 𝒂𝒓𝒆 𝒕𝒉𝒆𝒐𝒏𝒆𝒔 𝒘𝒉𝒐 𝒉𝒂𝒗𝒆 𝒍𝒂𝒑𝒕𝒐𝒑
𝒙
=
𝒏
𝟓𝟖𝟒
=
𝟖𝟎𝟎
=𝟎 . 𝟕𝟑 %Probability of success
This distribution can
be approximated as
Normal Distributed
𝒏=𝟖𝟎𝟎 𝒔𝒂𝒎𝒑𝒍𝒆𝒔
𝒙=𝟓𝟖𝟒 𝒏𝒐. 𝒐𝒇 𝒔𝒖𝒄𝒄𝒆𝒔𝒔 , 𝒔𝒊𝒏𝒄𝒆𝒕𝒉𝒆𝒚 𝒂𝒓𝒆 𝒕𝒉𝒆𝒐𝒏𝒆𝒔 𝒘𝒉𝒐 𝒉𝒂𝒗𝒆 𝒍𝒂𝒑𝒕𝒐𝒑
𝒙
=
𝒏
𝟓𝟖𝟒
=
𝟖𝟎𝟎
=𝟎 . 𝟕𝟑 %
^
𝒒=𝟏 ^
−𝒑
𝟏 − (𝟎 . 𝟕𝟑 )
= 𝟎 . 𝟐𝟕Probability of Failure
b. Conduct a 95% confidence interval to estimate the True
Population Proportion of students at University AJB who owns a
laptop and determine the margin of error
b.
𝒑^ − 𝒛 𝜶
𝟐 √ ^𝒒
𝒑
𝒏
^
^ +𝒛 𝜶
<𝒑< 𝒑
𝟐√^𝒒
𝒑
𝒏
^
( 𝟎 . 𝟕𝟑 ) −(𝟎 . 𝟎𝟑𝟎𝟖)< 𝒑< ( 𝟎 .𝟕𝟑 ) +( 𝟎 .𝟎𝟑𝟎𝟖 )
𝟔𝟗𝟗𝟐< 𝒑 <𝟎 . 𝟕𝟔𝟎𝟖

We are 95% certain that our
. 𝟕𝟎< 𝒑 <𝟎 . 𝟕𝟔
population mean is in the
interval 0.70 to 0.76
c. What sample size is needed to estimate the true
proportion with 2% margin of error at a 90% confidence
interval?
c. We need 1,333 samples to be 90%
confident that a certain interval
would contain our population mean

Estimation of Parameters

Uploaded by

Copyright:

Available Formats

Estimation of Parameters

Uploaded by

Document Information

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Estimation of Parameters

Uploaded by

Copyright:

Available Formats

EstimationofParameters

• Parameters are descriptive measures of an entire population.

• Point estimates are the single,

• in statistics, the degree of error in

• A confidence interval for the mean is a way of estimating the true

population mean. Instead of a single number for the mean, a

confidence interval gives you a lower estimate and an upper

{5,7}, where 5 is the lower estimate and 7 is the upper.

of 8 is not 100% accurate.

Is the formula we can use to

• 800 randomly selected students at university AJB were asked if

a. Can we approximate the Binomial Distribution with a Normal Distribution

b. Conduct a 95% confidence interval to estimate the True Population Proportion of

( 𝟎 . 𝟕𝟑 ) −(𝟎 . 𝟎𝟑𝟎𝟖)< 𝒑< ( 𝟎 .𝟕𝟑 ) +( 𝟎 .𝟎𝟑𝟎𝟖 )

𝟔𝟗𝟗𝟐< 𝒑 <𝟎 . 𝟕𝟔𝟎𝟖

You might also like