Lecture 5 Estimation of Parameters Part 2
Lecture 5 Estimation of Parameters Part 2
Lecture 5 Estimation of Parameters Part 2
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Statistics and Probability: Confidence Intervals
In this lesson, you will learn how to make a true estimate of a parameter,
what is meant by the margin of error, and whether or not the sample size is
large enough to represent all Filipinos.
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Statistics and Probability: Confidence Intervals
Statistical Inference
It refers to methods by which one uses sample information to
make inferences or generalizations about a population.
1. Estimation
2. Hypothesis Testing
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Statistics and Probability: Confidence Intervals
Estimation
One aspect of inferential statistics is estimation, which is the
process of estimating the value of a parameter from information
obtained from a sample.
Examples:
• the sample mean 𝑿 ഥ is an estimator of the population mean µ.
• the sample standard deviation 𝒔 is an estimator of the
population standard deviation 𝝈.
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Statistics and Probability: Confidence Intervals
Estimation
consistent
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Statistics and Probability: Confidence Intervals
Estimation
Properties of a Good Estimator:
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Statistics and Probability: Confidence Intervals
Types of Estimate
A point estimate is a specific numerical value estimate of a
parameter.
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Statistics and Probability: Confidence Intervals
Why????
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Statistics and Probability: Confidence Intervals
Confidence Intervals
For example, say the interval estimate from a survey have 90% confidence
level. This means that, if the survey were to be done for 100 times, then in 90 of
those times, the interval estimate will enclose or capture the true value of the
population parameter and not the other way around.
We must NOT say that there is a 90% chance or probability that the true
value of the parameter falls within the interval estimate, because it implies
that the parameter may be within this interval, or it may be somewhere else.
Confidence Intervals
There are three common confidence intervals that are used: the
90%, the 95%, and the 99% confidence intervals.
Remarks:
• Increasing the confidence level will increase the margin of
error resulting in a wider interval.
• Decreasing the confidence level will decrease the margin of
error resulting in a narrower interval
• If the confidence level is higher, the probability that an interval
estimate CAPTURES or ENCLOSES the true value of the
population parameter also increases.
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Statistics and Probability: Confidence Intervals
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Statistics and Probability: Confidence Intervals
NO YES
YES NO
S S S S
X − z / 2 , X + z / 2 X − t / 2 , X + t / 2
n n n n
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Statistics and Probability: Confidence Intervals
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Statistics and Probability: Confidence Intervals
Questions to Answer:
• Is the population standard deviation (σ) known?
YES, σ = 40
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Statistics and Probability: Confidence Intervals
Given:
𝑋ത = 780, σ = 40, n = 25
For 95% confidence level, use 𝑧𝛼Τ2 = 1.96
𝜎 𝜎
(𝑋ത − 𝑧𝛼ൗ ത
, 𝑋 + 𝑧𝛼ൗ )
2 𝑛 2 𝑛
40 40
(780 − (1.96) , 780 + (1.96) )
25 25
Questions to Answer:
• Is the population standard deviation (σ) known?
NO, σ is unknown.
• Is the sample size greater than or equal to 30?
NO, n = 20 which is less than 30.
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Statistics and Probability: Confidence Intervals
Given:
𝑋ത = 11.3, s = 2.45, n = 20
For 95% confidence level and v = n – 1 = 20 – 1 = 19,
𝑡𝛼Τ2 = 2.093
𝑠 𝑠
(𝑋ത − 𝑡𝛼ൗ ത
, 𝑋 + 𝑡𝛼ൗ )
2 𝑛 2 𝑛
2.45 2.45
(11.3 − (2.093) , 11.3 + (2.093) )
20 20
Questions to Answer:
• Is the population standard deviation (σ) known?
NO, σ is unknown.
• Is the sample size greater than 30?
YES, n = 100 which is greater than 30.
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Statistics and Probability: Confidence Intervals
Given:
𝑋ത = 23500, s = 3900, n = 100
For 99% confidence level, use 𝑧𝛼Τ2 = 2.576
𝑠 𝑠
(𝑋ത − 𝑧𝛼ൗ , 𝑋ത + 𝑧𝛼ൗ )
2 𝑛 2 𝑛
3900 3900
(23500 − (2.576) , 23500 + (2.576) )
100 100
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Statistics and Probability: Confidence Intervals
Estimating Proportions
DEFINITION:
𝒙
ෝ= ,
𝒑
𝒏
where
x is the number of sample units that possess the characteristics of interest
n is the sample size
The proportion of the sample that does not possess the characteristics of
interest is given by:
𝒏−𝒙
ෝ=
𝒒 𝐨𝐫 ෝ =𝟏−𝒑
𝒒 ෝ
𝒏
Note that 𝒑
ෝ (sample proportion) is an estimator of the population
proportion, p.
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Statistics and Probability: Confidence Intervals
Estimating Proportions
MAXIMUM ERROR OF THE ESTIMATES (Margin of Error)
ෝ𝒒
𝒑 ෝ
𝑬 = 𝒛𝜶ൗ
𝟐 𝒏
where 𝑧𝛼Τ2 depends on the confidence level of the interval.
ෝ𝒒
𝒑 ෝ ෝ𝒒
𝒑 ෝ
(ෝ
𝒑 − 𝒛𝜶ൗ ෝ + 𝒛𝜶ൗ
,𝒑 )
𝟐 𝒏 𝟐 𝒏
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Statistics and Probability: Confidence Intervals
Estimating Proportions
Example 1:
In a random sample of 200 students who enrolled in Math 17, there were 138 students
who passed on their first take. Construct a 95% confidence interval for the population
proportion of students who passed Math 17 on their first take.
Given:
n = 200
𝑥 138
𝑝Ƹ = = = 0.69
𝑛 200
𝑞ො = 1 − 𝑝Ƹ = 1 − 0.69 = 0.31
For 95% confidence level, 𝑧𝛼Τ2 = 1.96
𝑝ො𝑞ො 𝑝ො𝑞ො
(𝑝Ƹ − 𝑧𝛼Τ2 , 𝑝Ƹ + 𝑧𝛼Τ2 )
𝑛 𝑛
Estimating Proportions
Example 2:
A sample of 500 nursing applications included 60 applications from men. Find
the 90% confidence interval of the true proportion of men who applied to the
nursing program.
Given:
n = 500
𝑥 60
𝑝Ƹ = = = 0.12
𝑛 500
𝑞ො = 1 − 𝑝Ƹ = 1 − 0.12 = 0.88
For 90% confidence level, 𝑧𝛼Τ2 = 1.645
𝑝ො𝑞ො 𝑝ො𝑞ො
(𝑝Ƹ − 𝑧𝛼Τ2 , 𝑝Ƹ + 𝑧𝛼Τ2 )
𝑛 𝑛
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Statistics and Probability: Confidence Intervals
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Statistics and Probability: Confidence Intervals
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Statistics and Probability: Confidence Intervals
Length of Interval
𝜎 𝑠 𝑠
𝑒 = 𝑧𝛼ൗ , 𝑒 = 𝑧𝛼ൗ , 𝑒 = 𝑡𝛼ൗ
2 𝑛 2 𝑛 2 𝑛
are the maximum error of the estimate (margin of error) for the mean.
𝑝Ƹ 𝑞ො
𝐸 = 𝑧𝛼ൗ
2 𝑛
is the maximum error of the estimate (margin of error) for proportion.
Length of Interval
If asked to calculate for the length of interval, you can do two things:
• Subtract the Maximum and the Minimum value of the interval OR
• Calculate for:
• Length of Interval (Mean) = 2e
• Length of Interval (Proportion) = 2E
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Statistics and Probability: Confidence Intervals
Let’s go back to the SWS Survey in the introduction. The estimates given in
the survey were point estimates; however, since the sample proportion is
84% or 0.84 with a margin of error of 2% or 0.02, a confidence interval can
be constructed. The confidence interval would be 82%, 86% or (0.82, 0.86).
We don’t know whether this is a 90%, 95%, 99%, or some other confidence
level because this was not specified in the report.
We use a 95% confidence level (use 95% CL if not explicitly stated). Using the
formulas in Slide 27, a minimum sample size can be calculated. We can take
𝑝Ƹ = 0.84 and 𝑞ො = 0.16. For 95% confidence level, 𝑧𝛼Τ2 = 1.96
𝒛𝟐𝜶ൗ 𝒑
ෝ𝒒ෝ 𝟏. 𝟗𝟔 𝟐
𝟎. 𝟖𝟒 𝟎. 𝟏𝟔
𝟐
𝒏= = = 𝟏𝟐𝟗𝟎. 𝟕𝟕𝟕𝟔 ≈ 𝟏𝟐𝟗𝟏 (𝒓𝒐𝒖𝒏𝒅 𝒖𝒑)
𝑬𝟐 𝟎. 𝟎𝟐𝟐
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Statistics and Probability: Confidence Intervals
For a 95% confidence interval, then, the minimum sample size would be
1291. Since the survey used 4010 respondents, it can be said that we are at
least 95% confident that the interval (0.82, 0.86) encloses the true
proportion of Filipinos who believe that strict lockdown measures imposed
due to the pandemic “are worth it to protect people and limit the spread” of
the virus.
Note: Even if the minimum sample size was satisfied, this does not
automatically mean that this is real, or this is the truth. Many factors may
affect the results of the survey such as the respondent’s educational
attainment, political views, socio-economic status, location, gender, age, to
name a few. It is important that you will investigate if the study was done
in an unbiased, accurate and reliable way, before you believe it.
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Statistics and Probability
ESTIMATION OF PARAMETERS (PART 2)
Thank You!
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