Elements of Statistics

Module 6 Estimation

STAT 300 Elements of Statistics I

Objectives

At the end of the module, the student will be able to:

Construct and interpret point estimates and confidence intervals for the mean and proportion.

Determine the sample size needed to develop an estimated confidence interval for the mean or proportion.

Point estimator

Confidence limit

lower

Confidence limit

upper

Interval width

▪ A point estimator is a singular number.

▪ A confidence interval provides additional information about

the variability of the estimator.

Point and Interval Estimators

Point Estimators

Population parameter can be estimated…

With a sampling statistician (point

estimator)

Average

Proportion

μ

𝜋

▪ They answer how much uncertainty is associated with a

point estimate of a population parameter.

▪ An interval estimator provides more information about

the population characteristic than a point estimator.

▪ Interval estimators are known as confidence intervals.

Confidence Intervals

▪ An interval provides a range of values:

o Considers the sample variation for each sample.

o Based on observations from one (1) sample.

o Provides information on closeness to unknown

population parameters.

o Presented in terms of confidence levels

• 95% confidence interval, 99% confidence interval

• It is not possible to have a 100% confidence interval

Confidence Intervals

Population has µ = 368 y σ = 15.

= (356.42, 368.18)

If a sample is taken, n = 25, then 368 ± 1.96 15

25

,

= (362.12, 373.88) contains 95% of the sample means

▪ When µ is not known is used to approximate µ 15

25

Confidence Interval: Example

o If X = 362.3 the interval is

o Since 356.42 ≤ µ ≤ 368.18, the interval is based on the sample, it

correctly represents µ. If the interval contains the value of the

population parameter, it makes a good representation.

362.3 ± 1.96

General Formula

Point estimator ± (Critical value)(Standard error)

Where,

The point estimator is the sampling statistic that

estimates the population parameter.

The critical value is a value from the table based

on the sampling distribution of the point estimator

at the desired confidence level.

The standard error is the standard deviation of

the point estimator.

▪ Contains the unknown population parameter.

▪ It is expressed as a percentage and is less than 100%.

▪ If the confidence level is = 95% (1- ), then  = 0.05

▪ It means that 95% of all confidence intervals that can be

constructed will have the true unknown parameter.

▪ A specific interval may or may not contain the true

parameter.

Confidence Level,(1-)

Confidence Interval for μ (Known σ)

▪ Assumptions o The population standard deviation σ i is known. o Normally distributed population. o If the population is not normal, use large sample size.

▪ Confidence interval estimator:

/2 in each tail

σ

n X  Zα/2

Where,

X

Zα/2

is the point estimator

es is the critical value of the normal distribution for a probability of

is the standard error

σ/ n

Critical Value, Z α/2

Zα/2 = 1.96

Z units:

X units:

2

α = 0.025 2

α = 0.025

Zα/2 = -1.96 Lower confidence limit

Zα/2 = 1.96 Upper confidence limit

0

Point estimator

Developed by Professor Sylvia Y. Cosme Montalvo, MBA

▪ Consider 95% confidence interval

(1-α = 0.95, α=0.05)

Common Confidence Levels

Commonly used confidence levels are: 90%, 95% y 99%

Confidence

level Value Zα/2

80% 0.80 1.28

90% 0.90 1.645

95% 0.95 1.96

98% 0.98 2.33

99% 0.99 2.58

99.8% 0.998 3.08

99.9% 0.999 3.27

Confidence

coefficient,

1− 

Example

▪ A sample of 11 employees from a large normally distributed population has 2.2 days of vacation accrued monthly. It is known from experience that the population standard deviation of vacation is 0.35 days.

▪ Determine a 95% confidence interval for the true population mean of monthly accrued vacation days.

σ

n

= 2.20  1.96 (0.35/ 11)

= 2.20  0.2068

1.9932  μ  2.4068

X  Zα/2

Interpretation:

We are 95% confident that the true

mean of the population's cumulative

monthly vacation days is between

1.9932 and 2.4068. Although the true

mean may or may not be in the

interval, 95% of the intervals do

contain it.

▪ If the standard deviation σ is unknown, the sample standard deviation, s, can be

substituted.

▪ One has greater uncertainty since s, is variable from sample to sample.

▪ In these cases, the distribution t

o Assumptions

• Population standard deviation is unknown.

• Normally distributed population.

• If the population is not normal, use a large sample.

o Confidence interval estimator:

Confidence Interval for μ (Unknown σ)

(Where t α/2 is the critical value of the t-distribution with n -1 degrees of freedom and

area of α/2 in each tail).

X  tα / 2 S

n

Distribution t, Student

▪ The t refers to the family of distributions.

▪ The tα/2 value depends on the degrees of freedom (DF). o DF = n-1 is the number of observations that are free to vary after the sample mean is

calculated.

o Example: Suppose the mean of 3 numbers is 8.0.

Be X1 = 7, X2 = 8, X3 = ? If the mean is 8.0, then X3 = 9 n

= 3, so the degree of freedom = n – 1 = 3 – 1 = 2

(Two values can be any numbers, but the third value is not free to vary for a

given mean).

Distribution t, Student

t0

t (DF = 5)

t (DF = 13)They are bell-shaped and symmetrical with tails that are less fine than normal.

Normalized

standardized

(t with DF = ∞)

Note: t approaches Z as n increases

.10

Distribution Table t, Student

Upper tail area

DF

.

1

.05 0.025

3.078 6.314 12.706

2 1.886

3 1.638 2.353 3.182

t0 2.920 Body of the table

presents t-values,

not probabilities.

Be : n = 3

DF = n - 1 = 2

 = 0.10

/2 = 0.05

/2 = 0.054.3032.920

Example of t-Distribution Confidence Interval

A random sample n = 25 has mean = 50 and s = 8.

Construct a 95% confidence interval for μ.

DF = n – 1 = 24, therefore,

The confidence interval is:

tα/2 = t 0.025 = 2.0639

25

8

n

S = 50  (2.0639)X  tα/2

46.698 ≤ μ ≤ 53.302

Confidence Intervals for the Proportion of the Population, π

▪ An interval estimator for the population proportion π

can be calculated by adding an allowance for

uncertainty to the sample proportion ( p ).

▪ Recalling that the distribution of the sample proportion

approximates a normal one if the sample size is large

and the standard deviation is:

▪ It is estimated with sample data:

Developed by Professor Sylvia Y. Cosme Montalvo, MBA

Confidence Interval Extremes

▪ The lower and upper confidence interval bounds for the proportion of

the population are calculated with the formula:

▪ Where,

o Z α/2 is the standard normal value for the desired

confidence level

o p proportion of the sample

o n sample size

p(1 − p)

n p  Zα/2

Example

p  Z/ 2 p(1 − p)/n

▪ A random sample of 100 individuals shows that 25 are left- leaning.

▪ Develop a 95% confidence interval for the true proportion of left- leanings.

It must be ensured that the

sample is large enough.

There is a 95% confidence that the true percentage of lefties in the population is between 16.51% and 33.49%. Although the interval [0.1651, 0.3349] may or may not contain the true proportion, 95% of the intervals formed from samples of size 100 contain the true proportion.

= 25/100  1.96 0.25(0.75)/100

= 0 . 25  1 . 96 ( 0 . 0433 )

0 . 1651    0 . 3349

Sampling Error

▪ The required sample size can be found to obtain a desired

margin of error (e) with a specified confidence level (1 - ).

▪ The margin of error is also known as sampling error:

o Amount of imprecision in the estimate of the population

parameter.

o Amount added and subtracted from the point estimator to

construct the confidence interval.

▪ To determine the required sample size for the mean:

1. The level of confidence must be known (1 - ) that

determines the critical value, Zα/2

2. The acceptable sampling error, e, must be known.

3. The standard deviation, σ, must be known.

Sample Size Determination

Sample Size Required: Example

If  = 45, find the sample size to estimate the

mean to ± 5 with 90% confidence.

52e2 = 219.19=

(1.645)2 (45)2n = Z σ

2 2

Therefore, n = 220

It is always rounded

▪ If unknown, σ can be estimated by determining the required sample

size.

o Use a value for σ that is expected to be at least as large as the true σ.

o Select a pilot sample and estimate σ the sample standard deviation, s.

▪ To determine the sample size for a proportion:

o The level of confidence must be known (1 - α), that determines the

critical value, Zα/2

o Must know the acceptable sampling error, e.

o Must know the actual proportion of events of interest, π.

o π can be estimated with a pilot sample if necessary (or conservatively

use 0.5 as an estimator of n).

If σ is Unknown

▪ To determine the required sample size for the ratio, you

must know:

o The level of confidence must be known (1 - α), that

determines the critical value, Zα/2

o Must know the acceptable sampling error, e.

o Must know the actual proportion of events of interest, π.

• π can be estimated with a pilot sample, if necessary,

or you can use 0.5 as a conservative estimate.

Sample Size Determination

Required Sample Size: Example

Determine the sample size needed to estimate the true proportion of defects in a large population within an interval ±3%, with a confidence level of 95% (Assume pilot sample, p = 0.12).

Answer:

For 95% confidence

Zα/2 = 1.96

e = 0.03

p = 0.12 (it is used to estimate π)

Therefore, n = 451 (0.03)2e2

= 450.74n = Z

2 π (1 − π) (1.96)2 (0.12)(1 − 0.12) /2 =

1. A confidence interval estimator that reflects sampling

error should ALWAYS be included whenever a point

estimate is reported.

2. The confidence level should ALWAYS be reported. The

sample size should be reported.

3. An interpretation of the confidence interval estimator

should ALWAYS be provided.

Ethical Aspects

Summary

In this unit, we learned to:

1. Create point estimators.

2. Create confidence interval estimators for the mean when σ is known.

3. Create confidence interval estimators for the mean when σ is unknown.

4. Develop confidence interval estimators for a proportion.

5. Calculate the required sample size for the mean and proportion of

confidence interval estimators with a desired margin of error.

6. Consider ethical issues in confidence interval estimation.

References

Hesse C., Ofosu J. (2022). Statistical Methods for the Social Sciences. Akrong Publications Ltd.

Ghana. ISBN: 978–9988–2–6060–6

Howell David (2016). Fundamental Statistics for the behavioral sciences. Cengage Learning. ISBN-

10: 1305652975

Oja. (2022). PSYC 2200: Elementary Statistics for Behavioral and Social Sciences. [Vídeo]. Statistics

LibreTexts.

https://stats.libretexts.org/Courses/Taft_College/PSYC_2200:_Elementary_Statistics_for_Behavio

ral_and_Social_Sciences_(Oja)

Pelz, B. (s. f.). Statistics for the Social Sciences | Simple Book Publishing. [Vídeo]. Pressbooks.

https://courses.lumenlearning.com/suny-hccc-wm-concepts-statistics/

S. P. Mukherjee, Bikas K. Sinha Asis, Kumar Chattopadhyay (2018). Statistical Methods in Social

Science Research. Springer Nature Singapore Pte Ltd.

References Khan Academy. (n.d.). Confidence intervals | Statistics and probability. [Video]. Khan

Academy. https://www.khanacademy.org/math/statistics-probability/confidence-intervals-

one-sample

Khan Academy. (n.d.). Confidence intervals and margin of error. [Video]. Khan Academy.

https://www.khanacademy.org/math/ap-statistics/xfb5d8e68:inference-categorical-

proportions/introduction-confidence-intervals/v/confidence-intervals-and-margin-of-error

Khan Academy. (n.d.). Interpreting confidence levels and confidence intervals. [Video]. Khan

Academy. https://www.khanacademy.org/math/ap-statistics/xfb5d8e68:inference-

categorical-proportions/introduction-confidence-intervals/a/interpreting-confidence-levels-

and-confidence-intervals

Math and Stats Help. (2020, March 28). How to find the point estimate for the population

proportion [Video]. YouTube. https://www.youtube.com/watch?v=R0BYkGrQ6nY

Prof. Essa. (2021, May 13). Point Estimate Definition & example [Video]. YouTube.

https://www.youtube.com/watch?v=OTVwtvQmSn0

Congratulations you have reviewed the

theoretical summary of this week's topic!

Remember that to successfully build your learning it is important that:

Review as many times as required the information contained in the module folder

(includes this presentation).

Read the reference material to clarify any questions.

Carry out all the activities according to the instructions.

Submit assignments on the indicated date through the educational

platform.

Actively participate in collaborative sessions.