Azza Osman Mohamed

University of KhartoumFaculty of Mathematical Science

Department of Information Technology

Applied Statistics احص ) تطبيقي (301احصاء

Course component المقرر محتوى Statistical Estimates.

Test of Hypotheses . Correlation. Simple Linear Regression

Analysis. Analysis of Variance. Non-parametric Test. Statistic package SPSS.

االحصائي .التقدير. الفروض اختبارات. الخطي االرتباط الخطي االنحدار

البسيط.. التباين تحليل. الالمعلمية االختبارات االحصائية الحزمة

SPSS

Course aim: The aim of this course is to develop further understanding of

statistical methods. Outcome: By the end of this course you will be able to:

o Understand the inferential statistics.o Describing common measures of correlation and

association, and performing simple regression analysis.o understand the workings of the analysis of variance table

and its application to one-way ANOVA, and two-way ANOVA situations.

o understand the workings of the non-parametric methods.o Perform statistical analysis using SPSS.o Present and interpret the results.

Course evaluation:o Assignments.o Labs .o Mid-term exam.o Final exam.

Session 1

Learning Objectives At the end of session 1 and 2 you will be able to

State Estimation Process Introduce Properties of Point Estimates Explain Confidence Interval Estimates Compute Confidence Interval Estimation for

Population Mean ( known and unknown) Compute Confidence Interval Estimation for

Population Proportion

Introduction to Estimation

Point Estimation

Statistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

Statistical inference is the process by which we acquire information and draw conclusions about populations from samples.

In order to do inference, we require the skills and knowledge of descriptive statistics, probability distributions, and sampling distributions.

Parameter

Population

Sample

Statistic

Inference

Data

Statistics

Information

Statistical Inference…

Inference Process

Population

Sample

Sample Statistics

Estimates & Tests

X, Ps

Thinking Challenge

Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?

Estimation Methods

Estimation

PointEstimation

IntervalEstimation

Estimation… The objective of estimation is to determine the approximate value

of a population parameter on the basis of a sample statistic. An estimator is a method for producing a best guess about a

population value. An estimate is a specific value provided by an estimator. Example: We said that the sample mean is a good estimate of the

population meano The sample mean is an estimatoro A particular value of the sample mean is an estimate

Point Estimator…

Definition: A point estimator draws inferences about a population by

estimating the value of an unknown parameter using a single value or point.

Gives no information about how close value is to the unknown population parameter

Example: the sample mean ( ) is employed to estimate the population mean ( ).

Population Parameters Are

Estimated with Point Estimator

Estimate PopulationParameter

with SampleStatistic

Mean

Proportion p ps

Variance s2

Differences 12 1 2

2

X

X X

Point Estimator…

Question: Is there a unique estimator for a population parameter? For example, is there only one estimator for the population mean?

The answer is that there may be many possible estimators

Those estimators must be ranked in terms of some desirable properties that they should exhibit

Properties of Point Estimators The choice of point estimator is based on the following criteria

o Unbiasednesso Efficiencyo Consistency

Unbiased Estimators التحيز : عدمDefinition A point estimator is said to be an unbiased estimator of the

population parameter if its expected value (the mean of its sampling distribution) is equal to the population parameter it is trying to estimate

We can also define the bias of an estimator as follows

ˆE

ˆˆ EBias

Properties of Point Estimators To select the “best unbiased” estimator, we use the criterion of

efficiency

Efficiency:الكفاءة Definition An unbiased estimator is efficient if no other unbiased estimator of

the particular population parameter has a lower sampling distribution variance.

If and are two unbiased estimators of the population parameter , then is more efficient than if

The unbiased estimator of a population parameter with the lowest variance out of all unbiased estimators is called the most efficient or minimum variance unbiased estimator (MVUE).

1 21 2

21ˆˆ VV

Properties of Point EstimatorsConsistency :االتساق Definition: We say that an estimator is consistent if the probability of

obtaining estimates close to the population parameter increases as the sample size increases

One measure of the expected closeness of an estimator to the population parameter is its mean squared error

The problem of selecting the most appropriate estimator for a population parameter is quite complicated

References…..

Inferences Based on a Single Sample: Estimation with Confidence Intervals John J. McGill/Lyn Noble Revisions by Peter Jurkat

Chapter 10 Introduction on to Estimation Brocks/Cole , a division of Thomson learning, Inc.

Basic Business Statistics: Concepts & Applications Chapter 8 -

Confidence Interval Estimation Chapter 1, Point Estimation Algorithms , Department of Computer

science, University of Tennessee ,USA

Session 2

Introduction to Estimation

Interval Estimation

Estimation Methods

Estimation

PointEstimation

IntervalEstimation

Confidence Interval Estimation Process

Mean, , is unknown

Population Random SampleI am 95% confident that is between 40 & 60.

Mean X = 50

Interval Estimator… An interval estimator draws inferences about a population by estimating the value of

an unknown parameter using an interval.

Provide us with a range of values that we belive, with a given level of confidence, containes a true value.

That is we say (with some ___% certainty) that the population parameter of interest is between some lower and upper bounds.

Gives Information about Closeness to Unknown Population Parameter

Sample Statistic (Point Estimate)Confidence Interval

Confidence Limit (Lower)

Confidence Limit (Upper)

Point & Interval Estimation… For example, suppose we want to estimate the mean summer

income of a class of IT students. For n=25 students,

is calculated to be 400 $/week.

point estimate interval estimate

An alternative statement is:

The mean income is between 380 and 420 $/week.

Probability that the unknown population parameter θ falls within interval

للمعلمة الثقة فترة تسمي .θالفترة

probability that “true” parameter is in the interval is equaled to 1-.

1- is called confidence level.

1 - المعلمة على الفترة احتواء احتمال وهو الثقة معامل θيسمى.

Limits of the interval are called lower and upper confidence limits.

Confidence Interval )CI(.. فترة... الثقة

ul ˆ,ˆ

ul ˆ,ˆ

1)ˆˆ( ULP

ul ˆ,ˆ

ul ˆ,ˆ

Actual realization of this interval is called a (1- )% 100 of confidence interval.

بمقدار واثقين الفترة %(100- 1(نكون داخل تقع المجهولة المعلمة بأن.

We are 95% confident that the 95% confidence interval will include the population parameter

is probability that parameter is Not within interval

Typical values are 99%, 95%, 90%, …

Confidence Interval )CI(.. فترة... الثقة

ul ˆ,ˆ

Interval and Level of Confidence

Confidence Intervals

Intervals extend from

to

of intervals constructed contain ; 100% do not.

Sampling Distribution of the Mean

XX Z

X/ 2

/ 2

XX

1

XX Z

1 100%

/ 2 XZ / 2 XZ

Know Central Intervals of the Normal Distribution

X= ± Zx

90% Confidence

+1.65x-1.65x

95% Confidence

+1.96x-1.96x

99% Confidence

-2.58x+2.58x

Factors Affecting Interval Width

1. Data DispersionMeasured by X

2. Sample SizeX = X / n

3. Level of Confidence (1 - )Affects Z

Intervals Extend from

X - ZX toX + ZX

Confidence Interval Estimates

ProportionMean

x Unknown

ConfidenceIntervals

Variance

x Known

Estimating μ when σ is known…Known, i.e.

standard normal distribution

Known, i.e. sample mean

Unknown, i.e. we want to estimate

the population mean

Known, i.e. the number of items

sampled

Known, i.e. its assumed we

know the population standard

deviation…

Confidence Interval Estimator for μ

lower confidence limit (LCL)

upper confidence limit (UCL)

Usually represented with a “plus/minus”

( ± ) sign

Confidence Interval Estimator for μ

Four commonly used confidence levels… Confidence Level

Example …

A computer company samples demand during lead time over 25 time periods:

Its is known that the standard deviation of demand over lead time is 75 computers. We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels…

235 374 309 499 253421 361 514 462 369394 439 348 344 330261 374 302 466 535386 316 296 332 334

Example …

“We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels…”

Thus, the parameter to be estimated is the pop’n mean μ . And so our confidence interval estimator will be:

Example … In order to use our confidence interval estimator, we need the following pieces of data:

therefore: The lower and upper confidence limits are 340.76 and 399.56.

370.16

1.96

75

n 25 Given

Calculated from the data…

The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for X

if X = 100.

Thinking Challenge

92.5308.4625

1096.150

25

1096.150

2/2/

nZX

nZX

What is interval for sample size = 100?

Confidence Interval Estimates

ProportionMean

x Unknown

ConfidenceIntervals

Variance

x Known

Confidence Interval for Mean of a Normal Distribution with Unknown Variance

If the sample size is large n 30 ≤ : كبير العينة حجم حالة في The population variance is not be known The sample standard deviation will be a sufficiently good

estimator of the population standard deviation

Thus, the confidence interval for the population mean is:

n

sZ

n

sZX

n

sZX 2/2/

Confidence Interval for Mean of a Normal Distribution with Unknown Variance If the sample size is small and the population variance is unknown,

we cannot use the standard normal distribution

If we replace the unknown with the sample st. deviation s the following quantity

follows Student’s t distribution with (n – 1) degrees of freedom

The t-distribution has mean 0 and (n – 1) degrees of freedom

As degrees of freedom increase, the t-distribution approaches the standard normal distribution

ns

Xt

/

Zt

Student’s t Distribution

0

t (df = 5)

Standard Normal

t (df = 13)Bell-Shaped

Symmetric

‘Fatter’ Tails

Estimates the distribution of the sample mean, , when the distribution to be sample is normal

X

Confidence Interval for Mean of a Normal Distribution with Unknown Variance

a 100(1-)% confidence interval for the population mean when we draw small samples from a normal distribution with an unknown variance 2 is given by

n

stX n 2/,1

v t .10 t .05 t .025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

Student’s t Table

t values

Assume:n = 3df = n - 1 = 2 = .10/2 =.05

t0

/ 2

/ 2

t2.920

Estimation Example Mean ) Unknown(

/ 2 / 2

8 850 2.064 50 2.064

25 2546.69 53.30

S SX t X t

n n

A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for .

with 95% confidence

X

Thinking Challenge For a sample where the sample size = 9, the

sample mean = 28 and the sample s.d. = 3. What is the closest 95% confidence interval of the mean?

Select A for [27, 29] B for [26.5, 29.5] C for [26, 30] D for [25.25, 30.75]

E for [24.5, 31.5]

If we want to estimate the population proportion and n is large then:

العينة حجم وكان معلومة غير النجاح نسبة تكون ال ان المتوقع من كان اذافإن : كبير

and

Where x is the number of success .

Confidence Interval For the Population Proportion

2 2

ˆ ˆ ˆ ˆˆ ˆ

pq pqp z p p z

n n

Confidence interval estimate

npp

ppZ

1ˆ

ˆ

n

xp ˆ

A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p.

/ 2 / 2

ˆ ˆ ˆ ˆˆ ˆ

.08 .92 .08 .92.08 1.96 .08 1.96

400 400

.053 .107

pq pqp Z p p Z

n n

p

p

with 95% confidence

Example ….

Thinking Challenge

You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

/ 2 / 2

ˆ ˆ ˆ ˆˆ ˆ

.175 (.825) .175 (.825).175 1.645 .175 1.645

200 200

.1308 .2192

p q p qp z p p z

n n

p

p

with 90% confidence

p

Solution ….

References…..

Inferences Based on a Single Sample: Estimation with Confidence Intervals John J. McGill/Lyn Noble Revisions by Peter Jurkat

Chapter 10 Introduction on to Estimation Brocks/Cole , a division of Thomson learning, Inc.

Basic Business Statistics: Concepts & Applications Chapter 8 -

Confidence Interval Estimation.