Lesson 04 chapter 7 estimation

Inductive StatisticsInductive Statistics

Dr. Ning DING

[email protected]

I.007 IBS, Hanze

You’d better use the full-screen mode to view this PPT file.

Review:Chapter 5 Probability DistributionChapter 6 Sampling Distribution

Chapter 7 Estimation~Interval Estimation for Mean

* when σ is known* when σ is unknown* when σ is unknown AND n=<30

~Interval Estimation for Proportion

Chapter 8 Testing Hypothesis~Basics~Two- and One-tailed Test~Test for Mean

* when σ is known* when σ is unknown AND n=<30

~Test for Proportion

Table of Contents

Chapter 5: Probability Distribution

BionomialDistributionBionomial

DistributionPoisson

DistributionPoisson

Distribution

Discretewithin a range

Normal Distribution

Normal Distribution

continuousDiscretewith mean

λ: meanp: probability of success

q: probability of failure

q = 1- p

Review:-

Chapter 5 Probability Distribution

-Chapter 6 Sampling Distribution


* when σ is known

* when σ is unknown

* when σ is unknown AND n=<30



* when σ is known

* when σ is unknwn AND n=<30~Test for Proportion

60 80

The student is doing better in Chemistry than in biology. Is it correct?

Test Score Class mean SD z score

Biology 60 50 5 +2

Chemistry 80 90 10 -1

The student is actually doing better in biology.

Chapter 5: Probability DistributionChemistry

standardized scalestandardized scale

Review:-




* when σ is known





* when σ is known

* when σ is unknown AND n=<30~Test for Proportion

Chapter 6: Sampling Distribution

Sample size Sample size

Dispersion of sample meansDispersion of sample means

Standard ErrorStandard Error

Review:-




* when σ is known





* when σ is known


Infinitepopulation

Infinitepopulation

Finite population

Finite population

Chapter 6: Sampling DistributionReview:

-Chapter 5 Probability Distribution



* when σ is known





* when σ is known


confidence levelconfidence level

confidence intervalconfidence interval

Chapter 7: Estimation

Upper tailLower tail

Review:-




* when σ is known





* when σ is known


Interval Estimates & Confidence Intervals

= the probability that we associate with an interval estimate

It is the range of the estimate we are making.

-1.64σ-1.64σ +1.64σ+1.64σ

90%90%90% confident that our population mean will lie within this interval.

Review:-




* when σ is known





* when σ is known


Chapter 7 Estimation

Interval Estimates of the Mean

Interval Estimates of the Proportion

σ is known:σ is known:

σ is unknown:σ is unknown:

n <30 & σ is unknownn <30 & σ is unknown

•Degree of freedom •Confidence Level

Interval Estimates of the mean from Large SamplesEstimate the mean life of windshield wiper. The standard deviation of the population life is 6 months. We randomly select 100 wiper blades and know the mean is 21 months.

Example:Example:

Step 1: List the known variablesStep 1: List the known variables

Step 2: Calculate the standard error of the meanStep 2: Calculate the standard error of the mean

n=100 σ=6 months

Ch 7 Example P.361

Review:-




* when σ is known





* when σ is known



Example:Example:

Step 3: If we choose 95% confidence level, find the z scoreStep 3: If we choose 95% confidence level, find the z score

-1.96σ-1.96σ

95%95%

+1.96σ+1.96σ

Appendix Table 1Appendix Table 1

47.5%47.5% 47.5%47.5%

Ch 7 Example P.361

Review:-




* when σ is known





* when σ is known



Example:Example:

Step 4: Calculate the upper and lower limitsStep 4: Calculate the upper and lower limits

19.8219.82 22.1822.18

95%95%

Ch 7 Example P.361

Review:-




* when σ is known





* when σ is known


Interval Estimates of the mean from Large Samples

Suppose we don’t know the population standard deviation

Example:Example:

Find the interval estimate of the mean annual income of 700 families at 90% confidence level.

Step 1: Estimate the population standard deviation.Step 1: Estimate the population standard deviation.

Step 2: Find the standard error of the mean Estimate the mean. Step 2: Find the standard error of the mean Estimate the mean.

Ch 7 Example P.363

n/N = 50/700 = 0.0714 >.05 Use F.P.M.

Review:-




* when σ is known





* when σ is known



Suppose we don’t know the population standard deviation

Example:Example:

Find the interval estimate of the mean annual income of 700 families at 90% confidence level.

Step 2: Find the standard error of the mean Estimate the mean. Step 2: Find the standard error of the mean Estimate the mean.

Ch 7 Example P.363

Review:-




* when σ is known





* when σ is known



Step 3: Find the z scoreStep 3: Find the z score

-1.64σ-1.64σ +1.64σ+1.64σ


Example:Example:Find the interval estimate of the mean annual income of 700 families at 90% confidence level.

90%90%

45%45% 45%45%

Ch 7 Example P.363

Review:-




* when σ is known





* when σ is known



11,587.5011,587.50 12,012.5012,012.50

Example:Example:Find the interval estimate of the mean annual income of 700 families at 90% confidence level.


90%90%

45%45% 45%45%

Ch 7 Example P.363

Review:-




* when σ is known





* when σ is known


Interval Estimates Using t Distribution

• Sample size < 30

• Unknown population standard deviation

• Sample size < 30

• Unknown population standard deviation

1. Degree of Freedom1. Degree of Freedom

b=6

b=13

... ...

degree of freedomdegree of freedom

n-1n-1n-1n-1

Review:-




* when σ is known





* when σ is known


Interval Estimates Using t Distribution2. Using the 2. Using the tt Distribution Table Distribution Table Appendix Table 2Appendix Table 2Appendix Table 2Appendix Table 2

Confidence IntervalConfidence Intervaldegree

of

freedom

degree

of

freedom

77

0.050.05

2.365

Review:-




* when σ is known





* when σ is known


Interval Estimates Using t Distribution

Example:Example:The plant manager wants to estimate the coal needed for this year. He took a sample by measuring coal usage for 10 weeks.

Step 1: Calculate the standard error of the mean Step 1: Calculate the standard error of the mean

9

Step 2: Look in Appendix Table 2 to get the t valueStep 2: Look in Appendix Table 2 to get the t value t = 2.262

95%


The manager is 95% confident that the mean weekly usage of coal lies between 10,899 and 11,901 tons.

Ch 7 Example P.373

Review:-




* when σ is known





* when σ is known


Interval Estimates of the proportion from Large Samples

Mean of the Sampling Distribution of the Proportion

Standard Error of the Proportion

Esitmated Standard Error of the Proportion

unemployment rate is a proportion

Review:-




* when σ is known





* when σ is known



Example:Example:What proportion of employees prefer to provide their own retirement benefits in lieu of a company-sponsored plan?

n = 75 n = 75 Step 1: Conduct simple random samplingStep 1: Conduct simple random sampling

Step 2: QuestionnaireStep 2: Questionnaire Agree or Disagree

30 vs. 45

Step 3: Calculate the estimated error of the proportion

Step 3: Calculate the estimated error of the proportion

Ch 7 Example P.368

Review:-




* when σ is known





* when σ is known



Example:Example:What proportion of employees prefer to provide their own retirement benefits in lieu of a company-sponsored plan?

Step 4: Use the confidence level to find z scoreStep 4: Use the confidence level to find z score 99%99%

-2.58σ-2.58σ

99%99%

49.5%49.5% 49.5%49.5%

+2.58σ+2.58σ

Step 5: Estabilish of the upper and lower limitsStep 5: Estabilish of the upper and lower limits

0.2530.253 0.5470.547 99% sure that 25.3% to 54.7% of the employees agreed with this plan

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:BasicsVocabularyVocabulary

•hypothesis hypotheses

•Likelihood

•Significant

•Accept / Reject

hypothese Hypothese hipótesis гипотеза 假设 l'hypothèse

la probabilité 可能性 вероятность Wahrscheinlichkeit waarschijnlijkheid probabilidad

significativo signifikant significant significative значительный 显著

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:Basics

H0 H1There is no difference between the sample mean and the hypothesized population mean.

There is a difference between the sample mean and the hypothesized population mean.

Two-tailedtest

One-tailedtest

H0 : µ = 10

H1 : µ > 15

H1 : µ < 2

H1 : µ ≠ 10

For example:

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:Practice

Ch 8 No. 8-24 P.415

The STA department installed energy-efficient lights, heaters and air conditioners last year. Now they want to determine whether the average monthly energy usage has decreased. Should they perform a one- or two-tailed test?

If their previous average monthly energy usage was 3,124 kw hours, what are the null and alternative hypotheses?

Answer: One-tailed test lower-tailed test

8-248-24Review:

-Chapter 5 Probability Distribution



* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:two-tailed test

Ch 8 Example P.417

Assume that a manufacturer supplies axles which must withstand 80,000 pounds per square inch. Experiences show the standard deviation is 4,000 pounds. Being either lower or greater is not allowed. He sampled 100 axles from the production and found the mean is 79,600 pounds. Example:Example:


Step 2: Formulate the hypothesesStep 2: Formulate the hypotheses

Step 3: Calculate the standard error

Step 3: Calculate the standard error

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:two-tailed test

Ch 8 Example P.417

Assume that a manufacturer supplies axles which must withstand 80,000 pounds per square inch. Experiences show the standard deviation is 4,000 pounds. Being either lower or greater is not allowed. He sampled 100 axles from the production and found the mean is 79,600 pounds. Example:Example:

Step 4: Visualize the confidence levelStep 4: Visualize the confidence level

-1.96 +1.96

Step 5: Establish the limitsStep 5: Establish the limits

79,216 80,784

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:one-tailed test

Ch 8 Example P.419

Hospital uses large quantities of packaged doses of a drug. The individual dose is 100 cc. The body can pass off the excessive but insufficient doses will be problematic. The hospital knows the standard deviation of the supplier is 2 cc. They sampled 50 doses randomly and found the mean is 99.75 cc. Example:Example:

Step 1: List the known variablesStep 1: List the known variables Step 2: Formulate HypothesesStep 2: Formulate Hypotheses

Step 3: Calculate the standard errorStep 3: Calculate the standard error

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:one-tailed test

Ch 8 Example P.419

Hospital uses large quantities of packaged doses of a drug. The individual dose is 100 cc. The body can pass off the excessive but insufficient doses will be problematic. The hospital knows the standard deviation of the supplier is 2 cc. They sampled 50 doses randomly and found the mean is 99.75 cc. Example:Example:


Step 5: Calcuate the z valueStep 5: Calcuate the z value

-1.28

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses: Practice

Ch 8 No. 8-26 P.422

8-268-26 Step 1: List the known variablesStep 1: List the known variables

Step 2: Formulate HypothesesStep 2: Formulate Hypotheses




P=0.48 z=-2.05

-2.05

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses: Practice

Ch 8 No. 8-26 P.422

8-278-27 Step 1: List the known variablesStep 1: List the known variables





P=0.475 z= 1.96

-1.96 +1.96

Review:-




* when σ is known





* when σ is known


Ch 8 No. Example P.433

Example:Example:




The HR director thinks that the average aptitude test is 90. The manager sampled 20 tests and found the mean score is 80 with standard deviation 11.

If he wants to test the hypothesis at the 0.10 level of significance, what is the procedure?

Review:-




* when σ is known





* when σ is known



Example:Example:

Step 4: Visualize the confidence levelStep 4: Visualize the confidence level Step 5: Calcuate the t valueStep 5: Calcuate the t value

The HR director thinks that the average aptitude test is 90. The manager sampled 20 tests and found the mean score is 80 with standard deviation 11.

If he wants to test the hypothesis at the 0.10 level of significance, what is the procedure?


Review:-




* when σ is known





* when σ is known





Confidence IntervalConfidence Intervaldegree

of

freedom

degree

of

freedom

1212

0.05 0.10 0.05 0.10

1.782

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses-Summary

Ch 8 Example P.417

H0 H1There is no difference between the sample mean and the hypothesized population mean.

There is a difference between the sample mean and the hypothesized population mean.

Two-tailedtest

One-tailedtest

H0 : µ = 10

H1 : µ > 15

H1 : µ < 2

H1 : µ > 15 AND µ < 2

For example:

Mea

nM

ean

Proporti

on

Proporti

on

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:Proportion

Ch 8 Example P.427

HR director tell the CEO that the promotability of the employees is 80%. The president sampled 150 employees and found that 70% are promotable.

The CEO wants to test at the 0.05 significance level the hypothesis that 0.8 of the employees are promotable.

Example:Example:Step 1: List the known variablesStep 1: List the known variables


Step 3: Calculate the standard errorStep 3: Calculate the standard errorPro

portion

Proporti

on

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:Proportion

Ch 8 Example P.427

HR director tell the CEO that the promotability of the employees is 80%. The president sampled 150 employees and found that 70% are promotable.

The CEO wants to test at the 0.05 significance level the hypothesis that 0.8 of the employees are promotable.

Example:Example:Step 4: Visualize the confidence levelStep 4: Visualize the confidence level

Step 5: Calculate the z scoreStep 5: Calculate the z score

Proporti

on

Proporti

on

Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses:Practice

Ch 8 SC 8-9 P.431

.


Step 5: Calculate the z scoreStep 5: Calculate the z score

Proporti

on

Proporti

onStep 1: List the known variablesStep 1: List the known variables



SC 8-9SC 8-9Review:-




* when σ is known





* when σ is known


Chapter 8 Testing Hypotheses: Measuring Power of a Hypothesis Test

True Not True

Accept

Reject

H0

Type I Error

Type II Error

Review:-




* when σ is known





* when σ is known


Summary

Review:Chapter 5 Probability DistributionChapter 6 Sampling Distribution


* when σ is known* when σ is unknown* when σ is unknown AND n=<30



* when σ is known* when σ is unknwn AND n=<30

~Test for Proportion

Connection with BRM(Business Research Methods)

Connection with BRM(Business Research Methods)

P.354

The Normal DistributionSPSS 1st Assessment

The data can be downloaded from:

Blackboard – Inductive Statsitics STA2—SPSS--Week 3 Creating Graphs.sav

Lesson 04 chapter 7 estimation

Education

whenis unknown

sampling distribution

interval estimation

ning ding email

table of contents review

inductive statistics

screen mode

hanze youd