cabt shs statistics & probability - estimation of parameters (intro)

Estimating Parameters Intro

CABT Statistics & Probability Grade 11 Lecture Presentation

The session shall begin shortlyCABT Statistics & Probability Grade 11 Lecture Presentation

Lets Start!CABT Statistics & Probability Grade 11 Lecture Presentation

Estimation of ParametersA CABT Grade 11 Statistics and Probability Lecture

Inferential StatisticsCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of ParametersInferential statistics is concerned with drawing conclusions and/or making decisions concerning a population based only on sample data.Main functions of inferential statistics:estimate population parameterstest statistical hypotheses

http://www.gohomeworkhelp.com/admin/photos/what-is-inferential-statistics.jpgInferential StatisticsCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Parameter & StatisticA parameter is a descriptive measure that describes a population.A statistic is a descriptive measure that describes a sample.

Usually, parameters are denoted by lower-case GREEK letters (e.g. or ), while statistics use lower-case ROMAN letter (e.g. x and s).

Estimation of Population ParametersCABT Statistics & Probability Grade 11 Lecture PresentationAn estimator of a population parameter is a random variable that depends on sample information whose value provides an approximation to this unknown parameter.A specific value of that random variable is called an estimate.CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Properties of Good EstimatorsUNBIASED. The expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.CONSISTENT. As sample size increases, the value of the estimator approaches the value of the parameter being estimated.RELATIVELY EFFICIENT. Of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Estimation of Population Parameters

There are two types of estimates:Point estimate: It is a specific numerical value used to approximate a population parameter. Interval estimate: It is a range of values used to approximate a population parameter. Its also called a confidence interval. CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters


Note that:a point estimate is a single number.a confidence interval provides additional information about variability.

Point EstimateLower Confidence LimitUpperConfidence Limit

Width of confidence intervalCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters


Point EstimationPoint estimation is the process of finding a point estimate from a random sample of a population to approximate a parameter value. The statistic value that approximates a parameter value is called a point estimate.


The point estimate is the BEST GUESS or the BEST ESTIMATE of an unknown(fixed or random) population parameter.

Point EstimationCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

MEASUREPopulation Value(PARAMETER)Sample Statistic(POINT ESTIMATE)MeanStandard deviationsProportionp

Point EstimationCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Notes:Dont expect that the point estimate is exactly equal to the population parameter.Any point estimate used should be as close as possible to the true parameter.Sampling should be done at random, using a sample size that is as large as possible.Point EstimationCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

CABT Statistics & Probability Grade 11 Lecture PresentationThe following are some situations that use point estimates:CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Point Estimation

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(estimating a mean) A sample of 50 households is used to determine the average number of children in a household in a barangay.(estimating a proportion) A sample of 50 households is used to determine the percentage of households in a barangay watching a particular teleserye.

CABT Statistics & Probability Grade 11 Lecture PresentationThe SAMPLE MEAN is used to estimate the population mean .

The following are the lengths of seedlings in a plant box. We want to estimate the mean length of the seedlings.CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Point Estimation

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(Exercise 2 of the textbook)

Estimate the mean length using the following:average of the row averagesaverage of the column averagesusing the average of the first rowusing the average of the last two columnsPoint Estimation

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(Exercise 2 of the textbook)

To determine the average monthly income of factory workers of a CEPZ company, ten workers were randomly sampled. Their monthly incomes (in thousand pesos) are shown in the table. Calculate the point estimate for the average monthly income. CABT Statistics & Probability Grade 11 Lecture PresentationCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Point Estimation

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WorkerMonthly Income(thousand pesos)WorkerMonthly Income(thousand pesos)111.5611.521071239.5810.549911.5510109

Add all monthly incomes, then divide by 10. The answer is then multiplied by 1,000 to get the average monthly income.19

CABT Statistics & Probability Grade 11 Lecture PresentationFind the point estimate of the proportion of private school teachers who are LET passers in a city given that 480 out of a sample of 600 randomly selected teachers passed the LET. CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Point Estimation

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P = 480 / 600 = 0.8 or 80%20

CABT Statistics & Probability Grade 11 Lecture PresentationFind the point estimate of the proportion of the number of junior high school students who owns at least one cell phone given the following sample: CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Point Estimation

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GradeNumber of students surveyedNumber of students surveyed with at least one cell phone71098151192516102014

P = (total number of students with cellphone) / (total number of students surveyed)21

CABT Statistics & Probability Grade 11 Lecture PresentationAn interval estimate is a range of values used to approximate a population parameter. This estimate may or may not contain the actual value of the parameter being estimated.


Interval EstimationAn interval estimate has two components:a range or interval of values an associated level of confidence

CABT Statistics & Probability Grade 11 Lecture PresentationCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Interval EstimationWhy use an interval estimate instead?

Using a point estimate, while unbiased, poses a degree of uncertainty. There is no way of expressing the degree of accuracy of a point estimate. An interval estimate provides more information about a population characteristic than does a point estimate.

CABT Statistics & Probability Grade 11 Lecture Presentationconfidence n. a feeling or belief that you can do something well or succeed at somethingCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

(http://www.merriam-webster.com/dictionary/confidence )Confidence Levels and Intervals

CABT Statistics & Probability Grade 11 Lecture PresentationThe confidence level c of an interval estimate is the probability that the parameter is contained in the interval estimate.CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Confidence Levels

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CABT Statistics & Probability Grade 11 Lecture PresentationThe value of c is given byCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Confidence Levels

where represents a level of significance, which indicates the long-run percentage of confidence intervals which would include the parameter being estimated. The value of the level of significance is always between 0 and 1.

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CABT Statistics & Probability Grade 11 Lecture PresentationThe significance of the level of significanceCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Confidence LevelsThe level of significance represents a probability of lack of confidence; that is, the probability of NOT capturing the value of a population parameter in the interval estimate. The confidence level c = 1 , meanwhile represents the probability of confidence that the population parameter lies within the interval estimate.

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CABT Statistics & Probability Grade 11 Lecture PresentationThe significance of the level of significanceCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Confidence Levels

probability that lies in the interval estimate probability that does NOT lie in the interval estimate

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CABT Statistics & Probability Grade 11 Lecture PresentationA confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Confidence Intervals

http://blog.minitab.com/blog/adventures-in-statistics/understanding-hypothesis-tests:-confidence-intervals-and-confidence-levels

CABT Statistics & Probability Grade 11 Lecture PresentationNotes:CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

For a parameter , if P(a < < b) = 1 , then the interval a < < b is called a 100(1 )% confidence interval of . In repeated samples of the population, the true value of the parameter would be contained in 100(1 )% of intervals calculated this way. Confidence Intervals

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REGION OF CONFIDENCE100(1 - )% of all intervals contain the value of

Distribution of

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CABT Statistics & Probability Grade 11 Lecture PresentationIllustration:CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

A 95% confidence interval of a population mean means that 95% of the samples from the same population will produce the same confidence intervals that contain the value of .

http://www.statistica.com.au/confidence_interval.htmlAlso, this means that

so is the level of significance.


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CABT Statistics & Probability Grade 11 Lecture PresentationIllustration:CABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

a 95% confidence interval for the mean in a normally-distributed populationConfidence Intervals

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Determine the confidence level for the following levels of significance:Level of SignificanceConfidenceLevel

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Determine the levels of significance for the following confidence levels:Confidence LevelLevel of Significance

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PointEstimateLower Confidence LimitUpperConfidence Limit

Margin of ErrorMargin of Error

Width of confidence intervalImportant parts of a confidence intervalConfidence Intervals

CABT Statistics & Probability Grade 11 Lecture PresentationGeneral Formula for Confidence IntervalsCABT Statistics & Probability Grade 11 Lecture PresentationEstimation of Parameters

Confidence IntervalsThe general formula for all confidence intervals is given by: The value of the reliability factor depends on the desired level of confidence.

Wow!



https://onlinecourses.science.psu.edu/stat504/sites/onlinecourses.science.psu.edu.stat504/files/lesson01/simple_expres_CI.gif


Confidence IntervalsUsually, the general formula for a confidence interval is written as where is the estimate of the parameter and E is the margin of error.

In INEQUALITY FORM, the confidence interval of a parameter is given by

Estimation of Parameters

Population MeanUnknown

Confidence IntervalsPopulation ProportionKnown


Wow!

Do you have any QUESTIONs?

Confidence Intervals forthe Population Meanfor Known and Unknown Variances


Confidence Intervals forthe Population MeanTo construct an interval estimate for the population mean, we usea point estimate for the mean.a margin of error.


Confidence Intervals forthe Population MeanThe confidence interval for the population mean is given by

where E is the margin of error dependent on a given confidence level.

Wow!


Confidence Intervals forthe Population MeanIn the confidence interval

= lower confidence limit

= upper confidence limit

= width of the confidence interval

= margin of error


Confidence Intervals forthe Population Mean

PopulationMeanLower Confidence LimitUpper Confidence Limit

Margin of ErrorMargin of Error

Width of Confidence Interval

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Confidence Intervals forthe Population Mean

REGION OF CONFIDENCE100(1 - )% of all intervals contain the value of the population mean

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Confidence Intervals forthe Population MeanSuppose that a sample is taken from a normally-distributed population. If the sample mean is 10, the confidence interval for the population mean at a margin of error of 2 is

or

From the confidence interval, we have:Lower confidence limit: 8Upper confidence limit:12Width of confidence interval: 2E = 4 or 12 8 = 4

1

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Confidence Intervals forthe Population MeanFind the margin of error and the width of the following confidence intervals:

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ConfidenceIntervalWidth of Confidence IntervalMargin of Error

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Confidence Intervals forthe Population MeanIn constructing an interval estimate for the population mean, we consider two cases:CASE 1 the standard deviation of the population is knownCASE 2 the standard deviation of the population is not known


Confidence Intervals forthe Population MeanThe standard deviation of the population is known.

1CASEAconfidenceintervalfor a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution.


Confidence Intervals For the Population Mean

1CASEThe Central Limit Theorem A Throwback:

The mean and standard deviation of the distribution are, respectively, If random samples of size n are drawn from a population with replacement, then as n becomes larger, the sampling distribution of the mean approaches the normal distribution, regardless of the shape of the population distribution.



1CASEBecause of the Central Limit Theorem, we can think of the confidence level c = 1 as the area under the standard normal curve between two CRITICAL VALUES and .



1CASETo get a 100(1 )%confidenceinterval for a given level of significance , we must include the central (1 ) of the probability of the normal distribution, leaving a total area of in both tails, or /2 in each tail, of the normal distribution.

Confidence Intervals Intervals extend from100(1)%of intervals constructed contain ; 100()% do not.

Sampling Distribution of the Mean



1CASEto



1CASESampling ErrorThe difference between the point estimate and the actual parameter value is called the SAMPLING ERROR.For the sampling distribution of sample means, the sampling error is equal to



1CASEMargin of ErrorThe margin of error E is the maximum error of estimate given by

Wow!

or

where is the level of significance, is the population standard deviation, and n is the sample size.

The formula for the margin of error E used the formula for the value of z used in the Central Limit Theorem. E is the largest possible value of the sampling error.57



1CASESteps in Constructing a Confidence Interval for a Population Mean if the Standard Deviation is Known STEP 1 Calculate the sample mean. This is the point estimate for the population mean .STEP 2 Find the z-score (critical value) that corresponds to the confidence level .STEP 3 Calculate the margin of error E.STEP 4 Construct the confidence interval for :

Interpret the result.



1CASECommon Confidence Levels and the Corresponding z Values Confidence Level Confidence Coefficient c = 1 Level of SignificanceValue of

z-Value

80%0.800.200.101.2890%0.900.100.051.64595%0.950.050.0251.9698%0.980.020.012.3399%0.990.010.0052.57599.8%0.9980.0020.0013.0899.9%0.9990.0010.00053.27



1CASEA normally distributed population has standard deviation 1.5. A sample of size 36 is obtained from the population with sample mean 4. Find the margin of error for a 99% confidence interval for the population mean.

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1CASE

3

SolutionGiven:

Value of :

Value of z:

Value of E:

CABT Statistics & Probability Grade 11 Lecture PresentationCheck your understanding

Compute the margin of error for the estimation of the population mean for a 90% confidence with a sample of size 400 and population standard deviation of 58.Mean and Variance of Sampling Distributions of Sample MeansEstimation of Parameters

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1CASEA normally distributed population has standard deviation 2. A sample of size 25 is obtained from the population with sample mean 10. Construct a confidence interval for the mean of the population using90% confidence95% confidence

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1CASE

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a. SolutionGiven:

Value of :

Value of z:

Value of E:

Confidence limits:

Confidence interval:



1CASE

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What does our answer mean?

We are 90% confident that the true population mean lies between 9.34 and 10.66.

Wow!



1CASE

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b. SolutionGiven:

Value of :

Value of z:

Value of E:

Confidence limits:




1CASE

4

What does our answer mean?

We are 95% confident that the true population mean lies between 9.22 and 10.78.

Wow!



1CASETo determine the average amount of purchase of its customers, a convenience store samples 150 of its customers. The average purchase of the group is P 125. If the store knew that the standard deviation of all purchases is P 50, what is the 95% confidence interval for the average purchase in the store?

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1CASE

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SolutionGiven:

Value of :

Value of z:

Value of E:



1CASE

5

SolutionConfidence limits:


We are 95% confident that the actual average purchase is between P 117 andP 133. Conclusion:



1CASEA study of 400 kindergarten pupils showed that they spend on average 5,000 hours watching TV. The standard deviation of the population is 900.Find the 95% confidence level of the mean TV time for all pupils. If a parent claimed that his children watched 4,000 hours of TV, would the claim be valid? Why?

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1CASE

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a. SolutionGiven:

Value of :

Value of z:

Value of E:



1CASE

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a. Solution (continued)Confidence limits:


We are 95% confident that the actual average TV time is between 4,911.8 and 5,088.2 hours. Conclusion:



1CASE

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b. Question: Is the claim of the parent valid?Answer: NO, the claim of the parent is NOT valid because the average is NOT in the confidence interval.



1CASEIn a nutshell:Steps in Finding the Confidence Interval for Given:

Value of :

Value of z:

Value of E:

Confidence limits:




1CASEIn a nutshell:Steps in Finding the Confidence Interval for Conclusion:We are _____% confident that the true mean / average _____ is between _____ and _____.

Okay!


Solve Exercise 7(a) and 8(a) on page 166 of your textbook.Mean and Variance of Sampling Distributions of Sample Means

Estimation of Parameters

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1CASESample Size DeterminationThe MINIMUM sample size n needed to estimate the population mean iswhere is the level of significance, is the population standard deviation and E is the margin of error.

Okay!



1CASESample Size DeterminationSince the confidence interval widens as the confidence level increases, the precision of the interval estimate decreases. One way to increase the precision without changing c is to increase the sample size. The larger the sample size, the better.Why compute the sample size?



1CASEDetermine the minimum sample size needed to estimate the population mean with 95% confidence using a margin of error of 4. It is known that the population standard deviation is 8.

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1CASESolution:

7

Value of :

Value of z:

Minimum sample size:

Note:ROUND UPyour answerGiven:



1CASEIf the variance of a national accounting examination is 900, how large a sample is needed to estimate the true mean score within 5 points with 99% confidence?

8



1CASESolution:

8

Value of :

Value of z:

Minimum sample size:

Given:


Ehljie wants to conduct a study on the average number of hours a Grade 11 student spends in studying Statistics and Probability in a school week with 98% confidence and a margin of error of 2 hours. What sample size should Ehljie use for her study if the population standard deviation is 1.5 hours?Mean and Variance of Sampling Distributions of Sample MeansEstimation of Parameters

Okay!Gamitin mo yung formula na ibinigay ni Sir!

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Do you have any QUESTIONs?

Summing it up!

Thank you!

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cabt shs statistics & probability - estimation of parameters (intro)

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