Statistik 1_7 Estimasi & CI

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l GOALS Define a what is meant by a point estimate. Define the term level of confidence. Construct a confidence interval for the population mean when the population standard deviation is known. Construct a confidence interval for the population mean when the population standard deviation is unknown. Construct a confidence interval for the population proportion. Determine the sample size for attribute and variable sampling. Chapter Nine Estimation and Confidence Intervals

Point and Interval Estimates:

Point and Interval Estimates A point estimate is a single value (statistic) used to estimate a population value (parameter). A interval estimate is a range of values within which the population parameter is expected to occur. The interval within which a population parameter is expected to occur is called a confidence interval. The specified probability is called the level of confidence. The two confidence intervals that are used extensively are the 95% and the 99%.

Interval Estimation for The Population Mean:

Is the Population Normal Is n 30 or more ? Is the population standard deviation known ? Use a Non Parametric Test Use the Z Distribution Use the t Distribution Use the Z Distribution No No No Yes Yes Yes Interval Estimation for The Population Mean

Confidence Intervals:

Confidence Intervals The degree to which we can rely on the statistic is as important as the initial calculation. Remember, most of the time we are working from samples. And samples are really estimates. Ultimately, we are concerned with the accuracy of the estimate. Confidence interval provides Range of Values Based on Observations from 1 Sample Confidence interval gives Information about Closeness to Unknown Population Parameter Stated in terms of Probability Knowing Exact Closeness Requires Knowing Unknown Population Parameter

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Areas Under the Normal Curve Between: ± 1  - 68.26% ± 2  - 95.44% ± 3  - 99.74% µ µ-1 σ µ+1 σ µ-2 σ µ+2 σ µ+3 σ µ-3 σ If we draw an observation from the normal distributed population, the drawn value is likely (a chance of 68.26%) to lie inside the interval of (µ-1 σ , µ+1 σ ). P((µ-1 σ <x<µ+1 σ ) =0.6826.

P(µ-1σ <x<µ+1σ) vs P(x-1σ <µ <x+1σ):

P(µ-1 σ <x<µ+1 σ ) vs P(x-1 σ <µ <x+1 σ ) P(µ-1 σ <x<µ+1 σ ) is the probability that a drawn observation will lie between (µ-1 σ , µ+1 σ ). P(µ-1 σ <x<µ+1 σ ) = P(µ-1 σ -µ-x <x-µ-x<µ +1 σ -µ-x) = P(-1 σ -x <-µ<1 σ -x) = P(-(-1 σ -x )>-(-µ)>-(1 σ -x)) = P(1 σ +x >µ>-1 σ +x) = P(x-1 σ <µ <x+1 σ ) P(x-1 σ <µ <x+1 σ ) is the probability that the population mean will lie between (x-1 σ , x+1 σ ).

Elements of Confidence Interval Estimation:

Elements of Confidence Interval Estimation Confidence Interval Sample Statistic Confidence Limit (Lower) Confidence Limit (Upper) A probability that the population parameter falls somewhere within the interval.

Confidence Intervals:

Confidence Intervals 90% Samples 95% Samples 99% Samples  x _

Level of Confidence:

Level of Confidence Probability that the unknown population parameter falls within the interval Denoted (1 -  = level of confidence  Is the Probability That the Parameter Is Not Within the Interval Typical Values Are 99%, 95%, 90%

Interpreting Confidence Intervals:

Interpreting Confidence Intervals Once a confidence interval has been constructed, it will either contain the population mean or it will not. For a 95% confidence interval, if you were to produce all the possible confidence intervals using each possible sample mean from the population, 95% of these intervals would contain the population mean .

Intervals & Level of Confidence:

Intervals & Level of Confidence Sampling Distribution of Mean Large Number of Intervals Intervals Extend from (1 -  ) % of Intervals Contain  .  % Do Not.   x =  1 -   /2  /2 X _  x _

Point Estimates and Interval Estimates:

Point Estimates and Interval Estimates The factors that determine the width of a confidence interval are: The size of the sample (n) from which the statistic is calculated . The variability in the population, usually estimated by s. The desired level of confidence.

Point and Interval Estimates:

Point and Interval Estimates If the population standard deviation is known or the sample is greater than 30 we use the z distribution.

CONTOH:

CONTOH Penelitian dilakukan untuk mengetahui pendapatan bersih PKL di Surabaya. Dari 100 orang sampel random diketahui rata-rata pendapatan bersih per hari PKL Rp 50.000 dengan simpangan baku RP 15.000. Berdasarkan data tersebut lakukan estimasi pendapatan bersih PKL di Surabaya dengan tingkat keyakinan 95%.

Point and Interval Estimates:

Point and Interval Estimates If the population standard deviation is unknown and the sample is less than 30 we use the t distribution.

Student’s t-Distribution:

Student’s t-Distribution The t-distribution is a family of distributions that is bell-shaped and symmetric like the standard normal distribution but with greater area in the tails. Each distribution in the t-family is defined by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the normal distribution.

About Student Student is a pen name for a statistician named William S. Gosset who was not allowed to publish under his real name. Gosset assumed the pseudonym Student for this purpose. Student’s t distribution is not meant to reference anything regarding college students.

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Z t 0 t ( df = 5) Standard Normal t ( df = 13) Bell-Shaped Symmetric ‘Fatter’ Tails Student’s t-Distribution

Student’s t Table:

Upper Tail Area df .25 .10 .05 1 1.000 3.078 6.314 2 0.817 1.886 2.920 3 0.765 1.638 2.353 t 0 Student’s t Table Assume: n = 3 df = n - 1 = 2  = .10  /2 =.05 2.920 t Values  / 2 .05

Degrees of freedom:

Degrees of freedom Degrees of freedom refers to the number of independent data values available to estimate the population’s standard deviation. If k parameters must be estimated before the population’s standard deviation can be calculated from a sample of size n, the degrees of freedom are equal to n - k.

Degrees of Freedom (df ):

Degrees of Freedom ( df ) 1. Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated 2. Example Sum of 3 Numbers Is 6 X 1 = 1 (or Any Number) X 2 = 2 (or Any Number) X 3 = 3 (Cannot Vary) Sum = 6 degrees of freedom = n -1 = 3 -1 = 2

t-Values:

t-Values where: = Sample mean = Population mean s = Sample standard deviation n = Sample size

Estimation Example Mean ( Unknown):

Estimation Example Mean (  Unknown) A random sample of n = 25 has = 50 and S = 8 . Set up a 95% confidence interval estimate for  . X t S n X t S n n n                      / , / , . . . . 2 1 2 1 50 2 0639 8 25 50 2 0639 8 25 46 69 53 30

Central Limit Theorem:

Central Limit Theorem For a population with a mean  and a variance  2 the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed . The mean of the sampling distribution equal to  and the variance equal to  2 / n. The sample mean of n observation The population distribution

Standard Error of the Sample Means:

Standard Error of the Sample Means The standard error of the sample mean is the standard deviation of the sampling distribution of the sample means. It is computed by is the symbol for the standard error of the sample mean. σ is the standard deviation of the population. n is the size of the sample.

Standard Error of the Sample Means:

Standard Error of the Sample Means If  is not known and n  30 , the standard deviation of the sample, designated s , is used to approximate the population standard deviation. The formula for the standard error is:

95% and 99% Confidence Intervals for the sample mean :

95% and 99% Confidence Intervals for the sample mean The 95% and 99% confidence intervals are constructed as follows: 95% CI for the sample mean is given by 99% CI for the sample mean is given by

95% and 99% Confidence Intervals for µ :

95% and 99% Confidence Intervals for µ The 95% and 99% confidence intervals are constructed as follows: 95% CI for the population mean is given by 99% CI for the population mean is given by

Constructing General Confidence Intervals for µ :

Constructing General Confidence Intervals for µ In general, a confidence interval for the mean is computed by:

EXAMPLE 3:

EXAMPLE 3 The Dean of the Business School wants to estimate the mean number of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hours. What is the population mean? The value of the population mean is not known. Our best estimate of this value is the sample mean of 24.0 hours. This value is called a point estimate.

Example 3 continued:

Example 3 continued Find the 95 percent confidence interval for the population mean. The confidence limits range from 22.88 to 25.12. About 95 percent of the similarly constructed intervals include the population parameter.

Confidence Interval for a Population Proportion:

Confidence Interval for a Population Proportion The confidence interval for a population proportion is estimated by:

EXAMPLE 4:

EXAMPLE 4 A sample of 500 executives who own their own home revealed 175 planned to sell their homes and retire to Arizona. Develop a 98% confidence interval for the proportion of executives that plan to sell and move to Arizona.

CONTOH:

CONTOH Lembaga riset melakukan penelitian tentang perusahaan di Jawa Timur yang sudah menerapkan UMR. Data menunjukkan dari 50 sampel perusahaan, 40 diantaranya sudah memenuhi UMR. Buatlah confidence interval 90% untuk menduga persentase perusahaan yang sudah menerapkan UMR.

Finite-Population Correction Factor:

Finite-Population Correction Factor A population that has a fixed upper bound is said to be finite. For a finite population, where the total number of objects is N and the size of the sample is n , the following adjustment is made to the standard errors of the sample means and the proportion: Standard error of the sample means:

Finite-Population Correction Factor:

Finite-Population Correction Factor Standard error of the sample proportions: This adjustment is called the finite-population correction factor . If n / N < .05, the finite-population correction factor is ignored.

Finite-Population Correction Factor:

Finite-Population Correction Factor Standard error of the sample proportions: This adjustment is called the finite-population correction factor. Note : If n/N < 0.05, the finite-population correction factor is ignored. Interval Estimation for proportion with finite-pop

EXAMPLE 5:

EXAMPLE 5 Given the information in EXAMPLE 4 , construct a 95% confidence interval for the mean number of hours worked per week by the students if there are only 500 students on campus. Because n / N = 49/500 = .098 which is greater than 05, we use the finite population correction factor.

CONTOH:

CONTOH Pimpinan bank ingin mengetahui tentang kepuasan nasabah terhadap pelayanan bank. Dari jumlah nasabah 1000 orang, diambil sampel 100 orang untuk diwawancarai, Hasilnya 60 orang mengakui puas dengan pelayanan bank tersebut. Dengan  = 5%, berapa proporsi nasabah yang puas dengan pelayanan bank,

Selecting a Sample Size:

Selecting a Sample Size There are 3 factors that determine the size of a sample, none of which has any direct relationship to the size of the population. They are: The degree of confidence selected. The maximum allowable error. The variation in the population.

Selecting a Sample Size:

Selecting a Sample Size To find the sample size for a variable: where : E is the allowable error, z is the z- value corresponding to the selected level of confidence, and s is the sample deviation of the pilot survey.

EXAMPLE 6:

EXAMPLE 6 A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within \$5 using a 99 percent level of confidence. Based on similar studies the standard deviation is estimated to be \$20.00 . How large a sample is required?

Sample Size for Proportions:

Sample Size for Proportions The formula for determining the sample size in the case of a proportion is: where p is the estimated proportion, based on past experience or a pilot survey; z is the z value associated with the degree of confidence selected; E is the maximum allowable error the researcher will tolerate.

EXAMPLE 7:

EXAMPLE 7 The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet.

Two-sample Estimation:

Two-sample Estimation Mean : n 1 , n 2  30 n 1 , n 2 < 30 Proportion :

Contoh:

Contoh Perusahaan ban sedang memebandingkan daya pakai antara ban merek A dan Merek B. Dari sampel random 10 ban A diketahui rata-rata daya pakai 1.000 km dengan standar deviasi 100 km sedangkan dari sampel random 10 ban merek B rata-rata daya pakai 900 km dengan standar deviasi 90 km. Hitung perbedaan daya pakai antara ban merek A dan merek B dengan  = 0,05.

Contoh:

Contoh Sampel random menunjukkan dari 80 kendaraan di kota A, 60 diantaranya telah melunasi pajak sedangkan di kota B dari 70 kendaraan, 40 diantaranya telah melunasi pajak. Hitunglah perbedaan persentase pelunasan pajak kendaraan di kedua kota tersebut dengan tingkat keyakinan 95%.

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- END - Chapter Nine Estimation and Confidence Intervals 