# Sampling Theory and Some Important Sampling Distributions

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Sampling Theory and Some Important Sampling Distributions

Introduction to Sampling Distribution

Very simple random sample (VSRS)

Independently and identically distributed, i.i.d.
When X1, X2, …Xn are drawn from the same distribution and are independently distributed, they are said to be independently and identically distributed or i.i.d. 社會統計（上） ©蘇國賢2005

Interval Estimation區間估計

Interval Estimation區間估計

Interval Estimation區間估計

Interval Estimation區間估計

Interval Estimation區間估計

How reliable is this estimate? 社會統計（上） ©蘇國賢2005

(μ － 2 × 4.5 , μ ＋ 2 × 4.5) 社會統計（上） ©蘇國賢2005

To say that x-bar lies within 9 points of μis the same as saying that μ is within 9 points of x-bar

Statistical confidence
The language of statistical inference uses this fact about what would happen in the long run to express our confidence in the results of any one sample. 社會統計（上） ©蘇國賢2005

Interval Estimation區間估計

Interval Estimation區間估計

Confidence interval A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter. We must find the number z* such that any normal distribution has probability C within ± z* standard deviation of its mean. 社會統計（上） ©蘇國賢2005

Value of Zα Let Z be a standard normal random variable and let αbe any number such that 0<α<1. Then zαdenotes the number for which P(Z≧ zα) = α 社會統計（上） ©蘇國賢2005

Value of Zα 例題：α=.025，求zα? P(Z≧ zα) =.025 zα zα=1.96 Area=1-.025=0.975

Value of Zα 例題：求z.05? P(Z≧ z.05) =.05 z.05 zα=1.645 Area=1-.05=0.95

Value of Zα 例題：求z.005? P(Z≧ z.005) =.005 z.005 zα=2.58
Area=1-.005=.995 Area=.005 z.005 zα=2.58 社會統計（上） ©蘇國賢2005

Value of Zα P(Z≧ zα/2) =α/2 P(Z≦ -zα/2) =α/2 P(-zα/2 ≦Z≦ zα/2) =(1-α)
1-α/2-α/2 =1-α P(Z≧ zα/2) =α/2 P(Z≦ -zα/2) =α/2 P(-zα/2 ≦Z≦ zα/2) =(1-α) α/2 社會統計（上） ©蘇國賢2005

Confidence intervals for the mean with know population variance

Confidence intervals for the mean with know population variance

Confidence intervals for the mean with know population variance

Level of Confidence The level of confidence (C=1-α) of a confidence interval measures the probability that a population parameter will be contained in an interval calculated after a random sample has been selected from a population. 信賴度衡量從母體中抽取隨機樣本所建構出的信賴區間會含括母體參數的機率。 α 為信賴區間沒有正確涵蓋母體參數的機率。如α=.05，則信賴度1-α=.95，表示有5%的機率信賴區間無法包含母體參數。 社會統計（上） ©蘇國賢2005

Level of Confidence 一般常用「母體參數會落在信賴區間的機率」來定義信賴度是一種錯誤的說法。

Confidence intervals for the mean with know population variance
Suppose we take random sample of n observations from a normal population with mean u and variance σ2. If σ2is known and the observed sample mean is x, then the confidence interval for the mean with a level of confidence 100(1-α)% is given by: Where zα/2is the number for which P(Z≧ zα/2) =α/2 社會統計（上） ©蘇國賢2005

Confidence intervals for the mean with know population variance

Confidence intervals for the mean with know population variance

Formula for commonly constructed confidence intervals

Desirable Properties of Confidence Intervals

Margin of Error- The width of a confidence interval for u

Comparing Width of Confidence Intervals
Suppose we take a random sample of size n from population having known variance 2. Construct 99%, 95%, 90% CI for the population mean and compare their widths. W1比W2的寬度多32% W2比W3的寬度多19% 社會統計（上） ©蘇國賢2005

Comparing Width of Confidence Intervals
To decrease the width of confidence interval, we must either use a smaller level of confidence (1-), or increase the sample size n. 99% 95% 90% 80% 50% Width of CI Confidence coefficient 社會統計（上） ©蘇國賢2005

Confidence intervals for large samples

Student’s t distribution

Student’s t distribution

t分配的一些特性 t分配為中心點為零，介於- 至的對稱分配. t分配的形狀為類似標準常態分配的鐘形分配
t distribution的平均值為 0. t分配的機率密度函數決定於參數 (nu), 即自由度(degree of freedom) 。建構平均值的信賴區間時，自由度為樣本數減一degrees of freedom is =(n-1)。 社會統計（上） ©蘇國賢2005

Characteristics of t distribution
t distribution 的變異數為 /(-2) for >2，其值永遠大於1。v愈大（樣本越大），變異數越接近1，其形狀越接近標準常態分配。 社會統計（上） ©蘇國賢2005

Characteristics of t distribution
t分配是一群機率分配的組合，不同自由度對應不同的t distribution的密度函數，由於變異數較標準常態分配大，所以形狀較為矮胖。 Standard normal (d.f.=) d.f. =4 d.f. =2 d.f. =1 社會統計（上） ©蘇國賢2005

Value of t, The symbol t,denotes the value of t such that the area to its right is  and t has  degree of freedom. The value t, satisfies the equation: P(t > t, )= Where the random variable t has the t distribution with  degrees of freedom. 社會統計（上） ©蘇國賢2005

Value of t, P(t > t0.05,13 )=0.05找出t值？ 社會統計（上） ©蘇國賢2005

Confidence intervals for the mean with unknown population variance

Constructing confidence intervals using the t distribution
The area to the right of tα/2,υis α/2 for the t distribution having v degrees of freedom. Similarly, the area to the left of -tα/2,υ is α/2 . Thus, we obtain: 社會統計（上） ©蘇國賢2005

Constructing confidence intervals using the t distribution

Constructing confidence intervals using the t distribution

Confidence interval for the mean of a normal population with unknown population variance

If the variance is known, we use z =1.96 If the variance is unknown, we use t.025, 9 = 2.262 2.262/1.96=15%. The confidence interval based on the t value will be 15% wider than that based on the z value. 社會統計（上） ©蘇國賢2005

One-sided confidence intervals for the mean
Suppose that we wish to find the lower confidence limit (LCL) such that the probability (1-)that u exceeds LCL. The one-sided interval (LCL, ) is a left-sided confidence interval. The lower confidence limit is given by Suppose that we wish to find the upper confidence limit (UCL) such that the probability (1-)that u is less than UCL. The one-sided interval (-, UCL) is a right-sided confidence interval. The upper confidence limit is given by 社會統計（上） ©蘇國賢2005

One-sided confidence intervals for the mean

One-sided confidence intervals for the mean

One-sided confidence intervals for the mean
Take a random sample of n observations from some normal population having unknown mean u and unknown standard deviation σ. Suppose that we wish to find the lower confidence interval (LCL, ∞) is a left-sided confidence interval. The lower confidence limit is given by: Suppose that we wish to find the upper confidence interval (-∞, UCL) is a right-sided confidence interval 社會統計（上） ©蘇國賢2005

One-sided confidence intervals for the mean

Determining the sample size決定樣本大小
Confidence interval for the mean: Suppose an individual is interested in estimating the mean of a population having a known variance 2. How large a sample size must be taken if the investigator wants the probability to be (1-) that the sampling error |X - u| is less than some amount D? 社會統計（上） ©蘇國賢2005

Determining the sample size決定樣本大小

Confidence intervals for the mean with unknown population variance