Business Statistics CONTENTS Estimating parameters The sampling - PowerPoint PPT Presentation

𝜈 : ESTIMATES, CONFIDENCE INTERVALS, AND TESTS Business Statistics

CONTENTS Estimating parameters The sampling distribution Confidence intervals for 𝜈 Hypothesis tests for 𝜈 The 𝑢 -distribution Comparison of 𝑨 and 𝑢 Old exam question Further study

ESTIMATING PARAMETERS Central task in inferential statistics ▪ Estimation ▪ estimating a parameter (population value) from a sample ▪ Example ▪ what proportion of cars in Amsterdam is electric? ▪ population value: 𝜌 ▪ sample of size 𝑜 = 200 cars yields 26 electric cars 26 ▪ so, 𝑞 = 200 = 0.13 ▪ this suggests 𝜌 ≈ 0.13

ESTIMATING PARAMETERS Terminology ▪ Parameter ▪ a characteristic descriptive of the population ▪ e.g., 𝜈 , 𝜌 , 𝜏 (or 𝜏 2 ) ▪ Estimator ▪ a statistic derived from a sample to infer the value of a population parameter ▪ e.g., ത 𝑌 , 𝑄 , 𝑇 (or 𝑇 2 ) ▪ Estimate ▪ the value of the estimator in a particular sample ▪ e.g., ҧ 𝑦 , 𝑞 , 𝑡 (or 𝑡 2 )

ESTIMATING PARAMETERS

ҧ ESTIMATING PARAMETERS Estimator Estimate Population parameter 1 1 Mean 𝜈 ത 𝑜 𝑜 𝑌 = 𝑜 σ 𝑗=1 𝑌 𝑗 𝑦 = 𝑜 σ 𝑗=1 𝑦 𝑗 Standard 𝜏 1 1 𝑜 𝑌 𝑗 − ത 𝑜 𝑌 2 𝑦 2 𝑇 = 𝑜−1 σ 𝑗=1 𝑡 = 𝑜−1 σ 𝑗=1 𝑦 𝑗 − ҧ deviation 𝑦 𝑌 Proportion 𝜌 𝑞 = 𝑄 = 𝑜 𝑜

ESTIMATING PARAMETERS ▪ Another example (Amsterdam, 2015): ▪ what is the mean price of a glass of beer? ▪ population value: 𝜈 ▪ sample of size 𝑜 = 64 glasses of beer yields ҧ 𝑦 = 2.06€ ▪ this suggests that 𝜈 = 2.06€ ▪ But suppose we had taken a different sample ▪ again with sample size 𝑜 = 64 ▪ but now perhaps yielding ҧ 𝑦 = 2.13€ ▪ then we would estimate 𝜈 = 2.13€ ▪ Obviously there is sampling variation 𝑦 -values (the sampling distribution of ത ▪ so a distribution of ҧ 𝑌 ) ▪ Solution: point estimates and confidence intervals

THE SAMPLING DISTRIBUTION ▪ Example ▪ Consider a discrete uniform population consisting of the integers {0, 1, 2, 3} ▪ The population parameters are: ▪ 𝜈 = 1.5 ▪ 𝜏 = 1.118

THE SAMPLING DISTRIBUTION ▪ Sample 𝑜 = 2 values and calculate ҧ 𝑦 ▪ Do this for all possible sample of size 𝑜 = 2 𝑦 -values: the distribution ത ▪ You will get a distribution of ҧ 𝑌

THE SAMPLING DISTRIBUTION ▪ We will study the variance of the estimate of a population parameter from a sample statistic ▪ We will do so by studying how the sample statistic varies when you draw a different sample ▪ Example: ▪ GMAT score of MBA students ▪ 𝑂 = 2637 ▪ 𝜈 = 520.78 ▪ 𝜏 = 86.60

THE SAMPLING DISTRIBUTION ▪ Consider eight random samples, each of size 𝑜 = 5 ▪ the sample means ( ҧ 𝑦 8 = 582 ) 𝑦 1 = 504.0, ҧ 𝑦 2 = 576.0, … , ҧ tend to be close to the population mean ( 𝜈 = 520.78 ) ▪ sometimes a bit lower, sometimes a bit higher

THE SAMPLING DISTRIBUTION ▪ The dot plots show that the sample means ( ҧ 𝑦 8 ) 𝑦 1 , … , ҧ have much less variation than the individual data points ( 𝑦 1 , … , 𝑦 2637 )

THE SAMPLING DISTRIBUTION ▪ An estimator is a random variable since samples vary ▪ so we write it as a capital letter, e.g., 𝑌 , ത 𝑌 , 𝑇 , etc. ▪ The sampling distribution of an estimator is the probability distribution of all possible values the statistic may assume when a random sample of (a fixed) size 𝑜 is taken ▪ so we write 𝑌~𝑂 𝜈, 𝜏 , etc.

THE SAMPLING DISTRIBUTION ▪ The sampling distribution of ത 𝑌 ▪ for a population with 𝜈 = 𝜈 𝑌 and 𝜏 2 = 𝜏 𝑌 2 ▪ If the CLT holds 2 𝑌~𝑂 𝜈 𝑌 , 𝜏 𝑌 3 things: ത shape, mean, dispersion 𝑜 ▪ So, the statistic ത 𝑌 ▪ is normally distributed ▪ has mean 𝜈 𝑌 𝜏 𝑌 ▪ and has standard deviation 𝑜 ▪ Fortunately, the CLT holds pretty often

THE SAMPLING DISTRIBUTION ▪ The standard deviation of the distribution of sample means ത 𝑌 𝜏 𝑌 ▪ is given by 𝜏 ത 𝑌 = 𝑜 ▪ has a special name: standard error of the mean ▪ is often abbreviated as the standard error (SE) ▪ decreases with increasing sample size ▪ but only according to the “law of diminishing returns” ( 1/ 𝑜 ) ▪ is often calculated by software (SPSS, etc.) ▪ is the basis for confidence intervals and hypothesis tests (see later) That’s a bit confusing, because we will meet more standard errors later on

EXERCISE 1 What is the meaning of the standard error?

CONFIDENCE INTERVALS FOR 𝜈 ▪ A sample mean ҧ 𝑦 is a point estimate of the population mean 𝜈 ▪ it is the best possible estimate of 𝜈 To simplify notation, we will drop the “ 𝑌 ” from 𝜈 𝑌 now, ▪ but it will probably not be completely right and write just 𝜈 ▪ A confidence interval (CI) for the mean is a range of possible values for 𝜈 : 𝜈 lower ≤ 𝜈 ≤ 𝜈 upper ▪ such that the interval 𝐷𝐽 𝜈 = 𝜈 lower , 𝜈 upper contains the true value ( 𝜈 ) with a certain probability (e.g., 95% )

ҧ CONFIDENCE INTERVALS FOR 𝜈 ▪ From the CLT it follows that under certain conditions: the distribution of ത ▪ 𝑌 is normal the best estimate of ത ▪ 𝑌 of 𝜈 is ҧ 𝑦 𝜏 the standard deviation of ത 𝑌 is ▪ 𝑜 ▪ This implies that: 𝜏 𝜏 with probability 2.5% , ത 𝑜 ⇒ 𝜈 > ത ▪ 𝑌 < 𝜈 − 1.96 𝑌 + 1.96 𝑜 𝜏 𝜏 with probability 2.5% , ത 𝑜 ⇒ 𝜈 < ത ▪ 𝑌 > 𝜈 + 1.96 𝑌 − 1.96 𝑜 𝜏 𝜏 so with probability 95% , ത 𝑜 ≤ 𝜈 ≤ ത ▪ 𝑌 − 1.96 𝑌 + 1.96 𝑜 ▪ So, if we find a sample mean ҧ 𝑦 , we can construct the following 95% confidence interval for 𝜈 : 𝑦 − 1.96 𝜏 𝑦 + 1.96 𝜏 CI 𝜈,0.95 = 𝑜 , ҧ 𝑜

ҧ ҧ ҧ CONFIDENCE INTERVALS FOR 𝜈 Three notations for a confidence interval for 𝜈 𝜏 𝜏 ▪ 𝑦 − 1.96 𝑜 , ҧ 𝑦 + 1.96 𝑜 𝜏 𝜏 ▪ 𝑦 − 1.96 𝑜 ≤ 𝜈 ≤ ҧ 𝑦 + 1.96 𝑜 𝜏 ▪ 𝑦 ± 1.96 𝑜

ҧ CONFIDENCE INTERVALS FOR 𝜈 Example ▪ Population ▪ 𝜈 = 520.78 (unknown) ▪ 𝜏 = 86.60 (known) ▪ normally distributed (assumed) ▪ Sample ▪ 𝑜 = 5 (chosen) 𝑦 = 504.0 (estimated) ▪ ▪ Calculation 86.60 ▪ standard error of mean: 5 = 38.73 ▪ 1.96 × 38.73 = 75.91 ▪ 𝐷𝐽 𝜈,0.95 = 428.09, 579.91

EXERCISE 2 Write the confidence interval 428.09, 579.91 in two alternative ways.

ҧ CONFIDENCE INTERVALS FOR 𝜈 ▪ The factor 1.96 is of course related to the 95% probability ▪ Other confidence levels: Where 𝑨 𝛽/2 is such that 𝑄 𝑎 ≤ 𝑨 𝛽/2 = 𝛽 if 𝑎 is drawn from a 𝑎 -distribution ▪ General form of a 1 − 𝛽 × 100% confidence interval of the mean: 𝜏 𝜏 CI 𝜈,1−𝛽 = 𝑦 − 𝑨 𝛽/2 𝑜 , ҧ 𝑦 + 𝑨 𝛽/2 𝑜

CONFIDENCE INTERVALS FOR 𝜈

CONFIDENCE INTERVALS FOR 𝜈 ▪ Trade-off ▪ narrow CI  low confidence level ▪ wide CI  high confidence level ▪ Choice of confidence level depends on application ▪ more precision required for a refinery than for a dairy farm

CONFIDENCE INTERVALS FOR 𝜈 ▪ A confidence interval either does or does not contain 𝜈 ▪ The confidence level quantifies the risk ▪ Out of 100 confidence intervals, approximately 95% will contain 𝜈 , while approximately 5% might not contain 𝜈

HYPOTHESIS TESTS FOR 𝜈 ▪ We can use the standard error to perform a hypothesis test ▪ recall that 𝐷𝐽 𝜈,0.95 = 428.09, 579.91 ▪ Suppose we hypothesize 𝜈 = 550 ▪ The value 550 is inside the 95% confidence interval for 𝜈 ▪ therefore the sample statistic+confidence interval will not suggest that the hypothesis ( 𝜈 = 550 ) is wrong ▪ and we will not reject the hypothesis ▪ notice that we didn’t say that 𝜈 = 550 ; we only said that we can’t reject it (at a 5% significance level)

HYPOTHESIS TESTS FOR 𝜈 ▪ Another example: suppose we hypothesize that 𝜈 = 600 ▪ The value 600 is outside the confidence interval for 𝜈 ▪ finding a confidence interval not containing 𝜈 happens only in 5% of the cases ▪ so we conclude that 𝜈 ≠ 600 (at a 5% significance level) ▪ therefore the sample statistic+confidence interval will suggest that the hypothesis ( 𝜈 = 600 ) is wrong ▪ and we will reject the hypothesis Much more on hypothesis tests later on!

ҧ THE 𝑢 -DISTRIBUTION 𝜏 𝜏 ▪ A closer look at CI 𝜈,0.95 = 𝑦 − 1.96 𝑜 , ҧ 𝑦 + 1.96 𝑜 ▪ Given a sample mean ҧ 𝑦 , you can find a 95% confidence interval for the population mean 𝜈 ▪ Sounds great when you don’t know 𝜈 ... ▪ ... but it assumes you do know 𝜏 ! ▪ There are many situations in which you don’t know 𝜈 and you also don’t know 𝜏 ▪ So what to do?

THE 𝑢 -DISTRIBUTION ▪ A simple strategy ▪ If the population standard deviation 𝜏 is unknown, we can estimate it with the sample standard deviation 𝑡 𝑡 𝜏 ▪ Then we use ±1.96 𝑜 instead of ±1.96 𝑜 ▪ But we pay a price for that ▪ The reason is that 𝑡 is itself an estimate of 𝜏 , and therefore uncertain ▪ The price we pay is that the factor “ 1.96 ” must be somewhat larger