[PPT] - Business Statistics CONTENTS Estimating parameters The sampling PowerPoint Presentation

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𝜈: ESTIMATES, CONFIDENCE INTERVALS, AND TESTS

Business Statistics

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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 CONTENTS

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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 ▪ so, 𝑞 =

26 200 = 0.13

▪ this suggests 𝜌 ≈ 0.13

ESTIMATING PARAMETERS

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Terminology ▪ Parameter

▪ a characteristic descriptive of the population ▪ e.g., 𝜈, 𝜌, 𝜏 (or 𝜏2)

▪ Estimator

▪ a statistic derived from a sample to infer the value of a population parameter ▪ e.g., ത 𝑌, 𝑄, 𝑇 (or 𝑇2)

▪ Estimate

▪ the value of the estimator in a particular sample ▪ e.g., ҧ 𝑦, 𝑞, 𝑡 (or 𝑡2)

ESTIMATING PARAMETERS

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ESTIMATING PARAMETERS

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Estimator Estimate Population parameter Mean ത 𝑌 =

1 𝑜 σ𝑗=1 𝑜

𝑌𝑗 ҧ 𝑦 =

1 𝑜 σ𝑗=1 𝑜

𝑦𝑗 𝜈 Standard deviation 𝑇 =

1 𝑜−1 σ𝑗=1 𝑜

𝑌𝑗 − ത 𝑌 2 𝑡 =

1 𝑜−1 σ𝑗=1 𝑜

𝑦𝑗 − ҧ 𝑦 2 𝜏 Proportion 𝑄 =

𝑌 𝑜

𝑞 =

𝑦 𝑜

𝜌

ESTIMATING PARAMETERS

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▪ 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

▪ so a distribution of ҧ 𝑦-values (the sampling distribution of ത 𝑌)

▪ Solution: point estimates and confidence intervals ESTIMATING PARAMETERS

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▪ 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

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

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▪ 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

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▪ Consider eight random samples, each of size 𝑜 = 5

▪ the sample means ( ҧ 𝑦1 = 504.0, ҧ 𝑦2 = 576.0, … , ҧ 𝑦8 = 582) tend to be close to the population mean (𝜈 = 520.78) ▪ sometimes a bit lower, sometimes a bit higher

THE SAMPLING DISTRIBUTION

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▪ The dot plots show that the sample means ( ҧ 𝑦1, … , ҧ 𝑦8) have much less variation than the individual data points (𝑦1, … , 𝑦2637) THE SAMPLING DISTRIBUTION

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▪ 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

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▪ The sampling distribution of ത 𝑌

▪ for a population with 𝜈 = 𝜈𝑌 and 𝜏2 = 𝜏𝑌

2

▪ If the CLT holds ത 𝑌~𝑂 𝜈𝑌, 𝜏𝑌

2

𝑜 ▪ So, the statistic ത 𝑌

▪ is normally distributed ▪ has mean 𝜈𝑌 ▪ and has standard deviation

𝜏𝑌 𝑜

▪ Fortunately, the CLT holds pretty often THE SAMPLING DISTRIBUTION

3 things: shape, mean, dispersion

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▪ The standard deviation of the distribution of sample means ത 𝑌

▪ is given by 𝜏 ത

𝑌 = 𝜏𝑌 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)

THE SAMPLING DISTRIBUTION

That’s a bit confusing, because we will meet more standard errors later on

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What is the meaning of the standard error? EXERCISE 1

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▪ A sample mean ҧ 𝑦 is a point estimate of the population mean 𝜈

▪ it is the best possible estimate of 𝜈 ▪ but it will probably not be completely right

▪ A confidence interval (CI) for the mean is a range of possible values for 𝜈: 𝜈lower ≤ 𝜈 ≤ 𝜈upper

▪ such that the interval 𝐷𝐽𝜈 = 𝜈lower, 𝜈upper contains the true value (𝜈) with a certain probability (e.g., 95%)

CONFIDENCE INTERVALS FOR 𝜈

To simplify notation, we will drop the “𝑌” from 𝜈𝑌 now, and write just 𝜈

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▪ 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 ▪ the standard deviation of ത 𝑌 is

𝜏 𝑜

▪ This implies that:

▪ with probability 2.5%, ത 𝑌 < 𝜈 − 1.96

𝜏 𝑜 ⇒ 𝜈 > ത

𝑌 + 1.96

𝜏 𝑜

▪ with probability 2.5%, ത 𝑌 > 𝜈 + 1.96

𝜏 𝑜 ⇒ 𝜈 < ത

𝑌 − 1.96

𝜏 𝑜

▪ so with probability 95%, ത 𝑌 − 1.96

𝜏 𝑜 ≤ 𝜈 ≤ ത

𝑌 + 1.96

𝜏 𝑜

▪ So, if we find a sample mean ҧ 𝑦, we can construct the following 95% confidence interval for 𝜈: CI𝜈,0.95 = ҧ 𝑦 − 1.96 𝜏 𝑜 , ҧ 𝑦 + 1.96 𝜏 𝑜

CONFIDENCE INTERVALS FOR 𝜈

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Three notations for a confidence interval for 𝜈 ▪ ҧ 𝑦 − 1.96

𝜏 𝑜 , ҧ

𝑦 + 1.96

𝜏 𝑜

▪ ҧ 𝑦 − 1.96

𝜏 𝑜 ≤ 𝜈 ≤ ҧ

𝑦 + 1.96

𝜏 𝑜

▪ ҧ 𝑦 ± 1.96

𝜏 𝑜

CONFIDENCE INTERVALS FOR 𝜈

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Example ▪ Population

▪ 𝜈 = 520.78 (unknown) ▪ 𝜏 = 86.60 (known) ▪ normally distributed (assumed)

▪ Sample

▪ 𝑜 = 5 (chosen) ▪ ҧ 𝑦 = 504.0 (estimated)

▪ Calculation

▪ standard error of mean:

86.60 5 = 38.73

▪ 1.96 × 38.73 = 75.91 ▪ 𝐷𝐽𝜈,0.95 = 428.09, 579.91

CONFIDENCE INTERVALS FOR 𝜈

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

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▪ The factor 1.96 is of course related to the 95% probability ▪ Other confidence levels: ▪ General form of a 1 − 𝛽 × 100% confidence interval of the mean: CI𝜈,1−𝛽 = ҧ 𝑦 − 𝑨𝛽/2 𝜏 𝑜 , ҧ 𝑦 + 𝑨𝛽/2 𝜏 𝑜 CONFIDENCE INTERVALS FOR 𝜈

Where 𝑨𝛽/2 is such that 𝑄 𝑎 ≤ 𝑨𝛽/2 = 𝛽 if 𝑎 is drawn from a 𝑎-distribution

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CONFIDENCE INTERVALS FOR 𝜈

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▪ 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

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▪ 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 CONFIDENCE INTERVALS FOR 𝜈

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▪ We can use the standard error to perform a hypothesis test

▪ recall that 𝐷𝐽𝜈,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

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▪ 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

HYPOTHESIS TESTS FOR 𝜈

Much more on hypothesis tests later on!

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▪ A closer look at CI𝜈,0.95 = ҧ 𝑦 − 1.96

𝜏 𝑜 , ҧ

𝑦 + 1.96

𝜏 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

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▪ A simple strategy ▪ If the population standard deviation 𝜏 is unknown, we can estimate it with the sample standard deviation 𝑡 ▪ Then we use ±1.96

𝑡 𝑜 instead of ±1.96 𝜏 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 THE 𝑢-DISTRIBUTION

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▪ Recall that the CLT yields that

ത 𝑌−𝜈 𝜏/ 𝑜 ~𝑂 0,1

▪ where 𝑎 is the standard normal distribution

▪ Likewise, it can be shown that

ത 𝑌−𝜈 𝑡/ 𝑜 ~𝑢

▪ where 𝑢 is the 𝑢-distribution (or Student’s 𝑢-distribution) ▪ which has an even more complicated formula than the normal distribution ▪ 𝑔 𝑨 =

1 2𝜌 𝑓−1

2𝑨2 vs. 𝑔 𝑢; 𝜉 =

Γ 1

2 𝜉+1

𝜉𝜌Γ 1

2𝜉

1 +

𝑢2 𝜉 −1

2 𝜉+1

THE 𝑢-DISTRIBUTION

Arrrgh: forget quickly!

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▪ The confidence interval for 𝜈 with unknown 𝜏 is CI𝜈,1−𝛽 = ҧ 𝑦 − 𝑢𝛽/2 𝑡 𝑜 , ҧ 𝑦 + 𝑢𝛽/2 𝑡 𝑜 ▪ What is the 𝑢-distribution?

▪ quite similar to the 𝑎-distribution (𝜈 = 0, continuous, symmetric, bell- shaped, infinite range, ...) ▪ a little bit “fatter” tails ▪ it has 1 parameter, usually denoted with df or 𝜉, and called degrees of freedom

THE 𝑢-DISTRIBUTION

Where 𝑢𝛽/2 is such that 𝑄 𝑈 ≥ 𝑢𝛽/2 = 𝛽 if 𝑈 is drawn from a 𝑢-distribution

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▪ Graph of pdf of 𝑢-distribution THE 𝑢-DISTRIBUTION

𝑦

𝑎 (standard normal) distribution

𝑔 𝑦

𝑢-distribution with df = 13 𝑢-distribution with df = 5 𝑢-distribution with df = 1000

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▪ Different notations

▪ 𝑢13 ▪ 𝑢 df = 13 ▪ etc.

▪ And likewise

▪ 𝑢13;𝛽/2 ▪ 𝑢13 𝛽/2 ▪ etc.

▪ So altogether for the confidence interval CI𝜈,1−𝛽 = ҧ 𝑦 − 𝑢𝑜−1;𝛽/2 𝑡 𝑜 , ҧ 𝑦 + 𝑢𝑜−1;𝛽/2 𝑡 𝑜 THE 𝑢-DISTRIBUTION

Compare to ҧ 𝑦 − 𝑨𝛽/2 𝜏 𝑜 , ҧ 𝑦 + 𝑨𝛽/2 𝜏 𝑜

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THE 𝑢-DISTRIBUTION

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▪ How to choose the parameter df?

▪ it is a parameter based on the sample size that is used to determine the value of the 𝑢-statistic ▪ it tells how many observations are used to estimate 𝜏, less the number

f intermediate estimates used in the calculation

▪ the df for the 𝑢-distribution in the case of a confidence interval for 𝜈 when 𝜏 is unknown, is df = 𝑜 − 1 ▪ but in other cases, it may be different

▪ Properties of 𝑢

▪ as 𝑜 increases, the 𝑢-distribution approaches the shape of the normal distribution ▪ for a given confidence level 𝛽, 𝑢 is always larger than 𝑨, so a confidence interval based on 𝑢 is always wider than if 𝑨 were used

THE 𝑢-DISTRIBUTION

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▪ Reading the table of critical 𝑢-values

▪ e.g., 𝑢0.025 9 ▪ 𝑢 = 2.262

THE 𝑢-DISTRIBUTION

𝑒𝑔 = 9 𝛽/2 = 0.025

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Look carefully at tables for 𝑨 and 𝑢: ▪ 𝑨 usually runs from left to right

▪ 𝑄 𝑌 ≤ 𝑨 = ׬

−∞ 𝑨 𝑔 𝑦 𝑒𝑦

▪ 𝑢 usually runs from right to left

▪ 𝑄 𝑌 ≥ 𝑢 = ׬

𝑢 ∞ 𝑔 𝑦 𝑒𝑦

THE 𝑢-DISTRIBUTION

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▪ Background of 𝑢

▪ developed by William Gosset in 1908 ▪ while working at Guiness Brewery, Dublin ▪ published under the pen name “Student”

THE 𝑢-DISTRIBUTION

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Example for confidence interval ▪ Population

▪ 𝜈 = 520.78 (unknown) ▪ 𝜏 = 86.60 (unknown) ▪ normally distributed (assumed)

▪ Sample

▪ 𝑜 = 5 (chosen) ▪ ҧ 𝑦 = 504.0 (estimated) ▪ 𝑡 = 73.01 (estimated)

▪ Calculation

▪ standard error of mean:

𝑡 𝑜 = 32.65

▪ 2.776 × 32.65 = 90.65 ▪ 𝐷𝐽𝜈,0.95 = 413.35, 594.64

THE 𝑢-DISTRIBUTION

now we have a situation in which 𝜏 is not known to us

df=4

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▪ Repeat the hypothesis test for this case

▪ now CI𝜈,0.95 = 413.35, 594.65

▪ So we will reject the hypothesis 𝜈 = 600

▪ while we will not reject the hypothesis 𝜈 = 550

▪ Exactly the same reasoning as with the 𝑨-test, but with (slightly) different numbers THE 𝑢-DISTRIBUTION

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▪ When to use which?

▪ for a confidence interval for 𝜈 if 𝜏2 is known: use 𝑨 ▪ for a confidence interval for 𝜈 if 𝜏2 is unknown: use 𝑢, and estimate 𝜏2 by 𝑡2

▪ How to find?

▪ from a table with 𝑨-values: given 𝛽, look up 𝑨 ▪ from a table with 𝑢-values: given 𝛽 and df, look up 𝑢

▪ What is the difference?

▪ confidence intervals with 𝑢 are a bit wider than with 𝑨 ▪ the difference is small for 𝑜 ≥ 30 and negligible for 𝑜 ≥ 100

COMPARISON OF 𝑨 AND 𝑢

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▪ Example: 50 confidence intervals with 𝑨 and 𝑢 COMPARISON OF 𝑨 AND 𝑢

10 20 30 40 50 60

Based on 2 Based on s

2 (i )

50 Samples, sample size n=10 S i m u l a t e d f r o m : N (2,9) d i s t r i b u t i o n

Sample Number

i

 2 4 2 4