Stats for Engineers Lecture 10

Recap: Linear regression

Linear regression: fitting a straight line to the mean value of as a function of

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We measure a response variable at various values of a controlled variable

Least-squares estimates and :

�̂�=𝑆𝑥𝑦

𝑆𝑥𝑥∧�̂�=𝑦− �̂�𝑥

𝑆𝑥𝑥=∑𝑖𝑥 𝑖2−

(∑𝑖 𝑥𝑖)2

𝑛 =∑𝑖

(𝑥𝑖−𝑥 )2

𝑆𝑥𝑦=∑𝑖𝑥 𝑖 𝑦 𝑖−

∑𝑖𝑥 𝑖∑

𝑖𝑦 𝑖

𝑛 =∑𝑖

(𝑥𝑖−𝑥) ( 𝑦 𝑖− 𝑦 )❑

Sample means

�̂�=�̂�+�̂� 𝑥Equation of the fitted line is

Estimating : variance of y about the fitted line

¿𝑆𝑦𝑦 −�̂�𝑆𝑥𝑦

𝑛−2

Quantifying the goodness of the fit

Residual sum of squares

Predictions

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For given of interest, what is mean ?

Predicted mean value: .

It can be shown that

Confidence interval for mean y at given x

What is the error bar?

y 240 181 193 155 172 110 113 75 94x 1.6 9.4 15.5 20.0 22.0 35.5 43.0 40.5 33.0

Example: The data y has been observed for various values of x, as follows:

Fit the simple linear regression model using least squares.

�̂�=234.1−3.509 𝑥

Example: Using the previous data, what is the mean value of at and the 95% confidence interval?

𝑦=234.1−3.509 𝑥Recall fit was

Need

95% confidence for Q=0.975

Confidence interval is ,

⇒ .

Hence confidence interval for mean is

≈129±17

Extrapolation: predictions outside the range of the original data

𝑥𝑛𝑒𝑤

𝑦 𝑛𝑒𝑤

What is the prediction for mean at ?

𝑦 𝑛𝑒𝑤=�̂�+�̂� 𝑥𝑛𝑒𝑤

Extrapolation: predictions outside the range of the original data

𝑥𝑛𝑒𝑤

𝑦 𝑛𝑒𝑤

What is the prediction for mean at ?

Looks OK!

𝑦 𝑛𝑒𝑤=�̂�+�̂� 𝑥𝑛𝑒𝑤

Extrapolation: predictions outside the range of the original data

𝑥𝑛𝑒𝑤

𝑦 𝑛𝑒𝑤

What is the prediction for mean at ?

Quite wrong!

Extrapolation is often unreliable unless you are sure straight line is a good model

We previously calculated the confidence interval for the mean: if we average over many data samples of at , this tells us the interval we expect the average to lie in.

What about the distribution of future data points themselves?

Confidence interval for a prediction

Two effects:

- Variance on our estimate of mean at

- Variance of individual points about the mean

Confidence interval for a single response (measurement of at ) is

�̂� 2( 1𝑛+(𝑥−𝑥 )2

𝑆𝑥𝑥 )�̂� 2

Example: Using the previous data, what is the 95% confidence interval for a new measurement of at Answer

≈129±50

A linear regression line is fit to measured engine efficiency as a function of external temperature (in Celsius) at values . Which of the following statements is most likely to be incorrect?

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1. The confidence interval for a new measurement of at is narrower than at

2. Adding a new data at would decrease the confidence interval width at

3. If and accurately have a linear regression model, adding more data points at and would be better than adding more at and

4. The mean engine efficiency at T= -20 will lie within the 95% confidence interval at T=-20 roughly 95% of the time

Confidence interval for mean y at given x

Confidence interval for a single response (measurement of at )

- Confidence interval narrower in the middle (

- Adding new data decreases uncertainty in fit, so confidence intervals narrower ( larger)

- If linear regression model accurate, get better handle on the slope by adding data at both ends(bigger smaller confidence interval)

- Extrapolation often unreliable – e.g. linear model may well not hold at below-freezing temperatures. Confidence interval unreliable at T=-20.

AnswerThe confidence interval for a new measurement of at is narrower than at

The mean engine efficiency at T= -20 will lie within the 95% confidence interval at T=-20 roughly 95% of the time

Adding a new data at would decrease the confidence interval width at

If and accurately have a linear regression model, adding more data points at and would be better than adding more at and

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Correlation

Regression tries to model the linear relation between mean y and x.

Correlation measures the strength of the linear association between y and x.

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A

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Weak correlation Strong correlation

- same linear regression fit (with different confidence intervals)

If x and y are positively correlated:

- if x is high ( y is mostly high ()

- if x is low () y is mostly low ()

on average is positive

If x and y are negatively correlated:

on average is negative

- if x is high ( y is mostly low ()

- if x is low () y is mostly high ()

can use to quantify the correlation

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More convenient if the result is independent of units (dimensionless number).

𝑟=𝑆𝑥𝑦

√𝑆𝑥𝑥𝑆𝑦𝑦

r = 1: there is a line with positive slope going through all the points; r = -1: there is a line with negative slope going through all the points; r = 0: there is no linear association between y and x.

Range :

Pearson product-moment.

Define

If , then is unchanged ( Similarly for - stretching plot does not affect .

Example: from the previous data:

Hence

Notes:

- magnitude of r measures how noisy the data is, but not the slope

- finding only means that there is no linear relationship, and does not imply the variables are independent

Correlation

A researcher found that r = +0.92 between the high temperature of the day and the number of ice cream cones sold in Brighton. What does this information tell us?

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1. Higher temperatures cause people to buy more ice cream.

2. Buying ice cream causes the temperature to go up.

3. Some extraneous variable causes both high temperatures and high ice cream sales

4. Temperature and ice cream sales have a strong positive linear relationship.

Question from Murphy et al.

Correlation r

error

- not easy; possibilities include subdividing the points and assessing the spread in r values.

Error on the estimated correlation coefficient?

Causation? does not imply that changes in x cause changes in y - additional types of evidence are needed to see if that is true.

J Polit Econ. 2008; 116(3): 499–532.http://www.journals.uchicago.edu/doi/abs/10.1086/589524

Strong evidence for a 2-3% correlation.

- this doesn’t mean being tall causes you earn more (though it could)

1.

Correlation

Which of the follow scatter plots shows data with the most negative correlation ?

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0%

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1. No correlation2. Correct3. Not large4. positive

2.

3. 4.

Acceptance Sampling

Situation: large batches of items are produced. We want to sample a small proportion of each batch to check that the proportion of defective items is sufficiently low.

One-stage sampling plans

Sample items number of defective items in the sample

Reject batch if , accept if

How do we choose and ?

Operating characteristic (OC): probability of accepting the batch

Define proportion of defective items in the batch (typically small).

Then if the population the samples are drawn from is large.

N=100, c=3

Testing 100 samples and rejecting if more than 3 are faulty gives the OC curve on the right. Which of the following is the curve for testing 100 samples and rejecting if more than 2 are faulty?

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1. C=2 correct2. C=53. Wrong height

1. 2. 3.

Rejecting more than 2, rather than more than 3 makes it more likely to reject the batch (for any ). is higher.

is lower, lower

For standard acceptance sampling, Producer and Consumer must decide on the following:

Acceptable quality level:

(consumer happy, want to accept with high probability)

Unacceptable quality level:

(consumer unhappy, want to reject with high probability)

Ideally:

- always accept batch if - always reject batch if

i.e. and

- but can’t do this without inspecting the entire batch

Use a sampling scheme

Producer’s Risk: reject a batch that has acceptable quality

¿1−𝐿(𝑝1)

Consumer’s Risk: accept a batch that has unacceptable quality

¿𝐿(𝑝2)

Want to minimize:

𝑝2𝑝1

Operating characteristic curve

Consumer’s risk (probability of accepting when unacceptable quality )

Producer’s risk (probability of rejecting when acceptable quality )

If consumer and producer agree on - can then calculate and .

𝛽

Acceptance Sampling Tables: give for and

Example

In planning an acceptance sampling scheme, the Producer and Consumer have agreed that the acceptable quality level is 2% defectives and the unacceptable level is 6%. Each is prepared to take a 10% risk. What sample size is required and under what circumstances should the batch be rejected?

Answer

,

⇒𝑛=153 ,𝑐=5

Should sample 153 items and reject if the number of defective items is greater than 5.

In planning an acceptance sampling scheme, the Producer and Consumer have agreed that the acceptable quality level is 1% defectives and the unacceptable level is 3%. Each is prepared to take a 5% risk. What is the best plan?

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• sample 308 items and reject if the number of defective items is greater than 5

• sample 308 items and reject if the number of defective items is 5 or more

• sample 521 items and reject if the number of defective items is 9 or more

• sample 521 items and reject if the number of defective items is 10 or more

In planning an acceptance sampling scheme, the Producer and Consumer have agreed that the acceptable quality level is 1% defectives and the unacceptable level is 3%. Each is prepared to take a 5% risk. What is the best plan?

Sample 521 and reject if more than 9 (i.e. 10 or more)

Example – calculating the risks

It has been decided to sample 100 items at random from each large batch and to reject the batch if more than 2 defectives are found. The acceptable quality level is 1% and the unacceptable quality level is 5%. Find the Producer's and Consumer's risks.

, 5Answer

1. For the Producer's Risk: want probability of reject batch when

𝑃 (Reject batchwhen𝑝=0.01 )=1−𝑃 (accept batch)=1−𝐿 (0.01 )

¿1−𝑃 ( 𝑋=0 )− 𝑃 ( 𝑋=1 )−𝑃 (𝑋=2)

¿1−𝐶0100 0.010×0.99100−𝐶1

1000.01×0.9999−𝐶21000.012×0.9998

- 0.3660 - 0.3697 - 0.1849 = 0.079.

𝑋∼ 𝐵(100,0.01)

Example – calculating the risks

It has been decided to sample 100 items at random from each large batch and to reject the batch if more than 2 defectives are found. The acceptable quality level is 1% and the unacceptable quality level is 5%. Find the Producer's and Consumer's risks.

, 5Answer

2. For the Consumer’s Risk: want probability of accepting batch when

𝑃 (Accept when𝑝=0.05 )=𝐿 (0.05 )

¿𝑃 ( 𝑋=0 )+𝑃 (𝑋=1 )+𝑃 (𝑋=2)

¿𝐶0100 0.050×0.95100+𝐶1

1000.05×0.9599+𝐶2100 0.052×0.9598

¿0.118

𝑋∼ 𝐵(100,0.05)

It has been decided to sample 100 items at random from each large batch and to reject the batch if more than 2 defectives are found. The acceptable quality level is 1% and the unacceptable quality level is 5%.

Which of the following would increase the Consumer’s Risk?

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1. Increasing the acceptable quality level to 2%

2. Decreasing the unacceptable quality level to 4%

3. Rejecting if more than 1 defectives are found

It has been decided to sample 100 items at random from each large batch and to reject the batch if more than 2 defectives are found. The acceptable quality level is 1% and the unacceptable quality level is 5%.

Which of the following would increase the Consumer’s Risk?

Increasing the acceptable quality level

Decreasing the unacceptable quality level

Rejecting if more than 1 defectives are found

NO – Consumer’s Risk depends on the unacceptable quality level

YES –e.g. then more likely to accept when the defect probability is compared to

NO – more likely to get 1 or more, so less likely to accept batch lower Consumer’s Risk

Two-stage sampling planIdea: test some, reject if clearly bad, accept if clearly good, if not clear investigate further

1. Sample items, number of defectives in the sample

2. Accept batch if , reject if (where )

3. If , sample a further items; let number of defectives in 2nd sample

4. Accept batch if , otherwise reject batch.

Advantage: can require fewer samples than single-stage plan (for similar )

Distadvantage: more complicated, need to choose

Example

A two-stage sampling plan for a quality control procedure is as follows: Sample 75 items, accept if less than 2 defectives, reject if more than 3 defectives;otherwise sample 120 more and reject if more than 4 defectives in the new batch

Find the probability that a batch is rejected under this plan if the probability of any particular item being faulty is 2.

Answer

Let be number faulty in the first batch, be number faulty in second batch (if taken)

𝑋 1≤1

2,3

𝑋 1>3

Accept

Reject

𝑋 2≤ 4

𝑋 2>4

Accept

Reject

𝑃 (reject )=𝑃 (reject∈first stage )+𝑃 (𝑋 1=2,3 )𝑃 (reject∈second stage)

¿𝑃 (𝑋 1>3 )+[𝑃 (𝑋 1=2 )+𝑃 ( 𝑋1=3 ) ]𝑃 (𝑋 2>4)

¿1−∑𝑛=0

3

𝑃 (𝑋 1=𝑛)+[𝑃 (𝑋 1=2 )+𝑃 (𝑋 1=3 ) ][1−∑𝑛=0

4

𝑃 (𝑋 2=𝑛)]

-+…

099

defectives out of 75 defectives out of 120 more

𝑋 1≤1

2,3

𝑋 1>3

Accept

Reject

𝑋 2≤ 4

𝑋 2>4

Accept

Reject

Example (as before)

In planning an acceptance sampling scheme, the Producer and Consumer have agreed that the acceptable quality level is 2% defectives and the unacceptable level is 6%. Each is prepared to take a 10% risk. What sample size is required and under what circumstances should the batch be rejected?

Answer: single stage plan

,

𝑛=153 ,𝑐=5Sample 153 items and reject if the number of defective items is greater than 5.

Alternative answer: two-stage plan, as last example

𝑛1=75 ,𝑛2=120 ,𝑐1=1 , c2=3 , c3=4

- Always take 153 samples

- Sometimes takes only 75 samples, sometimes 120+75=195

- Mean number: (depending on ); more efficient!

Two-stage plan can have very similar OC curve, but require fewer samples

BUT: - not obvious how to choose ; example not optimal

Variation: better to include first sample with second sample for final decision

- less parallelizable (e.g. might care if testing is cheap but takes a long time)