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HNC MathsStatistics (5)

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Linear Regression

Example of ‘Line of Best Fit’

Problem: Work out the equation for the ‘Line of Best Fit’ for the data below

Step 1: Define the data

Step 2: Decide the Dependant (X) & Independent Data (Y)

Step 3: Tabulate the data

Step 4: Calculate (Multiply the data. Add the columns)

Step 5: See next Page

Frequency(Hz)

InductiveResistance

(Ohms)50 30

100 65150 90200 130250 150300 190350 200

FrequencyX

Ind. ResY X² XY Y²

50 30

100 65

150 90

200 130

250 150

300 190

350 200

Σ Σ Σ Σ Σ

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Linear RegressionStep 5: Fit the data into the Simultaneous Equations

Step 6: Solve for a0 and a1

Step 7: State the equation

ΣY = a0N + a1ΣX

ΣXY = a0ΣX + a1ΣX2

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Problem:

Work out the equation for the ‘Line of Best Fit’ for the data below. But in this case, make ‘Inductive Resistance’ the Dependant

Ind. Res.X

FrequencyY X² XY Y²

30 50

65 100

90 150

130 200

150 250

190 300

200 350

Σ Σ Σ Σ Σ

Linear Regression

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‘Confidence’ in DataConfidence

Level %Confidence

Coeffi cient zc

99 2.58

98 2.33

96 2.05

95 1.96

90 1.645

80 1.28

50 0.6745

Problem: Determine the confidence coefficient corresponding to a confidence level of 98.5%

In this case the area corresponds to 98.5% (area under the whole curve is 100%). An area of 98.5% represents a certain Standard Deviation +/-

Step 1: 98.5% is equivalent to 0.985 of the total area.

Step 2: On a normal distribution this means 0.985 divided by 2 each side of the mean line (centre).

Step 3: Look up 0.4925 in the Table. This is 2.43. [ This means +/- 2.43 Standard Deviations ]

Step 4: State the answer.

The Confidence Coefficient for a limit of 98.5% is 2.43(This means 98.5% of the data is within 2.43 standard deviations of the mean)

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Confidence TestingA company is producing components by machining. The diameters over a period of time were known to have a Standard Deviation of 0.18cm. On a certain day 100 components were sampled and these had a mean diameter of 0.476cm. If the machine produces 2500 components a day, find : -

a)the 90% confidence limitsb)the 97% confidence limits

Step 1: Define the data

Step 2: Define the confidence limits equation

Step 3: Look up the value for zc from the Table

Step 4: Insert the data and calculate

Confidence level = xzc s+-N√ √ ( Np – N )

( Np – 1 )

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Confidence Testing