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ORDP 20-110

ANALYSIS OF MEASUREMENT DATA

5-4.1.2.2 Give a (1 − a) Confidence Interval Estimate for a Single Point on the Line (i.e., the Mean Value of Y Corresponding to a Chosen Value of x = x')

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Note: An interval estimate of the intercept of the line (8.) is obtained by setting X' = 0 in the above procedure.

LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES

ORDP 20-110

5-4.1.2.3 Give a (1 - α)

Interval Estimate for a Single (Future) Value (Y') of Y Corresponding to a Chosen Value (x') of x.

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Procedure

5-4.1.3 What is the Confidence Interval Estimate for ß1, the Slope of the True Line y = ß。 + B2 x?

Example

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ORDP 20-110

ANALYSIS OF MEASUREMENT DATA

5-4.1.4 If We Observe n' New Values of Y (with Average Y'), How Can We Use the Fitted Regression Line to Obtain an Interval Estimate of the Value of x that Produced These Values of Y?

Example: Suppose that we obtain 10 new measurements of Young's modulus (with average, Ÿ' = 4500) and we wish to use the regression line to make an interval estimate of the temperature (x) at which the measurements were made.

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LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES

ORDP 20-110

5-4.1.5 Using the Fitted Regression Line, How Can We Choose a Value (x') of x Which We May Expect with Confidence (1 − a) Will Produce a Value of Y Not Less Than Some Specified

Value Q?

Example: What value (x') of temperature (x) can be expected to produce a value of
Young's modulus not less than 4300?

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ORDP 20-110

ANALYSIS OF MEASUREMENT DATA

5-4.1.6 Is the Assumption of Linear Regression Justified?

This involves a test of the assumption that the mean Y values (Ỹ1) for given x values do lie on a straight line (we assume that for any given value of x, the corresponding individual Y values are normally distributed with variance of, which is independent of the value of x). A simple test is available provided that we have more than one observation on Y at one or more values of x. Assume that there are n pairs of values (x,, Y.), and that among these pairs there occur only k values of x (where k is less than n).

For example, see the data recorded in Table 5-3 which shows measurements of Young's modulus (coded) of sapphire rods as a function of temperature.

Each x is recorded in Column 1, and the corresponding Y values (varying in number from 1 to 3 in the example) are recorded opposite the appropriate x. The remaining columns in the table are convenient for the required computations.

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