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