Exam 2 Thursday Correlation and Linear regression Hypothesis Testing Fourier Transforms Linear Systems 0th and 1st order Hypothesis Testing Small and large sample Relation to confidence intervals Population proportions Difference between 2 means Type 1 and Type II errors Level Testing and Power Equations Given General first order system dt y KF t dy Solution for sine input y t ce t KA 1 2 2 sin t arctan Euler Formulas 1 a0 T T 2 T 2 f t dt an 2 T T 2 T 2 f t cos n 2 t dt T bn 2 T Tables correlation coefficient t table z table T 2 T 2 f t sin n 2 t dt T In a sample of 150 households in a certain city 110 had high speed internet access Can you conclude that more than 70 of the households in this city have high speed internet access H 0 0 7 H 1 0 7 Variance p 1 p 0 7 0 3 0 21 Standard Deviation 0 21 0 4582 Your Sample Gave 110 0 7333 150 0 7333 0 7 z 0 89 0 4582 150 0 7333 0 7 z 0 89 0 4582 150 We get a P value of 0 1867 SO We can t conclude that more than 70 have access Example Shipments of nuts are checked for moisture content Five moisture measurements will be made on nuts chosen at random from the shipment Moisture content is reported in percent per weight the true moisture content is 12 and the standard deviation is 1 5 H 0 10 You are interested in testing the hypothesis at the 5 level H1 10 What is the power of the test Find the critical point x 10 2 132 1 5 5 x 11 43 t4 0 45 x 10 2 132 1 5 5 x 11 43 t4 0 45 11 43 12 12 11 43 0 849 1 5 5 P 0 55 t4 P P 0 275 2 The power is 0 5 0 275 0 775 Unchanged by multiplying each value of a variable by a constant Unchanged by adding a constant to each variable Unchanged by interchanging x and y For a given sample size we need to determine whether a given rxy is significant or a result of pure chance For practical problems we can consult a table General procedure Calculate rxy from the measured data Determine level of significance a required a gives the probability that an experimental value of rxy will be greater by pure chance a 0 05 rt value at which 5 chance due to random effects Compare rt to rxy if rxy rt then confidence level is confirmed 95 confidence often used in engineering Say we have a sample size of 18 and we get a correlation coefficient of 0 49 Does a linear relationship exist between our variables 95 Yes our correlation coefficient is greater than that required for 95 Confidence see table The standard error or uncertainty in the regression coefficients is 1 S S 0 n x2 S n 2 x x i 1 i 1 S n 2 x x i i 1 Now we are in a position to find the confidence interval 0 0 tn 2 P S 0 1 1 tn 2 P S 1 We can Find the best fit line slope and intercept Estimate the variance Estimate the standard error Report Confidence intervals for the slope and intercept y i 0 1 xi Best estimate of the mean value of yi The confidence interval on the mean value of yi yi 0 1 x tn 2 P SY Where 1 SY S n x x n 2 x x i 2 i 1 I m 95 sure that the next spring tested will fall within yi 0 1 xi tn 2 P S Pr ed S Pr ed 1 S 1 n x x 2 n 2 xi x i 1 Prediction interval Confidence interval y t KA y0 KA e Steady state t transient Here we can see the response is delayed and steady state not reached until the exponential burns out Consider the case of y0 0 transient term becomes t 0 KA KA 0 367 KA 2 71 KA t 2 0 135 KA 7 38 t SS y reaches 63 of its final value SS y reaches 87 of its final value The time constant gives the time that is takes for the system to reach 63 of its final value Can also look at this in a different way y t KA y0 KA e y t KA e y0 KA t t A thermocouple has a spherical junction with a diameter of 0 3 mm It is used to measure gas temperature in a combustion chamber When the flame is ignited it produces an approximate step change in T to 900 C The gas temperature before ignition is 300C The heat transfer coefficient is 500W m2 C The properties of platinum are r 21 450 kg m3 and c 134 J kg C Given the system is first order and follows the equation hA TG TT cm After how much time will the measurement error be less than 1 d y y KAH t dt dTT dt m is the bulb mass c is the bulb specific heat h is the heat transfer coefficient A is the surface area cm dTT TT TG H t hA dt Great so we know this solution from the previous slides T t TG T0 TG e t y t KA y0 KA e t Time constant Initial temperature Now we can solve for the error fraction and determine the wait time T t TG e 01 T0 TG t 4 605 287 1 32 s y t ce t KA 1 2 2 sin t arctan Long time response purely sinusoidal linear system but Amplitude modified by quantity 2 2 What Happens if 2 2 1 What Happens if 2 2 1 How about the Phase The output follows the input The output goes to zero What if 4 What is the time delay Note 2 f 1 2 The period is T f 2 s The time delay is 1 2 40 at 10 rad s y t A sin t System 1 K1 M 1 System 1 K1 M 1 y t A sin t y AK1M 1 sin t System 2 K 2 M 2 2 y AK1K 2 M 1 M 2 sin t 1 2 f t a0 an cos n t bn sin n t n 1 Find the Fourier Series representation of the periodic displacement signal shown above Neglect frequency components beyond 0 2 Hz Find Fourier coefficients using the Euler formulas and then plug in Fourier Series 2 2 f t a0 an cos n t bn sin n t T T n 1 The period of the signal is 8 seconds so the first n 1 Fourier Component will be at 0 125 Hz and the second at 0 250 Hz Therefore we only have to go to n 1 …