Probability distribution functions Probability mass functions Normal distribution find P determine m s determine P using z table P x m zs x m z s n Single measurement P mean Known population mean and standard deviation If we take n samples from a population 1 xs N N x i 1 i 1 N 2 sx x x i s N 1 i 1 Confidence interval on the mean n large Sx x x z n P n small Sx n P x x tv P 1 2 A manufacturer of vacuum pumps wishes to estimate the mean failure time of its product with 95 confidence Initially six pumps were tested with the results in hour of operation of 1272 1384 1443 1465 1350 and 1319 Estimate the sample mean and the 95 confidence interval of the true mean How many more data points would be needed to improve the confidence interval to be within 50 hours Find sample mean and standard deviation xs 1 N N xi 1372 2 hrs i 1 x x tv P Sx n P 1 2 sx x x i s N 1 i 1 N x 1372 2 t5 95 73 7 6 1 2 73 7 hrs 95 73 7 x 1372 2 t5 95 6 73 7 x 1372 2 2 571 6 95 95 x 1372 2 77 4 95 x 1372 2 50 0 95 tv 95 Sx 50 n Here we need to guess t10 95 73 7 49 50 11 t9 95 Works 73 7 52 71 NO 10 Sometimes we need to calculate confidence intervals on other parameters x and y Large sample statistics The true difference is d x y z s x2 nx s y2 ny P Small sample statistics The true difference is d x y tv P 2 S x2 S y nx n y The degrees of freedom is a bit complicated in this case P 2 S S n n y x v 2 2 2 2 S x S y nx n y nx 1 n y 1 2 x 2 y Example In an experiment involving the breaking strength of a certain type of thread used for PFD s one batch of thread was subjected to heat treatment for 60 seconds while another was treated for 120 seconds The breaking strength in N of 10 threads in each batch were measured 60 seconds 43 52 52 58 49 52 41 52 56 54 120 seconds 59 55 59 66 62 55 57 66 66 51 Find a 99 confidence interval for the difference in the mean strengths 60 seconds x 50 9 S x 5 32 120 seconds x 59 6 S x 5 30 5 322 5 302 d 59 6 50 9 tv 99 10 10 d 8 7 tv 99 2 37 99 99 2 S S n 4 n 2 37 x y v 17 9 17 2 2 2 0 87 0 890 S x2 S y nx n y nx 1 n y 1 2 x 2 y d 8 7 t17 99 2 37 99 d 8 7 6 9 N 99 We can be pretty sure that there is a difference here Prediction Intervals concerned with values that may be sampled from a population in the future Large sample x x zS x P Small sample x x tv P S x 1 1 n P Example A sample of 10 concrete blocks manufactured by a certain process had a mean compressive strength of 1312 MPa and a standard deviation of 25 MPa Find a 95 prediction interval for the next block to be measured 95 x 1312 2 262 25 1 10 95 x 1312 59 95 x 1312 t9 95 25 1 10
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