Swinburne University of Technology TUTORIAL 11: Normal distribution, confidence interval for the mean · 1. The random variable X is normally distributed according to X ~ N(110, 32). (a) Find b such that P(X ? b) = 0.8 (b) Find a such that P(X < a) = 0.2 (c) Determine P(X € [109, 111]). (d) Determine all quartiles of the X distribution. . 2. A soft-drink machine is regulated so that the amount of drink dispensed is approximately normally distributed with standard deviation equal to 0.15 deciliters. (a) Find a 90% confidence interval for the mean of all drinks dispensed by this machine if a random sample of 36 drinks has an average content of 2.25 deciliters. (b) How many drinks should be randomly sampled if we want the 90% confidence interval for the mean to be within 0.09 deciliters? · 3. Watching paint dry. The following measurements were recorded for the drying time, in hours, of a certain brand of paint: 3.4; 2.5; 4.8; 2.9; 3.6; 2.8; 3.3; 5.6; 3.7; 2.8; 4.4; 4.0; 5.2; 3.0; 4.8 Given that paint drying times are normally distributed with some unknown standard deviation, find a 98% confidence interval for the average time taken for the paint to dry. Hint: use your calculator to obtain sample statistics for the given data. . 4. (*) In case when X does NOT follow normal distribution, o is unknown and sample size n > 30, we use t-score t = X-A Vm (i.e. t-Table) to determine the confidence interval for the mean u = E(X), based on the sample statistics X = 1/n == 1 Xi and s2 = 1/(n - 1) }}=1(Xi - X)2. However, z-score z = n X-H (i.e. z-Table) can also be used as an approximation. Confirm this statement by inspecting the values of z* from the t-Table. Note that z* gives the value of the quantile of the normal distribution for the given confidence level. Compare z* for each value of the confidence level with the corresponding t-quantiles for n > 30.
Normal Distribution P Distribution function x The table gives probability P = @(x) =- 1Rx e-t2/2dt. For x<0 values of @(x) can be obtained from @(-x) = 1 - (x). -00 x 0 0.01 0.02 0.03 0 0.5000 0.5040 0.5080 0.5120 0.1 0.5398 0.5438 0.5478 0.5517 0.2 0.5793 0.5832 0.5871 0.5910 0.3 0.6179 0.6217 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.5 0.6915 0.6950 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.9 0.8159 0.8186 0.8212 0.8238 1 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0 1.3 0.9032 0.9049 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 1.5 0.9332 0.9345 0.9357 0.9370 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 1.8 0.9641 0.9649 0.9656