Question

A normal population has a mean of 18 and a standard deviation of 5. a. Compute the z value associated with 24. (Round your answer to 2 decimal places.) 2 1.20 b. What proportion of the population is between 18 and 24? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.) Proportion c. What proportion of the population is less than 13? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.) Proportion

Answer 1

Facts : 1) If X, Z be the normal and standard normal distibuted random variables respectively. To find the probabilites for X we convertX to Z by using Z = 4 Where u = mean(X) o = standard deviation(X) 2) P(- <Z < 0) = 0.5 and P(0 < Z < 3) P(-asz< 0) = P(O<Zsa) ) = 0.5 Let X is a normal random variable. Given that X follows normal distribution with mean(u) = 18 and standard deviation(o) = 5.

a) Now we have to find the Z-value associated with 24. Z = *-* = 24–18 = 1.2 : Z = 1.2 b) Now we have to find what proportion of the population is between 18 and 24. Z1 = *172 = 18518 = 0 Z2 = x2=* = 24–18 = 1.2 P(18 < X < 24) = P(Z <Z<Z2) = P(0 < Z < 1.2) = 0.3849 :: P(18 < X<24) = 0.3849

c) Now we have to find the what proportion of the population is less than 13. X-u 13 - 18 Z = - =- 5 =-1 P(X < 13) = P(Z <-1) = P(-0 <Z<0) - P(-1<Z<0) =P(-60 <Z<0) - P(0 << < 1) = 0.5 -0.3413 = 0.1587 :: P(X < 13) = 0.1587

Areas Under the One-Tailed Standard Normal Curve 0.1 0.8 This table provides the area between the mean and some Z score. O=1 For example, when Z score = 1.45 the area = 0.4265. 0.4265 u=0 1.45 Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 | 0.0120 | 0.0160 | 0.0199 | 0.0239 | 0.0279 0.0319 | 0.0359 0.0398 0.0438 0.0478 0.0517 | 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.0793 0.0832 0.0871 0.0910 | 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 | 0.1950 | 0.1985 | 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 | 0.2224 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 | 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 | 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 | 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 0.3643 0.3665 0.3686 0.3708 | 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 0.48210.4826 0.48300.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.49090.4911 0.4913 0.4916 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 | 0.4981 0.4982 0.49820.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 0.4987 0.4987 0.49870.4988 0.4988 0.4989 0.4989 0.4989 0.4990 | 0.4990 0.4990 0.4991 0.49910.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995 3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997 3.4 0.4997 0.4997 0.4997 0.4997 0.4997 | 0.4997 0.49970.4997 0.4997 0.4998 3.5 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 3.6 0.4998 0.4998 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.7 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.8 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.9 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 1.9 3.1