a Let Us Try! '\\ Hello! I hope you are having a good day. Before proceeding to Lesson 1, try to read and answer the activity below rst. Let us try rst your understanding of the concept. Get 1/2 crosswise sheet of paper and try to answer the following questions. 1. The total area under the normal curve is a. 1 b. 1 c. 0 d. 0.5 . The normal curve is bellshaped. a. True b. False c. Sometimes d. it depends. . Which part of the normal curve is extended indenitely in both directions along the horizontal axis, approaching but never touching it? a. Center b. Tail 0. Top d. Spread . According to the property of a Normal Probability Distribution, the mean is equal to a. Median and variance c. Median and Mode b. Mode and Standard deviation d. variance and standard deviation . Which of the following rules state that almost all data fall within the 1, 2, and 3 Standard Deviation of the Mean when the population is normally distributed? a. Empirical Rule c. Pascal's Triangle Rule b. Lottery Rule d. Sampling rule 0 Let Us Study What Is Normal? Scientic researchers have decided soknown as normal interval for someone's blood pressure, cholesterol, triglycerides (fatty oils), and so forth. For example, a person's normal interval of systolic blood pressure is l 10 to 140 and for a person's triglycerides is from 30 to 200 milligrams per deciliter (mg/ d1). By way of measuring these variables, a health practitioner can determine if a patient's vital statistics are within the normal interval or if some sort of treatment is needed to address a health condition and keep away from future ailments. What is Normal Random Variable? 0 Random variables can be either discrete or continuous. o a discrete random variable cannot assume all values between any two given values and they are countable. . Continuous random variable can assume all values between any two given values and they are measurable. Examples of continuous variables: 0 blood pressure 0 level of cholesterol 0 level of triglycerides o birth weights 0 heights of adult men 0 body temperatures, and many more. Many continuous variables, such as the examples just mentioned, have distributions that are bellshaped, and these are called approximately normally distributed variables. Thus, a continuous random variable X whose distribution has the shape of a normal curve is called a normal random variable. For example, a city health ofcer selected a random sample of 100 students from Pandapan Integrated School, measured their weights (in kg) and constructed a histogram similar to the one shown below. Fig. 1. Histogram for the Distribution of Weights from a Random Saranle of 100 Students Now, if the city health ofcer increases the sample size and decreases the width of the classes, the histograms will look like the ones shown in the succeeding gures. Fig. 2. Sample size increased and Fig. 3. Sample size increased and class Width decreased class Width decreased further If it were possible to measure exactly the weights of all high school students in Tagum City Division, the histogram would approach what is called a normal distribution as shown in gure 4. Fig. 4. Normal Distribution of the Weights of All High School Students in Tagum City Division Normal Distributions The normal distribution is a continuous, symmetric, bellshaped distribution of a random variable. Normal Distribution is also known as a bell curve or a Gaussian distribution, named for the German mathema tician Carl Friedrich Gauss (17771855), who derived the formula / equation for normal distribution. e-(X-u)2/(202) y 2 (N21: where, e m 2.718 ( z means "is approximatelyequal to") 1r z 3.14 [r = population mean 0' 2 population standard deviation Shapes of Normal Distributions Each normally distributed variable has its own normal distribution curve, which depends on the values two parameters; the mean and standard deviation. The larger the standard deviation, the more dispersed, or spread out, the distribution is. The gures below illustrate different normal distributions described by its means and standard deviations. Curve1-\\ Curve 2 f\" Curve 1 \\ Er1 > 0'2 #1 = #2 {a} Same means but different standard deviations #1 #2 "1 "2 (b) Different means but same standard deviations (a) Different means and different standard deviations Characteristics of a Normal Distribution 1. 2. 3. 4. (II A normal distribution curve is bellshaped. The mean, median, and mode are equal and are located at the center of the distribution. A normal distribution curve is unimodal (i.e., it has only one mode). The curve is symmetric about the mean (its shape is the same on both sides of a vertical line passing through the center). . The curve is continuous; that is, there are no gaps or holes. . The curve never touches the X axis (asymptotic to Xaxis no matter how far in either direction the curve extends, it never meets the x axisbut it gets closer and closer to xaxis). . The total area under a normal distribution curve is equal to 1.00, or 1 00%. The Empirical (Normal) Rule When a distribution is bellshaped (or what is called normal), the areas under the normal curve is determined by the empirical rule. 0 Approximately 68% of the data values will fall within 1 standard deviation of the mean. 0 Approximately 95% of the data values will fall Within 2 standard deviations of the mean. 0 Approximately 99.7% of the data values will fall Within 3 standard deviations of the mean. 2.28% p.-30' [1,20' ula' p, \"+10 p.+20' p,+30' | | ' l ; About 68% / % About 95% / % About 99.7% / Illustrative Example: We can use Empirical rule to complete the table to identify equivalent points based on the given average or mean and standard deviation in the area of shaded region. Mean Standard Area of the shaded re ion Deviation 68% 95% 99.70 1 50 3 47 to 53 44 to 56 41 to 59 41 44 47 50 59 56 59 I ; About68% J I About95% / * About 99.7% / For the 68% of the area of the shaded region: (Ll-0].) and (LH'nl) = (503) and (50+3) = 47 and 53 Therefore, 68% of the area of the shaded region is 47 to 53. For the 95% of the area of the shaded region: (11-02) and (11412) = (SO3(2)) and (50+3(2)) = 44 and 56 Therefore, 95% of the area of the shaded region is 44 to 56. For the 99.7% of the area of the shaded region: (11-03) and (Ll+03) = (SO3(3)) and (50+3(3)) = 41 and 59 Therefore, 99.7% of the area of the shaded region is 41 to 59. The same procedures are used for numbers 2 to 6. 2 30 15 15 to 45 0-60 15 to 75 3 43 5.5 37.548.5 3254 26.5 to 59.5 4 55 25 30 to 80 5105 20 to 130 5 62 20 42 to 82 22102 2 to 122 6 7O 19 51 to 89 32108 13 to 127 ZS Let Us Practice Activity 1. Use Empirical rule to complete the following table. Write on the respective column the range or interval of the scores based on the given parameters. You may construct a normal curve to help you answer the activity. The first two items item are done for you. Standard Deviation Area of the shaded region 44 47 so 53 56 I L About 68% / I % About 95% / % About 99.7% / 41 to 209 a Let Us Assess Direction: Read and analyze each item carefully. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following statements is the characteristic of a normal probability distribution? a. The mean, mode, and median are equal. b. The mean, mode, and median are not equal. C. The mean and mode are equal while the median is greater than zero. d. The mean and mode are equal While the median is less than zero. . Under the curve, the area to the right of the mean is a. 30% c. 50% b. 47.72% d. 68% . Suppose that distribution of data about the number of deaths of COVID 19 positive has a mean of 45 and a standard deviation of 18. How many standard deviations away from the mean is a value of 81? a. It is one standard deviation above the mean. b. It is two standard deviations above the mean. c. It is one standard deviation below the mean. (1. It is two standard deviations below the mean. . If the mean and the standard deviation of a continuous random variable that is normally distributed are 10 and 3, respectively, nd an interval that contains 99.7% of the distribution. a. [7, 16] b. [4, 16] c. [7,16] d. [1, 19] . Based on the empirical rule, the bellshaped distribution will have approximately 68% of the data Within What number of standard deviations of the mean? a. 0 c. 2 b. 1 d. 3 Let Us Try! Hello! I hope you are having a good day. Before proceeding to Lesson 2, try to read and answer the activity below rst. Let us try rst your understanding of the concept. Get 1/2 crosswise sheet of paper and try to answer the following questions. 1. What is the sum of the area that corresponds to the right of O and to the left of O? a. 0.5 c. 1.5 b. 1 d. 2 2. The Standard Normal Distribution Table is also known as a. Table of Contents b. The ZTable c. Table of Areas of Geometric Figure (1. Periodic Table of Elements 3. How will you describe this graph? 3 2 1 O 1 2 3 a. The shaded region of the normal curve is below the z = 1. b. The shaded region of the normal curve is above the z = 1. c. The shaded region of the normal curve is between 2:0 and z = 1. d. The shaded region of the normal curve is between z= -1 and z=1. 4. What will you do to nd the area of the region to the right of 2 value? a. Calculate the mean and standard deviation b. Rewrite the value obtained from 2 table c. Subtract the two obtained values from the 2 table. Let Us Study [3 What proportion of individuals are geniuses? What percentage of a particular brand of light emits 300 and 400 lumens? Is a systolic blood pressure of 110 unusual? We want to be able to answer some questions about variables that are normally distributed. d. Subtract the obtained z value from1. 5. Which of the following illustrates the region to the left of z = 1? a. C. - 3 -2 -1 0 2 3 - 3 -1 0 1 2 3 b. d. -2 -1 0 1 - 3 2 -2 -1 0 1 2 6. Find the area of the shaded region of the given figure. a. 0.8413 c. 0.3907 b. 0.3413 d. 0. 1587 - 3 -2 - 0 1 2 3 7. Find the area of the region to the right of z = -1.43. a. 0.8413 c. 0.9236 b. 0.0764 d. 0.2061 8. Find the area of the region to the left of z = 2.11. a. 0.9826 C. 0.4634 b. 0.0174 d. 0.9049 9. What is the area between z = -1.23 and z = 2? a. 0.0865 C. 0.8679 b. 0.4772 d. 0.8779 10. The area under the normal curve between z = 0 and z = 1 is the area under the normal curve between z = 1 and z = 2. a. less than b. greater than c. equal to d. a, b or c, depending on the value of the meanData can be distributed in many different ways. The most common distribution that applies to many reallife data is the normal distribution. Most measurable physical quantities like heights and weights of people, temperature, exam scores, and blood pressure all follow the normal distribution. In the normal distribution, all data tend to approach the mean as the amount of data increases. It is a continuous probability distribution of a normal random variable. How does the graph of a normal distribution look like? A normal distribution / normal curve has the following properties: 1. The graph of a normal distribution is bellshaped. \\10 (II-P03 00 > It depends on two factors: the mean and standard deviation. > The mean determines the center of the graph and the standard deviation determines its height and width. > Normal distributions with higher standard deviation create curves with smaller height and bigger width. A normal curve looks like this: [4-30 [.1-20 |.l-O p [4+0 p+20 \"+30 . The mean of a normal distribution lies in the center of the bell curve along the horizontal axis. . The total area under the normal curve is equal to one (1). . Its curve is symmetric about yaxis, the mean. . The mean, median, and mode coincide at the center of the distribution. . It is asymptotic with respect to the X-axis. . It has a maximum point at the mean. . About 68% of the area under the normal curve falls within one standard deviation from the mean. . About 95% of the area under the normal curve falls within two (2) standard deviations from the mean. 10. About 99.7% of the area under the normal curve falls within three (3) standard deviations from the mean. Standard Normal Distribution Since the appearance of the normal curve depends on the distribution's mean and standard deviation, there must be innitely many different normal curves. In order to make use of its properties, statisticians came up with a way of transforming every normal curve to what we call standard normal distribution/curve or z-distribution. A standard normal distribution is a normal distribution with mean of 0 and standard deviation of 1. Basically, any normal distribution could be transformed into this type. The gure below illustrates a standard normal distribution. The values at the horizontal axis are the values of the random variable Z, the transformed values of the random variable X. The values of Z can be computed using the formula, x 0' Z: where: u = mean a = standard deviation 3 2 'I [1 +11 +2 +3 zscore Standard Normal Distribution Standard Scores or Z-scores By using the normal distribution, the number of standard deviations of a value away from the mean can be found. However, by converting the values to standard scores (2 scores), it will be easier to make conclusions from the data through the standard Normal Distribution table (ztable). The most important part of the normal curve is the area under it bounded by some values. Basically, some knowledge of Integral Calculus is required to solve for the areas bounded by some values, lines, or curves. However, a table called Ztable which gives the area under the standard normal curve for any zvalue was developed to make the calculations easier. The z-table Table E The Standard Normal Distribution Cumulative Standard Normal Distribution Z 00 .01 02 .03 .04 05 06 07 08 09 3.4 .0003 0003 .0003 0003 0003 .0003 0003 0003 .0003 0002 -3.3 0005 0005 .0005 004 0004 .0004 0004 0004 .0004 0003 -3.2 0007 0007 0006 006 0006 0006 0006 0005 .0005 0005 -3.1 0010 0009 1009 0009 1008 0008 0008 0008 0007 0007 -3.0 0013 .0013 .0013 0012 1012 .001 0011 .0011 0010 0010 -2.9 0019 0018 0018 0017 0016 0016 0015 0015 0014 0014 -2.8 0026 0025 0024 0023 1023 1022 0021 .0021 0020 0019 -2.7 0035 0034 0033 032 0031 .0030 0029 0028 0027 026 -2.6 0047 0045 .0044 0043 .0041 .0040 0039 0038 0037 1036 -2.5 0062 .0060 1059 0057 1055 3054 0052 0051 0049 0048 -2.4 0082 .0080 .0078 0075 .0073 0071 0069 .0068 0066 0064 -2.3 0107 0104 .0102 0099 1096 .0094 0091 0089 0087 0084 -2.2 0139 0136 0132 0129 0125 0122 0119 0116 0113 0110 -2.1 0179 0174 .0170 0166 162 .0158 0154 .0150 0146 0143 -2.0 0228 0222 0217 0212 .0207 .0202 0197 192 0188 0183 - 1.9 0287 .0281 .0274 0268 .0262 0256 0250 .0244 0239 0233 -1.8 0359 0351 0344 0336 0329 0322 0314 0307 .0301 0294 -1.7 0446 0436 .0427 0418 0409 .0401 1392 0384 0375 0367 -1.6 0548 .0526 0516 0505 .0495 1485 0475 .0465 .0455 -1.5 0668 0655 .0643 0630 0618 .0606 0594 .0582 0571 0559 -1.4 0808 793 0778 0764 )749 .0735 172 0708 .0694 0681 -1.3 968 .0951 0934 0918 0901 .0885 0869 0853 838 0823 -1.2 1151 1131 1112 1093 1075 1056 1038 1020 003 0985 -1.1 1357 1335 1314 1292 1271 1251 1230 1210 1190 1170 -1.0 1587 .1562 1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379 -0.9 1841 1814 1788 1762 .1736 1711 1685 1660 1635 1611 -0.8 2119 2090 2061 2033 2005 1977 1949 1922 1894 1867 -0.7 2420 2389 .2358 2327 2296 2266 236 2206 .2177 .2148 -0.6 .2743 709 2676 2643 .2611 2578 .2546 .2514 .2483 2451 -0.5 3085 .3050 3015 .2981 .2946 2912 2877 .2843 .2810 .2776 -0.4 3446 $409 .3372 3336 3300 3264 3228 3192 3156 3121 -0.3 3821 3783 3745 3707 3669 3632 3594 3557 3520 3483 -0.2 4207 4168 4129 4090 1052 4013 $974 3936 .3897 .3859 -0.1 602 4562 4522 483 4443 4404 4364 4325 4286 4247 -0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641 For z values less than -3.49, use 0.0001. AreaTable E (continued) Cumulative Standard Normal Distribution Z 00 01 02 .03 .04 .05 .06 .07 08 .09 0.0 .5000 .5040 .5080 .5120 5160 .5199 5239 .5279 .5319 5359 0.1 .5398 5438 5478 5517 5557 5596 5636 5675 .5714 5753 0.2 5793 5832 .5871 .5910 5948 .5987 6026 5064 6103 .6141 0.3 6179 6217 6255 6293 6331 6368 6406 .6443 6480 .6517 0.4 6554 6591 .6628 6664 .6700 6736 6772 .6808 6844 6879 0.5 5915 6950 6985 019 7054 7088 7123 7157 190 7224 0.6 .7257 7291 . 7324 735 7389 .7422 7454 .7486 .7517 7549 0.7 7580 7611 7642 673 7704 7734 7764 7794 7823 7852 0.8 .7881 7910 .7939 796 7995 8023 8051 8078 8106 8133 0.9 3159 8186 3212 238 8264 .8289 8315 3340 .8365 .8389 1.0 .8413 8438 .8461 .8485 8508 .8531 8554 8577 8599 8621 1.1 .8643 8665 8686 8708 8729 8749 8770 8790 8810 8830 1.2 8849 8869 8888 8907 .8925 8944 8962 8980 .8997 9015 1.3 9032 .9049 .9066 082 .9099 .9115 .9131 .9147 9162 9177 1. 4 .9192 .9207 9222 .9236 9251 .9265 .9279 .9292 .9306 .9319 1.5 9332 9345 9357 9370 9382 .9394 9406 9418 .9429 9441 1.6 .9452 9463 9474 484 .9495 9505 9515 525 535 .9545 1.7 .9554 .9564 .9573 9582 9591 .9599 8096 9616 .9625 .9633 1.8 .9641 9649 .9656 9664 9671 .9678 9686 9693 .9699 9706 1.9 9713 719 9726 732 9738 9744 9750 756 .9761 9767 2.0 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817 2.1 .9821 826 9830 9834 2838 9842 9846 9850 .9854 9857 2.2 .9861 .9864 .9868 9871 .9875 .9878 9884 9887 .9890 2.3 9893 9896 .9898 .9901 .9904 .9906 9909 9911 .9913 .9916 2.4 .9918 9920 9922 9925 2927 9929 931 9932 .9934 9936 2.5 9938 9940 9941 9943 .9945 9946 9948 .9949 1951 9952 2.6 9953 955 9956 95 9959 9960 96 9962 .9963 9964 2.7 .9965 9966 9967 9968 9969 9970 9971 9972 1973 1974 2.8 9974 975 9976 977 9977 9978 9979 9979 9980 9981 2.9 9981 1982 9982 9983 9984 9984 9985 9985 1986 9986 3.0 .9987 .9987 .9987 98 9988 .9989 9989 .9989 2990 9990 3.1 9990 9991 .9991 9991 1992 9992 .9992 9992 .9993 9993 3.2 9993 9993 .9994 9994 .9994 .9994 .9994 9995 .9995 9995 3.3 9995 9995 9995 9996 .9996 9996 9996 9996 9996 9997 3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998 For z values greater than 3.49, use 0.9999. AreaFinding Areas Under the Standard Normal Distribution Curve To nd the area under the Standard Normal Curve, a twostep process is recommended With the use of the Procedural Table. The two steps are: Step 1. Draw the normal distribution curve and shade the area. Step 2. Find the appropriate gure in the Procedure Table and follow the directions given. Procedure Table: in Finding the Area Under the Standard Normal Distribution 1. To the left of any 2 value 0 Look up the 2 value in the table and use the given area. At: or A 0 +2 2 0 2. To the right of any 2 value Look up the 2 value and subtract the area from 1. 0r 3. Between any two z values 0 Look up both 2 values in the table and subtract the corresponding 0 2122 41220 Example: Given z = 1.39, find the area of the regions under the standard normal curve to the left of z = 1.39, to the right of z = 1.39 and between z = 0 and z = 1.39 Solution: Using the z-Table, the area to the left of a z value of 1.39 is found by looking up 1.3 in a. to the left of z =1.39 the left column and 0.09 in the top row. Where the two lines meet gives an area of 0.9177 0.9177 0.00 0.09 or 0.0 91.77% . 1.3 0.9177 0 1.39 Therefore, the area of the region to the left of z = 1.39 is 0.9177 or 91.77% b. to the right of z =1.39 Area to the right of z = 1.39 0.0823 = 1 - 0.9177 or 8.23% = 0.0823 Therefore, the area of the region to the right of z = 1.39 is 0.0823 or 8.23% 1.39 c. between z = 0 and z = 1.39 Solution. Subtract area of the curve 0.4177 or 41.77% under z=1.39 and z=0 Area = 0.9177 - 0.5000 0.5000 Area = 0.4177 or 41.77% 1.39 Therefore, the area of the region between z=0 and z=1.39 is 0.4177 or 41.77%a: Let Us Remember Awesome! Now, let us gather what we have learned. Copy and ll in the blanks using the words in the box to demonstrate understanding of the key concepts of the Standard Normal Distribution and the areas under the standard normal curve. symmetric ztable one (1) zero (0) zscores between right equal left The standard normal distribution is a distribution of standardized values of a normal random variable called the . It is the distribution that occurs when a normal random variable has a mean of and a standard deviation of The total area of the entire region under the standard normal curve is . The standard normal curve is . Thus, the area of the region to the left of z value is to the area of the region to the right of z value. Given the z value, the area of the region under the normal curve is found by looking at the If the region is to the of the zscore, then the ztable value is the area. If the region is to the of the zscore, then subtract the ztable value from 1 to get the area. If the region is two zscores, subtract the z-table value of the leftmost zscore from the ztable value of the rightmost zscore to get the area. a Let Us Assess Direction: Read and analyze each item carefully. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following is not a property of the standard normal distribution? a. It is symmetric. b. It has a mean of 1. c. It has a standard deviation of 1. d. 95% of the data falls within 2 standard deviations 2. What does it mean to have a negative zscore? a. The score is above the mean. b. The score is below the mean. c. The score is more than one standard deviation below the mean. (1. The score is more than one standard deviation above the mean. 3. To nd the area of a specic region under the standard normal curve, the is used. a. Table of Areas of Geometric Figure b. The ZTable c. Table of Contents (1. Periodic Table of Elements 4. Which of the following figures is the best illustration of the area to the right of z =1? a. C. - 3 -2 -1 0 1 3 - 3 -2 -1 0 1 2 3 b. d. - 3 -2 -1 - 3 - 2 0 1 2 5. What will you do to find the area between the two z values? a. Subtract the obtained value from the Z table to 1 b. Rewrite the value obtained from z table c. Subtract the two obtained values from the z table. d. Calculate the mean and standard deviation 6. How will you describe this normal curve? - 3 -1 0 3 a. The shaded region of the normal curve is below the z = 1 b. The shaded region of the normal curve is above the z =-1 c. The shaded region of the normal curve is between z=0 & z=1 d. The shaded region of the normal curve is between z=-1& z=1 7. Find the area of the shaded region of the given figure. - 3 -2 -1 0 1 2 a. 0. 1587 b. 0.3413 C. 0.3907 d. 084138. For the standard normal distribution, the area to the left of z = -1.4 is a. 0.0808 b. 0. 1616 c. 0.9192 d. 0.5075 9. The area of the region under the standard normal curve to the right value of z = -1.19 is a. 0.1170 b. 0.0287 C. 0.8643 d. 0.8830 10. What is the sum of the areas under the standard normal curve to the left of z = - 3.14 and to the right of z = -2.62. a. 0.0008 b. 0.9964 C. 0.9956 d. 0.0036e Let Us Try! '\\ Hello! I hope you are having a good day. Before proceeding to Lesson 3, try to read and answer the activity below rst. 1. Which of the following is not part of the calculation for a zscore? a. X score c. population standard deviation b. population mean d. degree of normality 2. For any normal distribution, what is the zscore corresponding to the mean? a. O c. 1 b. l (1. cannot be determined 3. If an entire population with u = 60 and o = 8 is transformed into 2 scores, then the distribution of zscores will have a mean of _ and a standard deviation of _. a. (O, 1) c. (0, 8) b. (60, 1) d. (60, 8) 4. If a normally distributed population, )1 = 40 and o = 5, what is the z score corresponding to X = 34? a. O.50 b. O.85 c. 1.20 d. 1.53 5. If )1 = 75 and o = 4, what X value corresponds to z = 1.75? a. 78 b. 68 c. 67 d. 59 6. The mean height of the candidates for \"Ms. SHS 2021\" is 63.6 in. and a standard deviation is 2.5 inches. The z score corresponding to the height of Miss LNHS is 1.56. How tall is Ms. LNHS? a. 61.6 in b. 65.3 in c. 67.5 in d. 68.7 in 7. The mean temperature for the month of February is 27 degrees with a standard deviation of 1.6 degrees. What is the zscore for a temperature of 29 degrees? a. 1.16 c.1.52 b. 1.25 d. 1.06 0 Let Us Study Suppose that the Math nal exam scores are normally distributed with a mean of 80 and a standard deviation of 5. If you got 85 in the Math nal exam, what is its equivalent zscore? Converting A Normal Random Variable to A Standard Normal Variable: If each normal random variable x in a normal distribution is converted to a standard deviation unit, called standard normal variable 2, the result will be the standard normal distribution. Nomi Distribution Standard Normal Distribution z)=1P(Zb) The probability of z is in the opposite direction of two values, say a and b 5 P(Xx) 7 P(ab) The probability of X is in the opposite direction of two values, say a and b Example 1. Find the probabilities (a) P (Z -0.88) for each of the following. Solution: (a) The probability P (Z -0.88) is the area of the region under the normal curve at the right ofz = -0.88. 1.000 - W 0.8106 To get the area, subtract the area of z = 0.88 from 1. P (z > O.88) = 1 P (z O.88) = 1 0.1894 P (z > O.88) = 0. 8106 or 81. 06%. 0.1894 .2 _ z=-0.88 Example 2. Let X be a normal random variable with mean M = 15 and standard deviation 0 = 3. Find the probabilities of the following: (a) P (X Probability to the (left, right) of Z. 2. P (Z > z) = 1 P (Z Probability to the (left, right) of z. 3. P (a Probability of z that is in (between, opposite direction) two otherz values a and b. 4. P (X Probability to the (left, right) of a normal random variable x. 5. P (X > x) >Probability to the (left, right) of a normal random variable x. a Let Us Assess It is amazing how you were able to study and answer the activities! Now it is time to try the fruit of your journey by answering the assessment below. Encircle the letter of the best answer. 1. Convert 0.8547 into percent. a. 8.547% b. 85.47% c. 47.85% d. .7458% 2. What is the equivalent of 26.4% in decimal? a. 26.4 b. 264 c. 2.64 d. 0.264 3. What is the product of 180 and 0.5784? (Round off your answer to nearest whole number) a. 104 b. 102 c. 1,004 d. 10.4 4. What is the 80% of 100? a. 20 b. 8 c. 50 d. 80 5. What is the total area of a normal curve? a. 1 b. 4 c. 2 d. 3 6. Find the area of the shaded region under the normal curve and make a brief and concise interpretation of the following graph. 18 20 22 24 26 28 30 32 34