NU Clark Senior High School Statistics & Probability PDF

Statistics & Probability COURSE MATERIAL NO. 3 Hazellene V. Bondoc, LPT Contributors Page | 1  WHAT WILL YOU LEARN? This module has been designed to help stude...

Statistics & Probability COURSE MATERIAL NO. 3 Hazellene V. Bondoc, LPT Contributors Page | 1  WHAT WILL YOU LEARN? This module has been designed to help students to; illustrates a normal random variable and its characteristics. NORMAL identifies regions under the normal curve corresponding to different DISTRIBUTION standard normal values. converts a normal random variable to a standard normal variable and vice versa. computes probabilities and percentiles using the standard normal table. This module deals with the properties of normal distribution. A four-step process in finding the area under the normal curve is also explained. Page | 2  LET’S BE ENGAGED! Data can be "distributed" (spread out) in different ways. It can be spread out Or more on the right Or it can be all more on the left jumbled up But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this: The “Bell Curve” is a normal distribution. It has a very important role in inferential statistics: It provides a graphical representation of statistical values. Describing the characteristics of populations as well as in making decisions. LET’S LEARN MORE! The normal probability distribution has six properties that you need to know: 1. The distribution curve is bell-shaped. NUC SHS Module for Statistics 11: The Normal Distribution | 3 2. The curve is symmetrical about its center. 3. The mean, median, and mode coincide at the center. They are equal. 4. The width of the curve is determined by the standard deviation of the distribution. 5. The tails of the curve flatten out indefinitely along the horizontal axis, always approaching the axis but never touching it. That is, the curve is ASYMPTOTIC to the baseline. 6. The area under the curve is approximately 1 or 100%. Understanding the Standard Normal Curve The Table of Areas under the Normal Curve is also known as the z-Table. (see page 13 of this module for your copy) The z- score is a measure of relative standing. It is calculated by the formula: The final result, the z-score, represents the distance between a given measurement X and the mean, expressed in standard deviations. Four-Step Process In Finding The Areas Under The Normal Curve Given A Z-Value 1. Express the given z-value into a three-digit form. 2. Using the z-table, find the first two digits on the left column. 3. Match the third digit with the appropriate column on the right. 4. Read the area(or probability) at the intersection of the row and the column. NUC SHS Module for Statistics 11: The Normal Distribution | 4 Example: Find the area that corresponds to z = 1. It means that the probability in that area is 34.13%. Before we move on to the application of z-scores, practice finding the corresponding area on your z-table: NUC SHS Module for Statistics 11: The Normal Distribution | 5  LET’S TRY THIS! DIRECTIONS: Find the corresponding area between 𝑧 = 0 and each of the following: z-score z = 0.96 z = 1.74 z = 2.18 z = 2.69 z = 3.00 area *The answer key is located on page 10 Now that you know how to get the area using your z-table, it’s time to solve and interpret the z- score! Before that, it’s important that you know how to differentiate z-score and raw score… Understanding the Z-Score z-score Raw Score It indicates how many standard It may be composed of large deviations an element is from the values, but large values cannot be mean. accommodated at the base line of z-score is stated to be a measure of the normal curve. relative standing. Raw scores are simply the number These scores represent distances questions or problems the student from the center measured in answered or solved correctly. standard deviation units. Without knowing how many There are SIX z-scores at the base questions were on the test or the line of the normal curve: three z point value of each question, raw scores to the left of the mean and scores are impossible to decipher three z-scores to the right of the in terms of percentile, grade, or mean. measured progress. Example: Given the mean, μ = 50 and the standard deviation, σ = 4 of a population of Reading scores. Find the z-value that corresponds to a score X = 58. NUC SHS Module for Statistics 11: The Normal Distribution | 6 This conversion from raw score to z-score is shown graphically, 38 42 46 50 54 58 62 From the diagram, we see that a score X = 58 corresponds to z = 2. It is above the mean. So we can say that, with respect to the mean, the score of 58 is above average. LET’S TRY THIS! DIRECTIONS: Find the z-value that corresponds to the score X: X μ σ z 50 5 40 40 8 52 36 6 28 74 10 60 82 15 75 *The answer key is located on page 10 Let’s try the following word problems. Example: On a final examination in Biology, the mean was 75 and the standard deviation was 12. Determine the standard score of a student who received a score of 60 assuming that the scores are normally distributed. Solution: 𝒙−𝒙̅ 𝒛= 𝒔 𝟔𝟎 − 𝟕𝟓 𝒛= 𝟏𝟐 𝒛 = −𝟏. 𝟐𝟓 This means that 60 is 1.25 standard deviation below the mean. NUC SHS Module for Statistics 11: The Normal Distribution | 7 Example: On the first quarterly exam in Statistics, the population mean was 70 and the population standard deviation was 9. Determine the standard score of a student who got a score of 88 assuming that the scores are normally distributed. Solution: 𝒙−𝝁 𝒛= 𝝈 𝟖𝟖 − 𝟕𝟎 𝒛= 𝟗 𝒛=𝟐 This means that 88 is 2 standard deviation above the mean. LET’S TRY THIS! Liza scored 90 in an English test and 70 in a Physics test. Scores in the English test have a mean of 80 and a standard deviation of 10. Scores in the Physics test have a mean of 60 and a standard deviation of 8. In which subject was her standing better assuming that the scores in her English and Physics class are normally distributed? NUC SHS Module for Statistics 11: The Normal Distribution | 8  Determining Areas Under Normal Curve The area under the curve is 1. So, we can make correspondence between area and probability. The following steps may help you determine the area under the curve, please note that you can have your own strategy too! NUC SHS Module for Statistics 11: The Normal Distribution | 9 Examples: a. Find the area to the left of z = –1.5. b. Find the proportion of the area above z = –1. DIRECTIONS: Determine each of the following areas and show these graphically. Use probability notation in your final answer. NUC SHS Module for Statistics 11: The Normal Distribution | 10  LET’S TRY THIS! 1. 𝑎𝑏𝑜𝑣𝑒 𝑧 = 1.46 2. 𝑏𝑒𝑙𝑜𝑤 𝑧 =– 0.58 3. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = –.78 𝑎𝑛𝑑 𝑧 = – 1.95 4. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = 0.76 𝑎𝑛𝑑 𝑧 = 2.88 5. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = –0.92 𝑎𝑛𝑑 𝑧 = 1.75 *The answer key is located on pages 10-11 NUC SHS Module for Statistics 11: The Normal Distribution | 11 *ANSWER KEY: DIRECTIONS: Find the corresponding area between 𝑧 = 0 and each of the following: z-score z = 0.96 z = 1.74 z = 2.18 z = 2.69 z = 3.00 area 0.3315 0.4591 0.4854 0.4964 0.4987 DIRECTIONS: Find the z-value that corresponds to the score X: X μ σ z 50 5 40 z = 1.13 40 8 52 z = 0.62 36 6 28 z = 1.07 74 10 60 z = 1.07 82 15 75 z = 0.89 Liza scored 90 in an English test and 70 in a Physics test. Scores in the English test have a mean of 80 and a standard deviation of 10. Scores in the Physics test have a mean of 60 and a standard deviation of 8. In which subject was her standing better assuming that the scores in her English and Physics class are normally distributed? For English: For Physics: 𝒙−𝒙̅ 𝒙−𝒙̅ 𝒛= 𝒛= 𝒔 𝒔 𝟗𝟎 − 𝟖𝟎 𝟕𝟎 − 𝟔𝟎 𝒛= 𝒛= 𝟏𝟎 𝟖 𝒛=𝟏 𝒛 = 𝟏. 𝟐𝟓 Her standing in Physics was better than her standing in English. Her score in English was one standard deviation above the mean of the scores in English whereas in Physics, her score was 1.25 standard deviation above the mean of the scores in Physics. NUC SHS Module for Statistics 11: The Normal Distribution | 12 DIRECTIONS: Determine each of the following areas and show these graphically. Use probability notation in your final answer. 1. 𝑎𝑏𝑜𝑣𝑒 𝑧 = 1.46 0.5-0.4279 =0.0721 2. 𝑏𝑒𝑙𝑜𝑤 𝑧 =– 0.58 0.5-0.2190 =0.2810 3. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = –0.78 𝑎𝑛𝑑 𝑧 = – 1.95 0.2823+0.4744 =0.7567 4. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = 0.76 𝑎𝑛𝑑 𝑧 = 2.88 NUC SHS Module for Statistics 11: The Normal Distribution | 13  0.4980-0.2764 =0.2216 5. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = –0.92 𝑎𝑛𝑑 𝑧 = 1.75 0.3212+0.4599 =0.7811 VII. References: Mercado, J. P. (2016). Next Century Mathematics Statistics & Probability. Quezon City: Phoenix Publishing House. Pierce, R. (2020). Math is Fun. Retrieved from www.mathisfun.com: https://www.mathsisfun.com/data/standard-normal-distribution.html Sal. (2018). Khan Academy. Retrieved from Khan Academy: https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of- data/more-on-normal-distributions/v/introduction-to-the-normal-distribution Vaidya, D. (2020). Wall Street Mojo. Retrieved from https://www.wallstreetmojo.com/z-score- formula/ NUC SHS Module for Statistics 11: The Normal Distribution | 14  Standard Normal Distribution Table z.00.01.02.03.04.05.06.07.08.09 0.0.0000.0040.0080.0120.0160.0199.0239.0279.0319.0359 0.1.0398.0438.0478.0517.0557.0596.0636.0675.0714.0753 0.2.0793.0832.0871.0910.0948.0987.1026.1064.1103.1141 0.3.1179.1217.1255.1293.1331.1368.1406.1443.1480.1517 0.4.1554.1591.1628.1664.1700.1736.1772.1808.1844.1879 0.5.1915.1950.1985.2019.2054.2088.2123.2157.2190.2224 0.6.2257.2291.2324.2357.2389.2422.2454.2486.2517.2549 0.7.2580.2611.2642.2673.2704.2734.2764.2794.2823.2852 0.8.2881.2910.2939.2967.2995.3023.3051.3078.3106.3133 0.9.3159.3186.3212.3238.3264.3289.3315.3340.3365.3389 1.0.3413.3438.3461.3485.3508.3531.3554.3577.3599.3621 1.1.3643.3665.3686.3708.3729.3749.3770.3790.3810.3830 1.2.3849.3869.3888.3907.3925.3944.3962.3980.3997.4015 1.3.4032.4049.4066.4082.4099.4115.4131.4147.4162.4177 1.4.4192.4207.4222.4236.4251.4265.4279.4292.4306.4319 1.5.4332.4345.4357.4370.4382.4394.4406.4418.4429.4441 1.6.4452.4463.4474.4484.4495.4505.4515.4525.4535.4545 1.7.4554.4564.4573.4582.4591.4599.4608.4616.4625.4633 1.8.4641.4649.4656.4664.4671.4678.4686.4693.4699.4706 1.9.4713.4719.4726.4732.4738.4744.4750.4756.4761.4767 2.0.4772.4778.4783.4788.4793.4798.4803.4808.4812.4817 2.1.4821.4826.4830.4834.4838.4842.4846.4850.4854.4857 2.2.4861.4864.4868.4871.4875.4878.4881.4884.4887.4890 2.3.4893.4896.4898.4901.4904.4906.4909.4911.4913.4916 2.4.4918.4920.4922.4925.4927.4929.4931.4932.4934.4936 2.5.4938.4940.4941.4943.4945.4946.4948.4949.4951.4952 2.6.4953.4955.4956.4957.4959.4960.4961.4962.4963.4964 2.7.4965.4966.4967.4968.4969.4970.4971.4972.4973.4974 2.8.4974.4975.4976.4977.4977.4978.4979.4979.4980.4981 2.9.4981.4982.4982.4983.4984.4984.4985.4985.4986.4986 3.0.4987.4987.4987.4988.4988.4989.4989.4989.4990.4990 3.1.4990.4991.4991.4991.4992.4992.4992.4992.4993.4993 3.2.4993.4993.4994.4994.4994.4994.4994.4995.4995.4995 3.3.4995.4995.4995.4996.4996.4996.4996.4996.4996.4997 3.4.4997.4997.4997.4997.4997.4997.4997.4997.4997.4998 3.5.4998.4998.4998.4998.4998.4998.4998.4998.4998.4998 Gilles Cazelais. 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