(Negative because it is below the mean. Characteristics: -It is symmetrical about the vertical line drawn through z = 0 -the curve is symptotic to the x-axis. z = 230 ÷ 150 = 1. e) less than 60? Solution: The Standard Normal Distribution. Standard and general normal distributions De nition (Standard normal distribution) A continuous random ariablev is a standard normal (written N(0;1)) if it has density f Z(x) = 1 p 2ˇ e x2=2: A synonym for normal is Gaussian. Normal Distribution - Download as a PDF or view online for free. For normalization purposes. It is tedious to integrate a new normal distribution for every single measurement, so use a standard normal distribution with tabulated areas. Additional probability concepts like independent and conditional probability, random variables, and transformations to the standardized normal distribution. P (X=a) = 0 P (a < x < b) = P (a < x < b) 15 The Normal PDF It’s a probability function, so no matter what the values of and , must integrate to 1! No matter what and are, the area between Sep 8, 2015 · This document discusses the preparation of various types of solutions. 7037. It expresses the probability of a number of events occurring in a fixed time if these events occur with a known average rate and are independent of the time since the last event. 2 The p. Jun 10, 2013 · The document discusses the normal distribution and standard deviation. Step 1: Subtract the mean from the x value. the Normal tables give the corresponding z-score as -1. Therefore, this is an example of a binomial distribution. Between μ – σ and μ + σ (in the center of the curve), the graph curves downward. Step 3: Square the differences from Step 1. 106. Normal Distribution Problems and Solutions. The random variable X = the number of successes obtained in the n independent trials. Sep 28, 2020 · Normal distribution. It is useful to share insightful information on Normal Distribution Examples Solutions This PPT slide can be easily accessed in standard screen and widescreen aspect ratios. Third Step : find the areas under ,using table E. The arithmetic mean, median, and mode of the distribution are equal and located at the peak. y Theorem f Definition: The Normal Distribution is also called the Gaussian distribution. Apart from this, you can download this well-structured PowerPoint template design in different formats Feb 10, 2013 · It is a distribution of a normal random variable with a mean equal to zero (μ = 0) and a standard deviation equal to one (μ = 1). 2 likes • 119 views. J. Part 2: Find the mean and standard deviation of the sampling distribution. Recall how we describe a distribution of data. It provides examples of calculating probabilities using these distributions. It explains that the normal distribution is a limiting case of the binomial distribution when the number of trials is large. Step 2Compute where i is the index (the position of the data value in the ordered list) and n is the number of observations. Application of Normal Dist 4. Normal Distribution 3. We would like to show you a description here but the site won’t allow us. Mar 24, 2019 · 3. 5 Properties of Normal Distributions. 4 Find probabilities for a normally distributed variable by transforming it into a standard normal variable. Mar 4, 2024 · Properties of a Normal Probability Distribution 1. Get in touch with our design agency. This bell-shaped curve is used in almost all disciplines. I. Read Feb 22, 2018 · pptx, 1004. The corresponding interval for the continuous normal We define the probability distribution function (PDF) of Y as f ( y) where: P ( a < Y < b) is the area under f ( y) over the interval from a to b. ÐÏ à¡± á> þÿ Ù Ý þÿÿÿñ ò ó ô õ Ü Û Ú Aug 17, 2018 · The normal distribution is a widely used probability distribution that is bell-shaped and symmetric around the mean. This is a PowerPoint presentation that can be used to introduce normal distributions. Many naturally-going on phenomena have a tendency to May 2, 2017 · The Normal distribution has three distinguishing features: • It is unimodal, in other words there is a single peak. 99. This means that both positive and negative ends approach Jun 17, 2024 · The normal distribution is produced by the normal density function, p ( x ) = e− (x − μ)2/2σ2 /σ Square root of√2π. deviation units. The graph below shows examples of Poisson distributions with Nov 24, 2016 · The document discusses the Poisson distribution, which describes the probability of rare events. If you wish to categorize your customers based on psychological and social elements, this bell curve template is perfect for you. Assuming the data follows a normal distribution, find:the percentage of sixth formers taller than (Note these are not tr. It is defined by two parameters mean ('average' m) and standard deviation (σ). The graph curves upward to the left of μ – σ and to the right of μ + σ. Normal Distribution. Use of DrFrostMaths for practice Register for free at: Practise questions by chapter, including past paper Edexcel questions and extension questions (e. To simplify this, statisticians use the standard normal distribution. Note: This PowerPoint is only a summary and your main source should be the book. Refer the below Gaussian distribution worked example. M = 1150. Note: For this standard deviation formula to be accurate, our sample size needs to be 10 % or less of the population so we can assume independence. )187 cm;8. The curve is symmetric about its mean. Nov 25, 2023 • Download as PPT, PDF •. PROPERTIES OF THE NORMAL DISTRIBUTION The total area of normal curve is 1. Step 4: Add up all of the squared differences from Step 3. f. 2. x = 2, μ = 3 and σ = 4. It provides examples of how to find probabilities and z-scores using the standard normal distribution table or calculator. The points at which the curve changes from curving upward to curving downward are called the inflection points. Helps to understand where a score lies in. A random variable is some outcome from a chance process, like how many heads will occur in a series of 20 flips, or how many seconds it took someone to read this sentence. com - id: 6fbd0c-OTI0Y 4 Definition The Poisson distribution is a discrete probability distribution. The Poisson distribution has only one parameter, λ (lambda), which is the mean number of events. For each example, it identifies the mean and standard deviation, converts values to z-scores using the Nov 25, 2023 · The Binomial,poisson _ Normal Distribution (1). Calculated using mean and standard deviation. One reason why it appears so often is that if one observes dif-ferent unrelated quantities then their sum, suitably normalized is close to the normal distribution. The document then explains the normal distribution as a theoretical probability distribution important in statistics. The normal distribution is described as the most important distribution in biostatistics. It has the shape of a bell and can entirely be described by its mean and standard deviation. Examples are provided to ÐÏ à¡± á> þÿ # ) þÿÿÿ May 20, 2014 · The document discusses approximating binomial probabilities with a normal distribution. It has a mean of zero and variance greater than one. 7% will be between 350 and 650. Since it is a continuous distribution, the total area under the curve is one. 53. Second Step: draw the figure and Shading the area. A normal random variable, X, may be transformed to a standard normal, Z, f(z) = 1 √ 2π e−y2/2, where µ = 0 and σ = 1 using following equation, Z = X − µ σ. Step 2: Divide the difference by the standard deviation. 1 hour ago We have an entire team of experts who can work on your custom presentation. It provides information on who discovered each distribution and examples of how they are applied. A theoretical frequency distribution for a set of variable data, usually represented by a bell-shaped curve symmetrical about the mean. Read more. The normal distribution is uniquely defined by its mean and standard deviation. 14. Unlike the discrete random variables, the pdf of a continuous random variable does not equal to P ( Y = y). The normal approximation involves using a normal distribution with mean np and standard deviation npq. 1 1 Identify the properties of a normal distribution. The outcomes of a binomial experiment fit a binomial probability distribution. Mar 23, 2023 · Template 1: Bell Curve Diagram Template. 3. 35 % + 13. 47)=0. ppt. because the distribution is symmetric, one-half (. Understanding the properties of normal distributions means you can use inferential statistics to compare May 24, 2024 · Normal Distribution Problems and Solutions. It provides examples of how to calculate standardized scores (z-values) and probabilities using the normal distribution. Errors for example often have normal distribution. 18 to 24 year old females have a mean of 64. Jul 10, 2021 · The central limit theorem states that the distribution of sample means approaches a normal distribution as sample size increases. The mean ( 𝑥 ) is equal the median (𝑥 ), and is also equal to the mode ( 𝑥 ). Presenting our Normal Distribution Examples Solutions Ppt Powerpoint Presentation Summary Show Cpb PowerPoint template design. It defines key terms like solute, solvent, saturation and types of solutions including percentage, molar, and normal solutions. 5; 97 Example Using a Correction for Continuity. Calculate probabilities and expected value of random variables, and look at ways to transform and combine random variables. Step 2: Add up all of the values from Step 1 then set this number aside for a moment. The standard normal distribution is given by ; -¥ < z< ¥ SlideTeam has published a new blog titled "Top 10 Learning Dashboard Templates with Examples and Samples". The parameters of the normal are the mean \(\mu\) and the standard deviation σ. relation to other scores on the distribution. In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. Using this PPT template, you can enlist various factors, such as the adoption profile of the customer, psychography Download ppt "Stats Yr2 Chapter 3 :: Normal Distribution". Dec 23, 2016 · The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. May 27, 2015 · - All normal distributions can be converted to the standard normal distribution (with μ=0 and σ=1) by subtracting the mean and dividing by the standard deviation. A function to determine the ordinates of the graph picturing the distribution b. A sample of size k is drawn and the rv of interest is X = number of successes. It provides examples and procedures for preparing solutions of specific concentrations using calculations of weight, volume, molarity or normality. Jul 9, 2021 · The document discusses the normal distribution and its key properties. 5. Dec 1, 2016 · NORMAL DISTRIBUTION. Probability and Normal Distributions Same area P ( x < 15) = P ( z < 1) = Shaded area under the curve = 0. Astronomers like It introduces the normal distribution as a continuous probability distribution where data values are clustered around the mean. Jan 23, 2021 • Download as PPTX, PDF •. Exercise7. The mean of the sample means equals the population mean, and the standard deviation of sample means is the population standard deviation Example. Actually, the normal distribution is based on the function exp (-x²/2). MAT). The document discusses key concepts related to probability distributions including the binomial, Poisson, and normal distributions. Aug 13, 2019 · It defines the standard normal distribution as having a mean of 0, a standard deviation of 1, and a bell-shaped curve. 5 Find specific data values for given percentages, using the standard normal Feb 9, 2021 · Example 1: Birthweight of Babies. The rst thing to do is to show that this is a (probability) densit. The binomial distribution is a discrete distribution. Teachers: you can create student accounts (or students can register themselves). About this unit. 5-* Drawing a Normal Probability Plot Step 3Find the z Make everyone drop their jaws with Normal Distribution Examples Solutions presentation templates and Google slides. In practical applications, one would have to have a table of areas under the curve for each variable. 6 KB. score in the distribution is in standard. 95% will be between 400 and 600. Many real-world variables are approximately normally distributed. This PowerPoint slide showcases four stages. The histogram of the birthweight of newborn babies in the U. Use a correction for continuity to convert the binomial intervals to a normal distribution interval. For more information, read Comparing Hypothesis Tests for Continuous, Binary, and Count Data. , original score = z-score µ+ z σ= 0 + z(1) = z (we use standard normal for statistical inference starting(we use standard normal for statistical inference starting in Chapter 6, where certain statistics are scaled to have a May 13, 2022 · A Poisson distribution is a discrete probability distribution. Because normally distributed variables are so common, many statistical tests are designed for normally distributed populations. plot the data (stemplot or histogram) look for the overall pattern (shape, peaks, gaps) and departures from it (possible outliers) calculate appropriate numerical measures of. Standard Normal Distribution. It is represented by a normal curve. The integral of the rest of the function is square root of 2xpi. It allows using a normal distribution for applications involving sample means. It’s well-documented that the birthweight of newborn babies is normally distributed with a mean of about 7. . x – M = 1380 − 1150 = 230. Jan 13, 2019 · The key points are: - A z-score represents the distance from the mean in terms of standard deviations and is used to convert a raw score to its standard normal equivalent. Standard deviation = 2. No portion of curve lies below x-axis so probability can never be negative. 5 Examples The number of emergency calls received by an ambulance control in an hour. 3) Properties of the normal distribution are outlined, including how to find probabilities using the standard normal distribution table and z-scores. Oct 3, 2016 · It explains that the normal distribution is a theoretical probability curve where the area under the curve equals 1. The lesson covers what a normal distribution is and then talks through how to convert to a standardised normal distribution (find the z-score). Find predesigned Normal Distribution Examples Solutions Ppt Powerpoint Presentation Summary Show Cpb PowerPoint templates slides, graphics, and image designs provided by SlideTeam. Fine-tune the curve by adjusting the points to create the desired shape of a normal curve. The idea is to encourage the students to understand exactly what they are finding, rather than just Dec 30, 2023 · 2) It provides examples of discrete probability distributions like the binomial and Poisson distributions. It is defined by its mean and standard deviation. It has 0 skew and a kurtosis of 3. - The standard normal distribution is useful because probability tables and computer programs provide the integral values, avoiding the need to calculate integrals manually. In a ordinary distribution the imply is 0 and the same old deviation is 1. Each individual can be characterized as a success or failure, m successes in the population. 7% within 3 standard deviations. The heights of. It then covers how to standardize a normal distribution by subtracting the mean and dividing by the standard deviation to obtain a standard normal distribution with mean 0 and variance 1. By the formula of the probability density of normal distribution, we can write; Hence, f(3,4,2) = 1. In this exponential function e is the constant 2. (b) the percentage of sixth formers smaller than 160 cm; cm. • It is symmetrical, one side is the mirror image of the other. It has one parameter, the mean (m), and is used when the number of trials is large but the probability of an individual success is small. Normal distribution : - The Normal Distribution, also called the Gaussian Distribution, is the most significant continuous probability distribution. The standard normal distribution refers to a normal distribution with a mean of 0 and standard deviation of 1. inches and a standard deviation of 2. Step 2: The diameter of 120 cm is one standard deviation below the mean. taller. x = 1380. Estimate the percentage of US males. Mar 6, 2024 · This completely editable PowerPoint graphic exhibits Standard Normal Distribution that will help you convey the message impactfully. The area to the left and the area to the right of the curve is 0. Mean = 4 and. If you try to graph that, you'll see Jul 1, 2020 · The normal distribution, which is continuous, is the most important of all the probability distributions. center and spread (5-number summary and/or mean. Drawing a Normal Probability Plot Step 1Arrange the data in ascending order. 0 mg/100 ml. The normal distribution is a continuous probability distribution that is symmetrical around its mean, most Jan 3, 2023 · Height, birth weight, reading ability, job satisfaction, or SAT scores are just a few examples of such variables. Sarabjeet Kaur. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode ), while the parameter is the normal distribution is ; x gt 157. Its graph is bell-shaped. Jan 22, 2013 · 4. The standard normal variate is defined as which follows standard normal distribution with mean 0 and standard deviation 1 i. Note! If Y is continuous P ( Y = y) = 0 for any given value y. The normal distribution is the most common continuous probability distribution and is characterized by a mean (μ) and standard deviation (σ). 1. If we can approximate the distribution of these grades by a normal distribution, what percent of the students should fail the test (i. You can click and drag these points to smooth out the curve and create the iconic bell shape of the normal distribution. 1 of 18. It defines the standard normal distribution as having a mean of 0, a standard deviation of 1, and a bell-shaped curve. This design is easy to understand and fully customizable. displays a bell-shape that is typically of the normal distribution: Jul 6, 2017 · The document discusses the standard normal distribution. SD = 150. 4. g. normal distribution with µ = 0, σ= 1 For that distribution, z = (y - µ)/σ= (y - 0)/1 = y i e original score = zi. 1 Introduction to Normal Distributions and the Standard Normal Distribution Important Concepts: Normal Distribution Standard Normal Distribution – A free PowerPoint PPT presentation (displayed as an HTML5 slide show) on PowerShow. This document discusses statistical concepts such as descriptive and inferential statistics, types of data, and properties of the normal distribution. nd boys is different. 5 pounds. It provides examples of using the normal distribution to calculate probabilities for various scenarios involving heights, spending amounts, newspaper waste, and blood pressure. All Z tests assume your data follow a normal distribution 23. May 5, 2015 · The document discusses several probability distributions including the normal, binomial, and Poisson distributions. e. The mean, μ, and variance, σ 2, for the binomial probability distribution are μ = np and σ 2 = npq. population of college-bound students (with range restricted to 200-800), and the average math SAT is 500 with a standard deviation of 50, then: 68% of students will have scores between 450 and 550. Presentation on theme: "Lesson 100: The Normal Distribution"— Presentation transcript: P (a < X < b) = area under the density curve between a and b. The standard normal distribution has a mean of 0 and standard deviation of 1. The expected proportion of observations less than or equal to the ith data value is f. and standard deviation 2. Hypergeometric Distribution. 50 or 50%) lies on either side of the mean. The curve becomes parallel to x-axis which is supposed to meet it at infinity. Let’s solve some problems on Normal Distribution. A normal distribution is a symmetric, bell-shaped curve that describes the distribution of continuous random variables. 1 of 42. A continuous random variable 2. Example 1: Find the probability density function of the normal distribution of the following data. Convert your measurement x to a standard score ; z (x - ?) / ? Use the standard normal distribution ? 0 and ? 1 ; areas tabulated in front of text; 11 Example Jul 27, 1998 · the normal distribution; 6 (C) The Normal Distribution. 2 Binomial Experiment A binomial experiment has the following properties: experiment consists of n identical and independent trials each trial results in one of two outcomes: success or failure P (success) = p P (failure) = q = 1 - p for all trials The random variable of interest, X, is the Apr 30, 2018 · The normal distribution, also known as the Gaussian distribution, is the most important probability distribution in statistics for independent, random variables. 8. 0 likes • 39 views. Other analyses can assess additional data types. of the normalIf you could work in only whole numbers of SDs, t. Aug 22, 2018 · It provides formulas and examples for calculating each measure. 7 inches. Half the area under the curve is above the peak, and the other half is below it. It provides key properties of the normal distribution including that 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99. It can be accessed with Google Slides and is available in both standard screen and widescreen aspect ratios. 3) Paired t-tests are used to compare the means of two related groups when samples are paired, while unpaired t-tests are used to compare unrelated groups or independent samples. Jan 23, 2021 · Normal distribution. Suppose SAT scores roughly follows a normal distribution in the U. d. Characteristics of normal distribution 5. The sampling distribution of a sample mean x ¯ has: μ x ¯ = μ σ x ¯ = σ n. 11 likes • 821 views. Elevated Apr 11, 2023 · It explains that the normal distribution is a theoretical probability curve where the area under the curve equals 1. The probability of a random variable falling within any given range of values is equal to the proportion of the 5. Normal Distribution Properties of Normal Distribution Empirical rule of normal distribution Normality limits Standard normal distribution (z-score/ SND) Properties of SND Use of z/normal table Solved examples. Solution ; The discrete midpoint values are ,60, 61, 62. , Z ~ N(0,1). 15 % + 2. Description involves two functions: a. Examples of Poisson distributions given include defects per box of screws and printing mistakes per page. The spread of the curve depends on the standard deviation of the distribution. Normal distributions are symmetrical, however now no longer all symmetrical distributions are ordinary. It provides overviews of each distribution including 1. 8413 15 μ = 10 P ( x < 15) μ = 10 σ = 5 Normal Distribution x 1 μ = 0 μ = 0 σ = 1 Standard Normal Distribution z P ( z < 1) 24. Dec 15, 2022 · The ordinary distribution is the right time period for a chance bell curve. Specifically, it shows how to find the probability of Population to be sampled consists of N finite individuals, objects, or elements. 5 % = 16 %. The z score for a value of 1380 is 1. the graph of the normal distribution is also referred to as the standard normal curve and one defining equation for the curve for our purposes is ; where z refers to a concept called the z score which takes into account the mean and standard deviation of a set of data ; 7 (C) The Normal Also known as a standard score. 5 inches. Indicates how far above or below the mean a given. • Similar in spirit to Binomial distribution, but from a finite. ) Hence the raw score is 3 Ie the lowest maximum length is 6. It is graphically represented by a normal curve. - The formula for converting a normal variable to standard is z = (x - μ) / σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. Example question: Calculate a paired t test by hand for the following data: Step 1: Subtract each Y score from each X score. AI-enhanced description. 1. Question A (Part 2) Let X be random variable which follows normal distribution with mean m and variance s2 . 0 mg/100 ml, and standard deviation of 1. So it must be normalized (integral of negative to positive infinity must be equal to 1 in order to define a probability density distribution). Your sample data follow a normal distribution, or you have a large sample size. Jan 31, 2024 · Step 2: Adjust the curve points. Most people recognize its familiar bell-shaped curve in statistical reports. 4cm Practice (Normal Distribution) 1 Potassium blood levels in healthy humans are normally distributed with a mean of 17. The standard deviation, σ, is then σ = n p q n p q. between 18 and 24 who are 6ft tall or. Common probability distributions like the binomial distribution and normal distribution. continuous probability distribution 2. Apr 12, 2017 · The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. Solution: Given, variable, x = 3. Ijaz Manzoor. • It is asymptotic, that is, it tails off very gradually on each side but the line representing the distribution never quite meets the horizontal axis. Step 3: Add the percentages in the shaded area: 0. Normal distribution 8. 4. We introduce normal distributions by first looking at another continuous probability distribution, the uniform distribution. Jan 23, 2014 · Properties of the Normal Probability Distribution • Slide 5 The normal is a family of Bell-shaped and symmetric distributions. Solution: Given, Variable (x) = 2; Mean = 3; Standard Deviation = 4; Using formula of probability density of normal distribution Nov 29, 2017 · This ppt is a part of Business Analytics course. A large group of students took a test in Physics and the final grades have a mean of about 70 and a standard deviation of 10. It defines the binomial distribution and states the requirements for the normal approximation are that np and nq must both be greater than or equal to 5. The normal probability distribution is bell-shaped. 39 Example : Solution: P (Z<0. Sep 27, 2020 • Download as PPTX, PDF •. 1 inches. Shade below that point. μ – 3σ μ + σ μ – σ μ For example, weight, height, and temperature are continuous. S – successes (probability of success) are the same – yes, the likelihood of getting a Jack is 4 out of 52 each time you turn over a card. 5. Aug 13, 2019 · The document discusses applications of the normal distribution. Nov 5, 2020 · The z score tells you how many standard deviations away 1380 is from the mean. 6808 Example 6-6: See page 329. Some key points: 1. It has a bell-shaped symmetrical curve centered around the mean. Jun 27, 2017 · The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. 71828…, is the mean, and σ is the standard deviation. Jagdish Powar. Jul 15, 2023 · The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. (see figure below) f (y) a b. x = 3, μ = 4 and σ = 2. Sep 25, 2020 · N – number of trials fixed in advance – yes, we are told to repeat the process five times. Click on the curve to reveal the adjustment points. Jul 27, 1998 · are approximately normal with mean 70. 2 3. For example, it shows how to find the probability of an event being below or above a given z-score, or The outcomes of a binomial experiment fit a binomial probability distribution. SDs. 645. 3 Learning Objectives 1. Solution: Step 1: Sketch a normal distribution with a mean of μ = 150 cm and a standard deviation of σ = 30 cm . 7-3 Characteristics of a Normal Distribution The normal curve is bell-shaped and has a single peak at the exact center of the distribution. 2 Identify distributions as symmetric or skewed 3 Find the area under the standard normal distribution, given various z values. It is the distribution which appears most often if data can take both positive and negative values. Probability and Normal Distributions Example : The average on a statistics test was 78 with a standard Mar 30, 2012 · Normal probability distribution function: This is the function for the normal (bell-shaped) curve - m 2 1 ( x ) 1 - = s 2 f ( x ) e s p 2 1. S. 1(Propertiesof theNormal Distribution) First Step: find the z , using the formula. Download now. It is characterized by two parameters: the mean (μ) and the standard deviation (σ). Each is characterized by a different pair of mean, , and variance, . Z 72 70. It gives the probability of an event happening a certain number of times ( k) within a given interval of time or space. ÐÏ à¡± á> þÿ • þÿÿÿþÿÿÿ{ | } ~ € ‚ ƒ „ … † ‡ ˆ ‰ Š ‹ Œ Ž ‘ ’ “ ” ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ 13 Standard Normal Distribution Since each normally distributed variable has its own mean and standard deviation, the shape and location of these curves will vary. The normal distribution describes how observations vary around a mean value and results in the familiar bell curve. 5 hours ago Apr 26, 2019 · 2) The t-distribution is used for calculating t-tests when sample sizes are small and/or variances are unknown. 1 . 13. The probability of getting less than 63 successes. Jun 27, 2017 · The Normal Distribution is a symmetrical probability distribution where most results are located in the middle and few are spread on both sides. Question 1: Calculate the probability density function of normal distribution using the following data. It outlines the characteristics of the normal curve and provides examples of using the normal distribution and calculating z-scores. kt cy em vu oz yz av tx po hm