H2 Mathematics - Normal Distribution (Basics Operations)

Title Junior College ‘A’ Levels H1/H2 Mathematics – Normal Distribution Author AprilDolphin Date 10/4/2025 In many situations out there, many types of continuous random variables, such as test scores, height and weight of students of a certain age would show the following characteristics once data got collected and plotted as it would be on a probability distribution. • You would realize that majority of the data are centred in the middle, with extreme small and big values tailing off symmetrically on the left and right of the average measurement respectively. When this happens, the random variable 𝑋 is said to follow a Normal Distribution with parameters 𝜇 and 𝜎2, where 𝑋~𝑁(𝜇, 𝜎2), with 𝜇 referring to the mean and 𝜎2 referring to the variance of the Normal Distribution. (*Be careful when dealing with notations, certain books, software and calculators deal with Normal Distribution using the Standard Deviation 𝜎 parameter rather than variance which is 𝜎2. In such cases, the obvious first step you should take is to square-root the variance to get the value of standard deviation 𝜎.) A basic visual look at the Normal Distribution and some properties to note. • The probability value is the area between the curve and the x-axis. (Which is also the definite integral of the Normal Distribution in question.) • The probability of getting a very specific value in a Normal Distribution is basically zero since area under curve cannot be created on a continuous random variable just with specific values. Instead of defining specific value on a Normal Distribution, we usually define a range of values to calculate probability in a Normal Distribution. [Therefore 𝑃(𝑋 = 𝑥) = 0]

• Any Normal Distribution is symmetrical at the mean 𝜇. (This property is important as certain questions you will encounter requires you to understand this symmetrical property of any Normal Distribution.) Understanding the concept of a Standard Normal Distribution • Any Normal Distribution can technically be transformed to a Standard Normal Distribution. • A Standard Normal Distribution has the property of which the area under curve from negative infinity to positive infinity is exactly 1. • A Standard Normal Distribution also has property of mean 𝜇 = 0 and 𝜎2 = 1, for this reason, a Standard Normal Distribution will also have standard deviation 𝜎 = 1 as well. (Note: In order to input “negative infinity” in Graphing Calculator, press -E99. In order to input a value of “positive infinity” in Graphing Calculator, press E99.) A visual look at the standard normal distribution. Understanding the concept of 𝑍-score in Standard Normal Distribution • 𝑍-score refers to the number of standard deviations from the mean in a standardized normal distribution where the 𝑍 −score can take on any finite values. (−∞ < 𝑍 < ∞) • 𝑍-score of any normal distribution can be computed using the below formula. 𝑍 = 𝑋 − 𝜇 𝜎 𝑍 refers to the 𝑍-score 𝑋 refers to the position of the normal random variable on the X-axis as it is in original unstandardized form. 𝜇 refers to the mean of the normal distribution as it is in original unstandardized form. 𝜎 refers to the standard deviation as it is in the original unstandardized form.

Process of Finding Probability in a Standard Normal Distribution: In order to find probability in a Standard Normal Distribution, we use the TI-84 graphing calculator in the following manner. Example 1. Find the probability of the following a) 𝑃(𝑍 < 1.96) b) 𝑃(𝑍 > −0.586) c) 𝑃(0.43 < 𝑍 < 1.23) Procedures (Example 1a) Output on Graphing Calculator Screen Press [2ND] followed by [VARS] DISTR DRAW 1:normalpdf ( 2:normalcdf ( 3:invNorm ( 4:invT ( 5:tpdf ( 6:tcdf ( 7:𝜒2pdf ( Look for “normalcdf” option and press [ENTER] normalcdf Lower: Upper: 𝜇: 0 𝜎: 1 Paste For probability values less than 𝑍 We key in -E99 in the field “lower” and we key in 1.96 in the upper field. Since Standard Normal Distribution has a value of 𝜇 = 0 and 𝜎 = 1, we input 𝜇 = 0 and 𝜎 = 1 normalcdf Lower: -E99 Upper: 1.96 𝜇: 0 𝜎: 1 Paste Press the down arrow after checking the inputs and press down arrow until the cursor is on “Paste” and press [ENTER] twice. The probability is 0.975 (3sf) normalcdf(-E99, 1.96, 0, 1) .9750021748

Procedures (Example 1b) Graphing Calculator Output Press [2ND] followed by [VARS] DISTR DRAW 1:normalpdf ( 2:normalcdf ( 3:invNorm ( 4:invT ( 5:tpdf ( 6:tcdf ( 7:𝜒2pdf ( Look for “normalcdf” option and press [ENTER] normalcdf Lower: Upper: 𝜇: 0 𝜎: 1 Paste For probability values more than 𝑍. We key in -0.586 in the “Lower” field and E99 into the “Upper” field. Since Standard Normal Distribution has a value of 𝜇 = 0 and 𝜎 = 1, we input 𝜇 = 0 and 𝜎 = 1. normalcdf Lower: -0.586 Upper: E99 𝜇: 0 𝜎: 1 Paste Press the down arrow after checking the inputs and press down arrow until the cursor is on “Paste” and press [ENTER] twice. The probability is 0.721 (3sf) normalcdf(-0.586, E99, 0, 1) .7210622905

Procedures (Example 1C) Graphing Calculator Output Press [2ND] followed by [VARS] DISTR DRAW 1:normalpdf ( 2:normalcdf ( 3:invNorm ( 4:invT ( 5:tpdf ( 6:tcdf ( 7:𝜒2pdf ( Look for “normalcdf” option and press [ENTER] normalcdf Lower: Upper: 𝜇: 0 𝜎: 1 Paste Since the question mentioned we have to find the probability for which the Z-score is in between 0.43 and 1.23. We key in 0.43 in the “Lower” field and 1.23 in the “Upper” field. We set 𝜇 = 0 and 𝜎 = 1. normalcdf Lower: 0.43 Upper: 1.23 𝜇: 0 𝜎: 1 Paste After checking that the values are correct, we can press down arrow key on calculator until the cursor is on “Paste”, press enter twice. The probability of obtaining Z-score between 0.43 and 1.23 is 0.224 (3sf) normalcdf (0.43, 1.23, 0, 1) .2242492346

Example 2 Find the probability of the following (a) 𝑃(|𝑍| < 1.234) (b) 𝑃(|𝑍| ≥ 2.17) (c) 𝑃(|𝑍 − 1| > 1.1389) 2(a) |𝑍| < 1.234 is to be rewritten as −1.234 < 𝑍 < 1.234 Steps taken Graphing Calculator Output Enter “normalcdf” functionality of the graphing calculator normalcdf lower: upper: 𝜇: 𝜎: Paste Set “lower” as −1.234 Set “upper” as 1.234 Set 𝜇 as 0 Set 𝜎 as 1 Press down arrow key until the cursor is on “Paste” normalcdf lower: -1.234 upper: 1.234 𝜇:0 𝜎:1 Paste Press [ENTER] twice and the following should appear. The probability is 0.783 (3sf) normalcdf(-1.234, 1.234, 0, 1) .7827969667 2(b) |𝑍| ≥ 2.17 is to be written as 𝑍 < −2.17 OR 𝑍 > 2.17 Steps Taken Graphing Calculator Output Enter “normalcdf” functionality of calculator normalcdf lower: upper: 𝜇: 𝜎: Paste

Input lower as - E99 Input upper as -2.17 Set 𝜇 = 0 and 𝜎 = 1 Press arrow down key until the cursor is at paste. normalcdf lower: -E99 upper: -2.17 𝜇:0 𝜎:1 Paste Press [ENTER] twice. normalcdf(-E99,2.17,0, 1) .150033693 Enter “normalcdf” functionality of calculator again. Input lower as 2.17 Input upper as E99 Set 𝜇 = 0 and 𝜎 = 1 Press arrow down key until the cursor is at paste. normalcdf lower: 2.17 upper: E99 𝜇:0 𝜎:1 Paste Press [ENTER] twice. normalcdf(2.17, E99, 0, 1) .150033693 𝑃(|𝑍| ≥ 2.17) = 0.150033693+0.150033693 = 0.300 (3sf) 2(c) Rewrite 𝑃(|𝑍 − 1| > 1.1389) as 𝑍 − 1 < −1.1389 as well as 𝑍 − 1 > 1.1389 which can be transformed as follows: 𝑍 < −1.1389 + 1 OR 𝑍 > 1.1389 + 1 𝑍 < −0.1389 OR 𝑍 > 2.1389

Steps Taken Graphing Calculator Output Enter “normalcdf” functionality of calculator normalcdf lower: upper: 𝜇: 𝜎: Paste Key in the values as follows Lower: -E99 Upper: -0.1389 𝜇:0 𝜎: 1 normalcdf lower:-E99 upper:-0.1389 𝜇:0 𝜎:1 Paste Press down arrow until the cursor is at “paste” and press [ENTER] key twice normalcdf(-E99, -0.1389,