Statistics Normal Distribution

University:
Bradley University
Course:
MTH 325 | Probability and Statistics I
Academic year:

2022
Views:

445

Pages:

3
Author:

blakestar2o1o

THE NORMAL DISTRIBUTION WHAT IS A NORMAL DISTRIBUTION? The normal distribution is a probability function that depicts the way values of a variable are distributed. It’s a symmetric distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean and taper off equally in both directions. Example - FINDING PROBABILITIES HOW TO FIND PROBABILITIES 1. Draw a picture of the normal distribution. 2. Translate the problem into one of the following: p(X < a), p(X > b), or p(a < X < b). Shade in the area on your picture. 3. Standardize a (and/or b) to a z-score using the z-formula: 4. Look up the z-score on the Z-table (see below) and find its corresponding probability. a.Find the row of the table corresponding to the leading digit (ones digit) and first digit after the decimal point (the tenths digit). b.Find the column corresponding to the second digit after the decimal point (the hundredths digit). c.Intersect the row and column from Steps (a) and (b) 5. a. If you need a “less-than” probability — that is, p(X < a) — you’re done. b. If you want a “greater-than” probability — that is, p(X > b) — take one minus the result from Step 4. c. If you need a “between-two-values” probability — that is, p(a < X < b) — do Steps 1–4 for b (the larger of the two values) and again for a (the smaller of the two values), and subtract the results. FINDING VALUES A critical value is a line on a graph that splits the graph into various sections. One or two of the sections is the “rejection region“; if your test value falls into that region, then you reject the null hypothesis. Step 1: Subtract the confidence level from 100% to find the α level: 100% – 90% = 10% Step 2: Convert Step 1 to a decimal: 10% = 0.10. Step 3: Divide Step 2 by 2 (this is called “α/2”). 0.10 = 0.05. This is the area in each tail. Step 4: Subtract Step 3 from 1 (because we want the area in the middle, not the area in the tail): 1 – 0.05 = .95. Step 5: Look up the area from Step in the z-table. The area is at z=1.645. This is your critical value for a confidence level of 90%. CENTRAL LIMIT THEOREM The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. Example -