7: Normal Distributions

Objectives

Upon successful completion of this lesson, you should be able to:
  • Describe the standard normal distribution
  • Determine the area under a normal distribution using Minitab
  • Determine the points that offset a given proportion of a normal distribution using Minitab
  • Summarize the Central Limit Theorem
  • Conducted a hypothesis test using a standardized test statistic
  • Construct a confidence interval using the standard form

Over the last three lessons you have approximated sampling distributions using bootstrapping and randomization methods. You may have noticed that many of the distributions that you constructed had similar shapes, such as those below:

Randomization Dotplot of Proportion

null = 0.5 samples = 5000 mean = 0.500 std. error = 0.014 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.5 100 120 80 40 60 20 0 Proportion Left Tail Two - Tail Right Tail Randomization Dotplot of Null hypothesis: p =

 

Bootstrap Dotplot of Mean

2871.636 samples = 5000 mean = 2871.636 std. error = 114.290 2500 2600 2700 2800 2900 3000 3100 3200 3300 100 120 80 40 60 20 0 Left Tail Two - Tail Right Tail Bootstrap Dotplot of Mean

Bootstrap Dotplot of Correlation

0.107 samples = 8000 mean = 0.107 std. error = 0.239 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 127 150 175 100 50 75 25 0 Left Tail Two - Tail Right Tail Bootstrap Dotplot of Correlation

Randomization Dotplot of \(\bar{x}_1-\bar{x}_2\)

null = 0 samples = 5000 mean = 0.466 std. error = 221.886 -800 -600 -400 -200 0 200 400 600 800 100 120 80 40 60 20 0 Left Tail Two - Tail Right Tail Randomization Dotplot of ⅹ 1 - ⅹ 2 , Null hypothesis: µ 1 = µ 2

These are all approximately normally distributed. You were first introduced to the normal distribution in Lesson 2 as a special type of symmetrical distribution. In this lesson, we'll review normal distributions, learn how to use Minitab to construct plots of normal distributions, and learn how the Central Limit Theorem allows us to apply what we know about the normal distribution to construct confidence intervals and conduct hypothesis tests without using simulations.