Statistics Chapter 6: Normal Curves and Sampling Distribution

Understandable Statistics, 13th Edition – Chapter 6: Normal Curves and Sampling Distribution

Section 6.1: Introduction

  • This chapter focuses on Normal Curves and Sampling Distributions, fundamental concepts in statistics.

Section 6.2: Standard Units and Areas under the Standard Normal Distribution

Section Objectives

  • By the end of this section, you should be able to:

    • Convert raw data to z-scores given the mean ($μ$) and standard deviation ($σ$).

    • Convert z-scores back to raw data using $μ$ and $σ$.

    • Graph the standard normal distribution.

    • Find areas under the standard normal curve.

Standard Normal Distribution

  • The standard normal distribution is defined as a normal distribution with:

    • Mean ($μ$) = 0

    • Standard Deviation ($σ$) = 1

  • A random variable that follows a standard normal distribution is denoted as:
    Zext N(0,1)Z ext{~} N(0, 1)

Properties of Z-scores

  • The property of the standard normal distribution indicates that any value of $z$ represents the number of standard deviations that value is from the mean:

    • For example:

    • If z=2z = 2, the value is 2 standard deviations above the mean.

    • If z=−1.53z = -1.53, the value is 1.53 standard deviations below the mean.

Finding Areas Under the Curve

Probability Interpretations

  • The probabilities of the areas under the standard normal distribution can be accessed through the Standard Normal Table.

  • The table provides values for: P(Z≤z)P(Z ≤ z)

    • This value corresponds to the area to the left of the z-score on the standard normal curve.

How to Use the Standard Normal Table

  • To look up a value for P(Z≤z)P(Z ≤ z):

    • The header column indicates the first decimal place.

    • The header row specifies the second decimal place.

    • Example:

    • For P(Z≤0.32)=0.6255P(Z ≤ 0.32) = 0.6255

    • For P(Z≤1.25)=0.8944P(Z ≤ 1.25) = 0.8944

  • Important to note that answer values should be rounded to four decimal places.

Continuous Random Variables

  • For continuous random variables, it follows that: P(Z ≤ z) = P(Z < z)

    • Thus,
      P(Z ≤ 0.32) = P(Z < 0.32) = 0.6255

Example 1 – Using the Standard Normal Table

  • Objectives for this example include finding:

    • a) P(Z < 1.52)

    • b) P(Z < 2.60)

    • c) P(Z < -0.75)

    • d) P(1.00 < Z < 2.10)

  • A diagram should be drawn for each case to illustrate the region being found.

Solution for Example 1

  • For part d, the area is determined by subtracting the corresponding entries from the table:

    • P(1.00 < Z < 2.10) = P(Z < 2.10) - P(Z < 1.00)

    • Using table values:

    • P(Z < 2.10) = 0.9821

    • P(Z < 1.00) = 0.8413

    • Therefore:

    • P(1.00 < Z < 2.10) = 0.9821 - 0.8413 = 0.1408

Activity 1 – Standard Normal

  • Reminder to draw sketches if unsure about the areas being calculated.

Inverse Normal Distribution

  • The Inverse Normal Distribution allows you to find the z-value corresponding to a specific probability using the Standard Normal Table.

Example 2 - Inverse Normal Distribution

  • In this section, specific examples illustrate how to find corresponding z-values given probabilities.

Activity 2 – Inverse Normal Solutions

  • Solutions for the exercises provided are to be consulted to verify the method and outcomes of inverse calculations.

Activity 3 – Additional Questions

  • Students should ensure sketches accompany their answers when necessary to visualize problems better.

Summary

  • Review the chapter’s objectives for completeness and understanding of core concepts and numerical problems involving standard normal distributions and their applications in statistical analysis.