half-bounded interval

(noun)

A set for which one endpoint is a real number and the other is not.

Related Terms

Examples of half-bounded interval in the following topics:

  • Interval Notation

    • An interval is said to be bounded if both of its endpoints are real numbers.
    • Bounded intervals are also commonly known as finite intervals.
    • An interval that has only one real-number endpoint is said to be half-bounded, or more descriptively, left-bounded or right-bounded.
    • For example, the interval $(1, + \infty)$ is half-bounded; specifically, it is left-bounded.
    • Use interval notation to show how a set of numbers is bounded

  • Working Backwards to Find the Error Bound or Sample Mean

    • When we calculate a confidence interval, we find the sample mean and calculate the error bound and use them to calculate the confidence interval.
    • If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.
    • Subtract the error bound from the upper value of the confidence interval
    • Suppose we know that a confidence interval is (67.18, 68.82) and we want to find the error bound.
    • If we know the error bound: = 68.82 − 0.82 = 68

  • Summary of the Types of Intervals

    • An augmented interval is one half step larger than the perfect or major interval.
    • A diminished interval is one half step smaller than the perfect or minor interval.
    • To find the inversion's number name, subtract the interval number name from 9.
    • Inversions of major intervals are minor, and inversions of minor intervals are major.
    • Inversions of augmented intervals are diminished, and inversions of diminished intervals are augmented.

  • Summary of Formulas

    • ( lower value,upper value ) = ( point estimate − error bound,point estimate + error bound )
    • Formula 8.2: To find the error bound when you know the confidence interval
    • error bound = upper value − point estimate OR error bound = (upper value − lower value)/2
    • The confidence interval has the format ($\bar{x}$ − EBM, $\bar{x}$ + EBM) .
    • The confidence interval has the format (p' − EBP, p' + EBP) .

  • Classifying Intervals

    • So the second step to naming an interval is to classify it based on the number of half steps in the interval.
    • The minor interval is always a half-step smaller than the major interval.
    • If an interval is a half-step larger than a perfect or a major interval, it is called augmented.
    • An interval that is a half-step smaller than a perfect or a minor interval is called diminished.
    • Both are six half-steps, or three whole tones, so another term for this interval is a tritone.

  • Changing the Confidence Level or Sample Size

    • Increasing the confidence level increases the error bound, making the confidence interval wider.
    • Decreasing the confidence level decreases the error bound, making the confidence interval narrower.
    • What happens to the error bound and the confidence interval if we increase the sample size and use n=100 instead of n=36?
    • Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.
    • Decreasing the sample size causes the error bound to increase, making the confidence interval wider.

  • Interval (Class)

    • Pitch intervals are the distance between pitches as measured in half steps.
    • Thus, the interval from G4 to A-sharp5 = +15.
    • Think of it like this: if you are G4, how many half steps do you need to move to get to A-sharp5?
    • The ordered pitch interval from G4 to B-flat5 is +15, but the ordered pitch interval from A-sharp5 to G4 is -15.
    • Using various combinations of pitch interval, pitch-class interval, ordered, and unordered, we arrive at four different conceptions of interval.

  • Lab 2: Confidence Interval (Place of Birth)

    • The student will determine the effects that changing conditions have on the confidence interval.
    • Calculate the confidence interval and the error bound. i.
    • Error Bound:
    • Using the above information, construct a confidence interval for each given confidence level given.
    • Does the width of the confidence interval increase or decrease?

  • Introduction to Confidence Intervals

    • State why a confidence interval is not the probability the interval contains the parameter
    • These intervals are referred to as 95% and 99% confidence intervals respectively.
    • An example of a 95% confidence interval is shown below:
    • There is good reason to believe that the population mean lies between these two bounds of 72.85 and 107.15 since 95% of the time confidence intervals contain the true mean.
    • It is natural to interpret a 95% confidence interval as an interval with a 0.95 probability of containing the population mean.

  • Half Steps and Whole Steps

    • In Western music, the small interval from one note to the next closest note higher or lower is called a half step or semi-tone.
    • The interval between C and the F above it is 5 half steps, or two and a half steps.
    • Identify the intervals below in terms of half steps and whole steps.
    • Three half-step intervals: between C and C sharp (or D flat); between E and F; and between G sharp (or A flat) and A.
    • All intervals in a chromatic scale are half steps.