open interval

(noun)

A set of real numbers that does not include its endpoints.

Related Terms

Examples of open interval in the following topics:

  • Interval Notation

    • An open interval does not include its endpoints and is indicated with parentheses.
    • The image below illustrates open and closed intervals on a number line.
    • Bounded intervals are also commonly known as finite intervals.
    • For example, the interval $(1,10)$ is considered bounded; the interval $(- \infty, + \infty)$ is considered unbounded.
    • Representations of open and closed intervals on the real number line.

  • The Mean Value Theorem, Rolle's Theorem, and Monotonicity

    • The MVT states that for a function continuous on an interval, the mean value of the function on the interval is a value of the function.
    • The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
    • More precisely, if a function $f$ is continuous on the closed interval $[a, b]$, where $a < b$, and differentiable on the open interval $(a, b)$, then there exists a point $c$ in $(a, b)$ such that
    • Rolle's Theorem states that if a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and f(a) = f(b), then there exists a c in the open interval $(a, b)$ such that $f'(c)=0$.
    • For any function that is continuous on $[a, b]$ and differentiable on $(a, b)$ there exists some $c$ in the interval $(a, b)$ such that the secant joining the endpoints of the interval $[a, b]$ is parallel to the tangent at $c$.

  • Interpreting confidence intervals

    • A careful eye might have observed the somewhat awkward language used to describe confidence intervals.
    • Incorrect language might try to describe the confidence interval as capturing the population parameter with a certain probability.
    • Another especially important consideration of confidence intervals is that they only try to capture the population parameter.
    • Our intervals say nothing about the confidence of capturing individual observations, a proportion of the observations, or about capturing point estimates.
    • Confidence intervals only attempt to capture population parameters.

  • Confidence interval for the difference

    • When the data indicate that the point estimate $\bar{x}_1-\bar{x}_2$ comes from a nearly normal distribution, we can construct a confidence interval for the difference in two means from the framework built in Chapter 4.
    • Using this information, the general confidence interval formula may be applied in an attempt to capture the true difference in means, in this case using a 95% confidence level:
    • Construct the 99% confidence interval for the population difference in average run times based on the sample data.
    • 5.5: If we were to collected many such samples and create 95% confidence intervals for each, then about 95% of these intervals would contain the population difference, µw −m.
    • (If the selection of z* is confusing, see Section 4.2.4 for an explanation. ) The 99% confidence interval: 14.48 ± 2.58×2.77 → (7.33, 21.63).

  • Introduction to confidence intervals

    • Subscribe to the OpenIntroOrg channel to stay up-to-date.
    • This video was created by OpenIntro (openintro.org) and provides an overview of the content in Section 4.2 of OpenIntro Statistics, which is a free statistics textbook with a $10 paperback option on Amazon.
    • This video introduces confidence intervals for point estimates, which are intervals that describe a plausible range for a population parameter.

  • Two sample t confidence interval

    • To answer this question, we can use a confidence interval.
    • Using df = 8, create a 99% confidence interval for the improvement due to ESCs.
    • Thus, we only need identify $t^*_8$ to create a 99% confidence interval: $t^*_8$ = 3.36.
    • The 99% confidence interval for the improvement from ESCs is given by: point estimate ± $t^*_8$SE → 7.83 ± 3.36×1.95 → (1.33,14.43).

  • The normal approximation breaks down on small intervals

    • Caution: The normal approximation may fail on small intervals The normal approximation to the binomial distribution tends to perform poorly when estimating the probability of a small range of counts, even when the conditions are met.
    • Notice that the width of the area under the normal distribution is 0.5 units too slim on both sides of the interval.
    • The normal approximation to the binomial distribution for intervals of values is usually improved if cutoff values are modified slightly.The cutoff values for the lower end of a shaded region should be reduced by 0.5, and the cutoff value for the upper end should be increased by 0.5.
    • While it is possible to apply it when computing a tail area, the benefit of the modification usually disappears since the total interval is typically quite wide.

  • Confidence intervals exercises

    • Will this new interval be smaller or larger than the 95% confidence interval?
    • Will this new interval be smaller or larger than the 95% confidence interval?
    • (e) A 90% confidence interval would be narrower than the 95% confidence interval.
    • Estimate the average age at first marriage of women using a 95% confidence interval, and interpret this interval in context.
    • The margin of error is half the width of the interval, and the sample mean is the midpoint of the interval.

  • Capturing the population parameter

    • A plausible range of values for the population parameter is called a confidence interval.
    • Using only a point estimate is like fishing in a murky lake with a spear, and using a confidence interval is like fishing with a net.
    • On the other hand, if we report a range of plausible values – a confidence interval – we have a good shot at capturing the parameter.
    • If we want to be very certain we capture the population parameter, should we use a wider interval or a smaller interval?
    • Likewise, we use a wider confidence interval if we want to be more certain that we capture the parameter.

  • Analyzing 12-tone music

    • Any row form that is the same as, or a strict transposition of, that opening prime form is also a prime form.
    • Once you have determined (or decided) which row is prime, analyze its interval content in ordered pitch-class intervals.
    • Its intervals (ascending pitch-class intervals, modulo12) are:1, 2, 6, 5, 9, 5, 6, 9, 1, 9, 1
    • Retrograde intervals are tricky (at first).
    • To get the interval succession of the retrograde forms, invert the retrograde interval progression, or reverse the prime interval progression:1, 9, 1, 9, 6, 5, 9, 5, 6, 2, 1