closed interval
(noun)
A set of real numbers that includes both of its endpoints.
Examples of closed interval in the following topics:
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Interval Notation
- A closed interval includes its endpoints and is denoted with square brackets rather than 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.
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Inverting Intervals
- To invert any interval, simply imagine that one of the notes has moved one octave, so that the higher note has become the lower and vice-versa.
- Because inverting an interval only involves moving one note by an octave (it is still essentially the "same" note in the tonal system), intervals that are inversions of each other have a very close relationship in the tonal system.
- To name the new interval, subtract the name of the old interval from 9.
- The inversion of a major interval is minor, and of a minor interval is major.
- The inversion of an augmented interval is diminished and of a diminished interval is augmented.
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Lab 3: Confidence Interval (Womens' Heights)
- Now write your confidence interval on the board.
- Using the class listing of confidence intervals, count how many of them contain the population mean µ; i.e., for how many intervals does the value of µ lie between the endpoints of the confidence interval?
- Is the percent of confidence intervals that contain the population mean µ close to 90%?
- Suppose we had generated 100 confidence intervals.
- Is this percent close to 90%?
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Significance Testing and Confidence Intervals
- Explain why a confidence interval makes clear that one should not accept the null hypothesis
- There is a close relationship between confidence intervals and significance tests.
- All values in the confidence interval are plausible values for the parameter whereas values outside the interval are rejected as plausible values for the parameter.
- Looking at non-significant effects in terms of confidence intervals makes clear why the null hypothesis should not be accepted when it is not rejected: Every value in the confidence interval is a plausible value of the parameter.
- Since zero is in the interval, it cannot be rejected.
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Hypothesis Tests or Confidence Intervals?
- Hypothesis tests and confidence intervals are related, but have some important differences.
- What is the difference between hypothesis testing and confidence intervals?
- When we use confidence intervals, we are estimating the parameters of interest.
- Confidence intervals are closely related to statistical significance testing.
- Explain how confidence intervals are used to estimate parameters of interest
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Classifying Intervals
- The physics of sound waves (acoustics) shows us that the notes of a perfect interval are very closely related to each other.
- (For more information on this, see Frequency, Wavelength, and Pitch and Harmonic Series. ) Because they are so closely related, they sound particularly good together, a fact that has been noticed since at least the times of classical Greece, and probably even longer.
- (Both the octave and the perfect fifth have prominent positions in most of the world's musical traditions. ) Because they sound so closely related to each other, they have been given the name "perfect" intervals.
- Seconds, thirds, sixths, and sevenths can be major intervals or minor intervals.
- The minor interval is always a half-step smaller than the major interval.
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Histograms
- To create this table, the range of scores was broken into intervals, called class intervals.
- There are three scores in the first interval, 10 in the second, etc.
- Placing the limits of the class intervals midway between two numbers (e.g., 49.5) ensures that every score will fall in an interval rather than on the boundary between intervals.
- Your choice of bin width determines the number of class intervals.
- Sturges' rule is to set the number of intervals as close as possible to $1 + log_{2}(N)$, where $log_{2}N$ is the base 2 log of the number of observations.
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Interpreting a Confidence Interval
- For users of frequentist methods, various interpretations of a confidence interval can be given.
- Typically a rule for constructing confidence intervals is closely tied to a particular way of finding a point estimate of the quantity being considered.
- Descriptive statistics - This is closely related to the method of moments for estimation.
- Bootstrapping - In situations where the distributional assumptions for the above methods are uncertain or violated, resampling methods allow construction of confidence intervals or prediction intervals.
- This figure illustrates a 90% confidence interval on a standard normal curve.
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Lab 1: Confidence Interval (Home Costs)
- Calculate the confidence interval and the error bound. i.
- Some students think that a 90% confidence interval contains 90% of the data.
- Is this percent close to 90%?
- Explain why this percent should or should not be close to 90%.
- Does the width of the confidence interval increase or decrease?
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What Is a Confidence Interval?
- A confidence interval is a type of interval estimate of a population parameter and is used to indicate the reliability of an estimate.
- A confidence interval can be used to describe how reliable survey results are.
- We realize that the point estimate is most likely not the exact value of the population parameter, but close to it.
- A confidence interval is a type of estimate (like a sample average or sample standard deviation), in the form of an interval of numbers, rather than only one number.
- Bayesian inference provides further answers in the form of credible intervals.