Examples of interval in the following topics:
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- The two numbers are called the endpoints of the interval.
- 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.
- Use interval notation to show how a set of numbers is bounded
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- 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.
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- 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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- 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.
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- And combining quality with a generic interval produces a specific interval.
- Note that both generic interval and chromatic interval are necessary to find the specific interval, since there are multiple specific diatonic intervals for each generic interval and for each chromatic interval.
- The intervals discussed above, from unison to octave, are called simple intervals.
- Any interval larger than an octave is considered a compound interval.
- A compound interval takes the same quality as the corresponding simple interval.
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- Because intervals are dependent upon the pitches that create them, the consonance and dissonance of intervals in tonal music is determined by tonality itself.
- In the context of G minor, this is a consonant interval.
- Thus, the interval from G4 to A-sharp5 = +15.
- 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.
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- 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:
- If repeated samples were taken and the 95% confidence interval computed for each sample, 95% of the intervals would contain the population mean.
- It is natural to interpret a 95% confidence interval as an interval with a 0.95 probability of containing the population mean.
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- A prediction interval is an estimate of an interval in which future observations will fall with a certain probability given what has already been observed.
- A prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter.
- Prediction intervals are often used in regression analysis.
- The prediction interval is conventionally written as:
- Formulate a prediction interval and compare it to other types of statistical intervals.
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- The proportion of confidence intervals that contain the true value of a parameter will match the confidence level.
- If confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will match the confidence level.
- This is guaranteed by the reasoning underlying the construction of confidence intervals.
- Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter .
- In applied practice, confidence intervals are typically stated at the 95% confidence level.
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- The student will calculate a 90% confidence interval using the given data.
- 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?
- Suppose we had generated 100 confidence intervals.
- When we construct a 90% confidence interval, we say that we are 90% confident that the true population mean lies within the confidence interval.