parameter

Examples of parameter in the following topics:

• Interpreting confidence intervals

  • Incorrect language might try to describe the confidence interval as capturing the population parameter with a certain probability.
  • This is one of the most common errors: while it might be useful to think of it as a probability, the confidence level only quantifies how plausible it is that the parameter is in the interval.
  • Another especially important consideration of confidence intervals is that they only try to capture the population parameter.
  • Confidence intervals only attempt to capture population parameters.

• Capturing the population parameter

  • A plausible range of values for the population parameter is called a confidence interval.
  • If we report a point estimate, we probably will not hit the exact population parameter.
  • 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.

• Three-Dimensional Coordinate Systems

  • A three dimensional space has three geometric parameters: $x$, $y$, and $z$.
  • Each parameter is perpendicular to the other two, and cannot lie in the same plane. shows a Cartesian coordinate system that uses the parameters $x$, $y$, and $z$.
  • Each parameter is labeled relative to its axis with a quantitative representation of its distance from its plane of reference, which is determined by the other two parameter axes.
  • The cylindrical system uses two linear parameters and one radial parameter:
  • Identify the number of parameters necessary to express a point in the three-dimensional coordinate system

• Introduction to confidence intervals

  • A point estimate provides a single plausible value for a parameter.
  • Instead of supplying just a point estimate of a parameter, a next logical step would be to provide a plausible range of values for the parameter.
  • In this section and in Section 4.3, we will emphasize the special case where the point estimate is a sample mean and the parameter is the population mean.
  • In Section 4.5, we generalize these methods for a variety of point estimates and population parameters that we will encounter in Chapter 5 and beyond.
  • This video introduces confidence intervals for point estimates, which are intervals that describe a plausible range for a population parameter.

• Parametric Equations

  • ., $x$ and $y$) are expressed in terms of a single third parameter.
  • is a parametric equation for the unit circle, where $t$ is the parameter.
  • The notion of parametric equation has been generalized to surfaces of higher dimension with a number of parameters equal to the dimension of the manifold (dimension one and one parameter for curves, dimension two and two parameters for surfaces, etc.)
  • For example, the simplest equation for a parabola $y=x^2$ can be parametrized by using a free parameter $t$, and setting $x=t$ and $y = t^2$.
  • This is a function of the derivatives of $x$ and $y$ with respect to the parameter $t$.

• Level of Confidence

  • The proportion of confidence intervals that contain the true value of a parameter will match the confidence level.
  • Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter .
  • However, in infrequent cases, none of these values may cover the value of the parameter.
  • This value is represented by a percentage, so when we say, "we are 99% confident that the true value of the parameter is in our confidence interval," we express that 99% of the observed confidence intervals will hold the true value of the parameter.
  • After a sample is taken, the population parameter is either in the interval made or not -- there is no chance.

• Estimating the Target Parameter: Point Estimation

  • Point estimation involves the use of sample data to calculate a single value which serves as the "best estimate" of an unknown population parameter.
  • The point estimate of the mean is a single value estimate for a population parameter.
  • A popular method of estimating the parameters of a statistical model is maximum-likelihood estimation (MLE).
  • The approach is called "linear" least squares since the assumed function is linear in the parameters to be estimated.
  • Contrast why MLE and linear least squares are popular methods for estimating parameters

• Review

• Estimation

  • Estimating population parameters from sample parameters is one of the major applications of inferential statistics.
  • One of the major applications of statistics is estimating population parameters from sample statistics.
  • It is rare that the actual population parameter would equal the sample statistic.
  • Instead, we use confidence intervals to provide a range of likely values for the parameter.
  • We know that the estimate $\hat { \theta }$ would rarely equal the actual population parameter $\theta $.

• Hypothesis Tests or Confidence Intervals?

  • When we conduct a hypothesis test, we assume we know the true parameters of interest.
  • When we use confidence intervals, we are estimating the parameters of interest.
  • It is worth noting that the confidence interval for a parameter is not the same as the acceptance region of a test for this parameter, as is sometimes assumed.
  • The confidence interval is part of the parameter space, whereas the acceptance region is part of the sample space.
  • Explain how confidence intervals are used to estimate parameters of interest