Given the following values of x̅, s and n, form a 90 % confidence interval for a^2. a. x⃗=21, x=

Given the following values of x̅, s and n, form a 90 %...

Key Concepts

Chi-Square Distribution

The chi-square distribution is used when constructing confidence intervals for a variance-related parameter, particularly under the assumption of normality. It provides the critical values needed to bound the sampling distribution of a statistic that is proportional to the variance, allowing the formation of an interval estimate for parameters that are functions of variance (such as a squared parameter).

Degrees of Freedom

Degrees of freedom represent the number of independent values that can vary in the computation of a statistic while estimating a parameter. In the context of estimating variance or parameters based on sample data, degrees of freedom (often equal to n-1 for a sample of size n) are used to determine the appropriate critical values from reference distributions such as the chi-square or t-distribution.

Transformation of Parameters

Sometimes the parameter of interest is a transformation of another estimated parameter. For instance, if one is interested in a^2 rather than a, a confidence interval for a can be transformed (by applying a monotonic function, such as squaring) to obtain a corresponding interval for a^2. This approach depends on the property that a monotonic transformation preserves the interval’s coverage probability.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. In statistical inference, it quantifies the uncertainty associated with an estimate, such that, for example, a 90% confidence interval implies that 90% of similarly constructed intervals would capture the true parameter value.

Sample Statistics

Sample statistics such as the sample mean and sample standard deviation summarize the characteristics of observed data. They serve as estimates for the underlying population parameters and are fundamental in constructing confidence intervals, especially when the population parameters are unknown.

Transcript

00:01 So we're given some information and we'll kind of figure out these values.

00:06 We know that in part a we have a sample size, well our standard deviation is 2 .7 and our sample size is 50, which means our degrees of freedom, is 49.

00:18 And we need our kye squared values with alpha over 2 and our kye squared values of our kai squared values of 1 minus alpha over 2.

00:30 Over 2 so that we can determine our interval.

00:32 And so when i looked these up, and we're having a 90 % confidence interval in each case, which means the alpha level is 0 .1, or we're using 0 .05.

00:44 And i have these values here to be 66 .3 and 33 .9.

00:51 I had to use software for that.

00:53 And so when we calculate that interval, we're going to take one less than the sample size.

00:58 So we're going to take 49, times our 2 .7 squared divided by that first number, 66 .3.

01:06 And our variance will be lying in here, and then we'll take that 49 times that 27 .2 squared, and then divided by that 33 .9.

01:20 And let me do a little ratio there.

01:26 All right, so let's get those calculations.

01:28 So 49 times 2 .7 squared, and then divided by is 66 .3.

01:35 So the first variance is between 5 .39 and and then we can just go back up and change that one value to 33 .9.

01:48 And we end up getting 10 .54.

01:53 Now let's find for part b.

01:56 I'll just put a little line there because i see that we have to go back to this and i'll use a different color.