Find the lower confidence limit for the population variance for each of the following normal pop

Find the lower confidence limit for the population variance...

Key Concepts

Population Variance

Population variance is a measure of the spread or dispersion of a set of values in the entire population. It quantifies how much the individual observations deviate from the population mean. In inferential statistics, the true population variance is often unknown and must be estimated from sample data, leading to the use of confidence intervals to provide a range of plausible values.

Confidence Interval for Variance

A confidence interval for the variance provides a range of values within which the true population variance is likely to lie with a specified level of confidence. When samples are drawn from a normal population, the interval is constructed using the chi-square distribution. The lower confidence limit represents the smallest value of the variance that is consistent with the data at the chosen level of confidence.

Chi-Square Distribution

The chi-square distribution is a probability distribution that arises in the estimation of variances for normally distributed populations. It is used to relate the sample variance to the population variance through its quantiles. The shape of the chi-square distribution depends on the degrees of freedom and is used to determine the critical values needed to construct the confidence interval for the variance.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in the calculation of a statistic. In the context of estimating variance from a sample, the degrees of freedom are typically equal to the sample size minus one (n-1). This parameter is essential in the chi-square distribution for variance estimation and influences the width of the confidence interval.

Significance Level and Lower Confidence Limit

The significance level (?) specifies the probability of rejecting a true hypothesis and is used to determine the corresponding critical value from the chi-square distribution. For a confidence interval, particularly the lower confidence limit for the variance, the selected ? is used to identify the chi-square percentile that provides the boundary below which the true variance is unlikely to fall. This ensures that the interval has the desired level of confidence.

Transcript

00:01 So we're finding confidence intervals for the population variance.

00:04 And in part a, we have a sample size that is 21, and we have an alpha level of 0 .05, and we have our variance from our sample is 16.

00:15 We want to calculate the lower confidence limit, which for a variance is going to be taking n -1 times the standard deviation squared, or the variance, and the sample variance, and then the kai squared for a.

00:29 N minus 1 degrees of freedom and alpha over 2.

00:33 So we're actually looking here, ironically, for the lower limit, we have to find where this kai squared value will be dividing by a larger value, which will cause the variance to be lower than what this value is.

00:48 So our lcl here is going to be to take 20 times the variance, which is 16.

00:56 And i know my alpha level is 0 .05, so i need the kye squared statistic that has 20 degrees of freedom, and it has 0 .025 in that upper tail.

01:10 And when i look that up, i believe i get 34 .170.

01:16 And when we take 320 and divide it by that 34 .17, that gives me a 9 .1 .7, that gives me a 9 .1 .7, that gives me, 0 .365 for my lower estimate.

01:32 And again, it is lower than the 16.

01:35 Now on part b, we have a sample size of 16, the same alpha level, and we have a sample standard deviation that is 8.

01:47 And again, we want that same kai squared statistic that has 15 degrees of freedom, and we want 0 .025 to be in that upper tail.

01:58 And i believe that that value, if i looked it up correctly, is 27 .488.

02:06 So our lower confidence limit will end up being our sample size, less 1, 15, times the variance, which will be 64 of the sample variance, and then divided by 27 .488...

Please give Ace some feedback