Critical Value

statistics

quantile function : the opposite of cdf (aka $Q={\text{cdf}}^{-1}$).

The critical value is the region of a sampling distribution of a test statistic. In hypothesis tests, critical values determine whether the results are statistically significant. For confidence intervals, they help calculate the upper and lower limits.

Use t score if:

1. sample size is < 30
2. you don’t know the population standard deviation.

Critical z value

We have a population. We take a random sample. We then calculate the statistic. Confidence interval says “x% of the statistic will fall within the confidence interval range”.

critical value = number of standard deviations to go to cover the stat. You can look up the value in a z-table to determine the confidence available.

so (100% - $confidenceInterval) / 2 (b/c 2 tails) Then find the z value in ^^^

So for an 87% confidence interval, we get (100-87)/2=6.5 100-6.5 = 93.5 Then we’d calculate the quantile function of 93.5 to get the critical value.

For ti-89, you can calculate z-value by doing an inverse normal with mean = 0 and stddev = 1, with the area you want in the top.

Critical $t$ value

You look up the confidence interval and the degrees of freedom (df) in a t-table which will output the value.

Using a calculator, you can determine this by doing an “inverse t” lookup. Note that the area is:

• split? 100-((100-$confidence)/2)
• left or right? (100-(100-confidence))

critical value for chi square

Require: significance level, tail’dness, and degrees of freedom

1. draw curve
2. draw rejection region
3. label areas
4. compute critical values using “inverted chi”.