Critical Values

Critical values are used for construction of parametric CIs.

For a two-sided 95% CI, they are the x-values that correspond to the locations on the x-axis, beyond which 5% of the total probability density lies.

In other words, the shaded regions in the tails of the figure below add up to an area of 5%.

In yet other words, the shaded area in the center has an area of 95%.

Critical Values: Standard Normal

Critical values for the Standard Normal distribution are also called critical Z values.

A 95% CI constructed from a Standard Normal distribution has critical Z values of \(\pm 1.96\):

How do I know?

I can use the qnorm() function to find the 2.5% and 97.5% quantiles of the standard normal:

qnorm(c(0.025, 0.975))
## [1] -1.959964  1.959964

Note that to calculate the quantiles I used endpoints of 0.025 and 0.975 instead of 0.05 and 0.95. Why?

Q1 90% Standard Normal Critical Values

  • Q1 (1 pt.): Calculate the critical z-values for a 90% CI of the standard normal distribution (not a 95% interval). Show the R-code you used to perform the calculation.

Q2-4 95% T-distribution critical values

Recall the t-distribution with it’s parameter: the degrees of freedom.

  • Q2 (1 pt.): Consult the help entry for qt() and calculate the critical values of a 95% CI for df = 10. Show the R-code you used to perform the calculation.

  • Q3 (2 pts.): How many degrees of freedom are required for the 2.5% lower critical value of a t-distribution to match the 2.5% lower critical z-value (from the standard normal) to within one decimal place (i.e. -2.0)? Show the R-code you used to perform the calculation.

  • Q4 (1 pt.): How many degrees of freedom are required for the 2.5% lower critical value of a t-distribution to match the 2.5% lower critical z-value (from the standard normal) to within two decimal places (i.e. -1.96)? Show the R-code you used to perform the calculation.

Q5-6 Building a 95% CI of the mean.

Recall the general procedure for constructing a CI? Check out the last section of slide deck 5 if you need a refresher.

Suppose you know that the sample standard deviation for a group of 50 measurements is 3.14. The mean value is 10.0.

  • Q5 (2 pts.): What are the critical t-values for a 95% CI on the mean?

  • Q6 (3 pts.): Construct the interval. Show the R-code you used to perform the calculation.

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