Question

how do you calculate standardized test statistic

how do you calculate critical value?

Answer 1

1.) Standardized test statistic

Standardized test statistics are a way for you to compare your results to a “normal” population. Z-scores and t-scores are very similar, although the t-distribution is a little shorter and fatter than the normal distribution. They both do the same thing. In elementary statistics, you’ll start by using a z-score. As you progress, you’ll use t-scores for small populations. In general, you must know the standard deviation of your population and the sample size must be greater than 30 in order for you to be able to use a z-score. Otherwise, use a t-score.

Example -

You take the SAT and score 1100. The mean score for the SAT is 1026 and the standard deviation is 209. How well did you score on the test compared to the average test taker?

Step 1: Write your X-value into the z-score equation. For this example question the X-value is your SAT score, 1100.

Step 2: Put the mean, μ, into the z-score equation.

Step 3: Write the standard deviation, σ into the z-score equation.

Step 4: Find the answer using a calculator:
(1100 – 1026) / 209 = .354. This means that your score was .354 std devs above the mean

-------------------------Critical value ------------------------------

A critical value is a line on a graph that splits the graph into sections. One or two of the sections is the “rejection region“; if your test value falls into that region, then you reject the null hypothesis.

The critical value of z is term linked to the area under the standard normal model. Critical values can tell you what probability any particular variable will have.


The above graph of the normal distribution curve shows a critical value of 1.28. The graph has two parts:

• Central region: The z-score is equal to the number of standard deviations from the mean. A score of 1.28 indicates that the variable is 1.28 standard deviations from the mean. If you look in the z-table for a z of 1.28, you’ll find the area is .3997. This is the region to the right of the mean, so you’ll double it to get the area of the entire central region: .3997*2 = .7994 or about 80 percent.
• Tail region: The area of the tails (the red areas) is 1 minus the central region. In this example, 1-.8 = .20, or about 20 percent. The tail regions are sometimes calculated when you want to know how many variables would be less than or more than a certain figure.

• A critical value of z is sometimes written as z_a, where the alpha level, a, is the area in the tail. For example, z_.10=1.28.

  When are Critical values of z used?
  A critical value of z (Z-score) is used when the sampling distribution is normal, or close to normal. Z-scores are used when the population standard deviation is known or when you have larger sample sizes. While the z-score can also be used to calculate probability for unknown standard deviations and small samples, many statisticians prefer to use the t distribution to calculate these probabilities.