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Z Critical Value Calculator + Online Solver With Free Steps

The Z Critical Value Calculator is an online tool that helps to calculate the critical value for the z statistic (normal distribution), choose the normal distribution, and enter the mean and standard deviation.

A z test is performed on a normal distribution when the population standard deviation is known and the sample size is more significant than or equal to 30.

What Is a Z Critical Value Calculator?

A Z Critical Value Calculator is a calculator that computes the critical values for various hypothesis testsThe test statistic distribution and the degree of significance can be used to interpret the crucial value of a certain test.

A test named a two-tailed test has two critical values, whereas a one-tailed test will only have one critical value. You must understand the distribution of your test statistic under the null hypothesis to calculate crucial levels.

Critical values are defined as the values on the plot at the significance level that have the same probability as your test statistic. At such crucial values, it is expected that these values are at least as extreme.

To determine what at least an extreme means, the alternative hypothesis is conducted.

For example, if the test is one-sided, there will only be one critical value; if the test is two-sided, there will be two critical values:

  • One to the right and the other to the left of the distribution’s median value.

Critical values are readily represented as points whose area under the density curve of the test statistic from those points to the tail’s equals:

  • Left-tailed test: The critical value’s critical value is equal to the area under the density curve on the left
  • The area covered under the density curve taken from the critical value to the right side is equivalent to the right-tailed test’s result.
  • The area covered under the density curve considered from the left critical value to the left side is equal to α2, as it is the area under the curve from the right critical value to the right; so, total area equals

How To Use a Z Critical Value Calculator?

You can use the Z-Critical-Value Calculator by following the given detailed guide. The calculator will provide the desired results if the steps are followed properly. You can therefore follow the given instructions to get the confidence interval for the provided data points.

Step 1

Fill the specified boxes with the given data and enter the number of tails and directions.

Step 2

Now, press the “Submit” button to determine the Z Critical Value of the given data points, and also the whole step-by-step solution for the Z Critical Value calculation will be displayed.

How Does a Z Critical Value Calculator Work?

The Z Critical Value Calculator works based on the function Q called the Quantile function, which is determined by taking the inverse of the Cumulative Distribution Function. Therefore, it can be defined as:

\[ Q = cdf^{-1} \]

Once the value of α has been selected, the critical value formulae are the following:

  1. left-tailed test: \[(- \infty, Q(\alpha)] \]
  2. right-tailed test: \[[Q(1 – \infty), \infty)\]
  3. two-tailed test: \[ (-\infty, Q(\frac{\alpha}{2})] \cup [Q(1 – \frac{\alpha}{2}), \infty) \]

For the distributions that are symmetric about 0, the critical values for the two-tailed test are symmetric as well:

\[ Q(1 – \frac{\alpha}{2}) = -Q(\frac{\alpha}{2})\]

Unfortunately, the most common probability distributions used in hypothesis testing contain cdf formulas that are a little challenging to understand.

Manually identifying critical values would need the use of specialized software or statistical tables. This calculator provides you access to a wider range of potential values to deal with while replacing the use of a Z value table.

For finding the test’s critical value based on your selected alpha level, a z score table is used. Do not forget to change the alpha $\alpha$ value depending on whether you are conducting a single- or two-tailed test.

Since the typical normal distribution is symmetric around its axis in this situation, we may simply divide the value of alpha in half.

From there, looking up the correct row and column in the Table will allow you to identify the critical values for your test. All you need to do to use our critical values calculator is enter your alpha value, and the tool will automatically determine the critical values.

Solved Examples

Let’s explore some examples to better understand the Z Critical Value Calculator.

Example 1

Find the critical value for the following:

Consider a left tailed z-test where $\alpha = 0.012 $.

Solution

First, subtract $\alpha$ from 0.5.

Thus

 0.5 – 0.012 = 0.488 

Using the z distribution table, the value of z is given as:

 z = 2.26

Since this is a left-tailed z test, so the z is equivalent to -2.26.

Answer

Therefore, the critical value is given as:

Critical value = -2.26 

Example 2

Find the critical value for a two-tailed f test conducted on the following samples at a $ \alpha$ = 0.025.

Sample 1

Variance = 110

Sample size = 41

Sample 2

Variance = 70

Sample size = 21

Solution

n1= 41, n2 = 21 

n1 – 1= 40, n2 – 1 = 20

Sample1 df = 40

Sample2 df = 20 

Using the F distribution table for $\alpha$= 0.025, the value at the intersection of the $40^{th}$ column and $20^{th}$ row is

F(40, 20) = 2.287 

Answer

The critical value is given as:

Critical Value = 2.287 

Example 3

Find $Z_{\frac{\alpha}{2}}$ for 90% confidence.

Solution

90% written as a decimal is 0.90.

\[ 1 – 0.90 = 0.10 = \alpha \] and \[ \frac{\alpha}{2} = \frac{0.10}{2}= 0.05\]

Look for 0.05 = 0.0500 or two numbers surrounding it in the body of the Table.

Since 0.0500 is less than 0.5, the number 0.0500 is not in the Table, but it is between 0.0505 and 0.0495, which are in the Table.

Next, check the differences between these last two numbers and 0.0500 to see which number

is closer to 0.0500$\cdot$ 0.0505 – 0.0500 = 0.0005  and 0.0500 – 0.0495 = 0.0005.

Since the differences are equal, we average the corresponding standard scores.

Because 0.0505 is to the right of -1.6 and under 0.04, its standard score is -1.64.

Because 0.0495 is to the right of -1.6 and under 0.05, its standard score is -1.65.

\[ (-1.64 + \frac{-1.65}{2} )= -1.645 \]

Thus $Z_{\frac{\alpha}{2}} = 1.645$ for 90% confidence.

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