The Critical Value Approach
This type of hypothesis testing is similar to the P-value test and yields similar results. In this approach, we look at a critical value and see where it lies and draw our conclusion. The steps are as follows:
- Determine the null hypothesis, Ho and alternate hypothesis Ha
- Find the test statistic
- Find the critical value
- Compare test statistic with critical value
- Conclusion
Critical Value
The area under the normal curve is equal to 1. If we are 95% confident about our analysis, we say that the confidence level is 0.95. The remaining 5% is our alpha. Alpha is the probability of rejecting the null hypothesis when it is true. In other words is the probability of making a Type I error.
- So, confidence level + alpha = 1
- If alpha is 0.05 the corresponding Z-score is called the critical value, Zc or Zα. In the graph the red region is alpha and the blue line at the start of alpha has a Z critical value 1.644
- The same method can be applied the Tα
Conclusion: Compare Test Statistic to Critical Value
The critical value of the test will be used as a sort of "limit line." For example, a right-tailed test at α=0.05 has a critical value of 1.65 as shown in the above picture. Because this is a right-tailed test, any z0 value obtained that is larger than 1.65 will be inside of the "reject region (shown in red, alpha)," which should then lead to a rejection of the null hypothesis.
- Right tailed: If test statistic > critical value, then do not reject the null.
- Left Tailed: If test statistic < critical value, then reject the null.
- Two-tailed: test statistic is NOT between the two critical values, then reject the null
HINT: It must be past the start of the colored in rejection area to reject. See image above. The blue line indicates the critical value. To reject, the Z or T stat must be past that red line, in the red colored in part, which is the rejection area.
Tech Help
In StatCrunch
Let’s try to find the Z- critical value such that the area to the right if it (α) is 0.10. To find Z-critical value:
- Go to Stat, then Calculators, then Normal For T critical value: Stat, then Calculators, then T Select the ’≥’ symbol and enter the value 0.10 on the right side after the equal sign
In Excel
NORMSINV(p)returns the Z-critical value for a known probability or alpha value
- Excel calculates Z-critical from the negative infinity end. So, when the area under the curve is 0.10 from the left, the Z critical is negative. By symmetry, when the area is 0.10 from the right then the Z-critical would be 1.28
- If we want a positive Z-critical for alpha on the right then then we could calculate it by setting the area under the curve as 0.90 instead of 0.10
*Note: this guide may use symbols you aren’t familiar with. View our complete list of Statistics Symbols here.
In Python
The scipy.stats library must be imported to use these functions.
For a normal distribution with the specified mean and standard deviation, norm.ppf(p, mean, sd) returns the critical z-score for which the probability of z being below that z- score is p
Both norm.ppf() and norm.isf() can also be used with non-standard normal distributions as well.(when mean is not 0 and standard deviation is not1)
For a normal distribution with the specified mean and standard deviation norm.isf(p, mean, sd) returns the critical z-score for which the probability of z being above that z- score is p
For a T-distribution, t.ppf(p, df, mean, sd) returns the critical t-statistic for which the probability of t being below that t-score is p, with the specified degrees of freedom, mean, and standard deviation.
Now You Try! Practice Problem
A detergent manufacturing company conducted a survey in 2017 and found that 77% of its customers were satisfied with their product.
- In 2019, they conducted the survey again and found that out of 894 customers 652 customers were satisfied. Is their evidence that their customer’s satisfaction rate has changed?
- Test this at alpha =0.05 level of significance.
1. First determine the null and alternate hypothesis:
Since this problem involves proportions we have to find out the proportion of customers who were satisfied. Since 652 out of 894 were satisfied the sample proportion is p = 652/894 =0.7293
- Null hypothesis, Ho: p = 0.77
- Alternate hypothesis, H1: p ≠ 0.77
- This is a two-tailed test
2. Find the test statistic:
Given p0 = 0.77; p =0.7293; n, number of observations = 894
3. Find the Critical Value:
Use technology or the Z-score chart to find the critical value is 1.64
4. Compare Z critical to test statistic:
The test statistic is -2.89. Z critical value is -1.64 and 1.64 by symmetry. We have to check if the test statistic lies between these 2 critical values. It does not. This value lies to the left of -1.64. This is in the ‘rejection region’, beyond the critical value point established.
5. Conclusion:
Since the Z test statistic is less than the critical value and lies in the rejection region, we reject the null hypothesis. Therefore, we have evidence to support the null hypothesis. There is evidence to suggest the customer’s satisfaction rate has changed.
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