7.6
Critical Values
A critical value is a definite value obtained from a particular probability distribution at a predecided confidence level (or a predecided significanc…
Confidence limits at a confidence level—say at 95%—cover 95% of the area under the curve.
Remaining 5% of the area, which is designated by α, is distributed equally on either tail of the data distribution. In the case of 95% level, it would be 2.5%.
In other words, α/2 = 0.025.
Recall that calculating estimates for population parameters require z scores obtained using the z distribution.
Such a z score calculated at the right tail of the distribution—that is—at positive α/2 is known as the critical value. It is denoted as zα/2.
A critical z value for a given confidence level is a fixed value. It does not change over any number of samples or for a statistic.
To calculate the critical z value for any confidence level, look for 1−α/2 value in the z table.
For the 95% level, look for 0.975, not 0.95, to note the value of 1.96.
Similarly, for 90% and 99% confidence levels, the critical z values are 1.645 and 2.575, respectively.
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Q1: What is a critical value in statistics?
A critical value is a fixed value obtained from a probability distribution at a predetermined confidence level that demarcates sample statistics likely to occur from those unlikely to occur. For normal distributions, critical values are z scores obtained from the z distribution table. Critical values are essential for calculating confidence limits and do not change across different samples or statistics.
Q2: How do you find the critical z value for a 95% confidence level?
To find the critical z value for 95% confidence, look up 1−α/2 in the z table, which equals 0.975 for this level. This lookup yields a critical z value of 1.96. Similarly, 90% confidence produces 1.645, and 99% confidence produces 2.575. These fixed values remain constant regardless of sample size.
Q3: Why is alpha divided by two when calculating critical values?
Alpha (α) represents the total area outside the confidence interval. Dividing by two distributes this area equally on both tails of the distribution. For 95% confidence, α equals 0.05, so α/2 equals 0.025 on each tail. This two-tailed approach is standard for interval estimates requiring both positive and negative critical values.
Q4: What distributions are used to find critical values for non-normal data?
When data are not normally distributed, critical values come from alternative probability distributions. The t distribution is used for small samples or unknown standard deviations, the F distribution for variance comparisons, and the Chi-square distribution for categorical data. The choice depends on the sampling distribution and parameter being estimated.
Q5: Can critical values be calculated at different locations in a distribution?
Yes, critical values can be calculated at the right tail, left tail, or both tails. Right-tail critical values are positive, while left-tail values are negative. For interval estimates, critical values are typically calculated at both tails, generating paired positive and negative scores used in confidence interval calculations.
Q6: What factors determine the value of a critical value?
Critical value magnitude depends on the nature of the hypothesis, the population parameter being estimated, and the sampling distribution type. In some cases, sample size also influences the critical value. The confidence level chosen directly affects which critical value is selected from the probability distribution table.
Q7: Why are critical values essential for confidence interval estimation?
Critical values are crucial because they define the boundaries of confidence intervals and enable calculation of confidence limits. Without a critical value, you cannot determine how far sample estimates should extend from the point estimate to capture the true population parameter at a specified confidence level.