Given below are two statements: Statement I: For a given sample size and level of significance (a), the critical value of Student's t always exceeds that of z Statement II: For a given level of significance (a), the critical value of Student's t increases as n increases. In the light of the above statements, choose the correct answer from the options given below:
Statement I is true but Statement II is false
Understanding Critical Values: Student's t vs. Z Distribution
This question asks us to evaluate two statements regarding the critical values of Student's t-distribution and the standard normal (z) distribution, which are fundamental concepts in inferential statistics, particularly in hypothesis testing.
Analysis of Statement I: Comparing t and Z Critical Values
Statement I says: "For a given sample size and level of significance ($\alpha$), the critical value of Student's t always exceeds that of z".
Let's consider the characteristics of the two distributions:
- The standard normal (z) distribution is a symmetric distribution with a mean of 0 and a standard deviation of 1.
- Student's t-distribution is also a symmetric distribution with a mean of 0, but its shape depends on the sample size. It has heavier tails than the z-distribution, especially for small sample sizes.
The critical value represents the point on the distribution beyond which a certain area (equal to $\alpha/2$ for a two-tailed test or $\alpha$ for a one-tailed test) lies. Because the t-distribution has fatter tails than the z-distribution, to capture the same area in the tails (corresponding to the same level of significance $\alpha$), you need to go further out from the mean (0) in the t-distribution compared to the z-distribution.
Thus, for any finite sample size (and hence finite degrees of freedom, which is typically n-1), the critical value of Student's t will indeed be larger than the critical value of z for the same level of significance $\alpha$. As the sample size increases, the t-distribution approaches the z-distribution, and the t-critical value approaches the z-critical value, but for any finite sample size, the t-critical value is greater.
Therefore, Statement I is true.
Analysis of Statement II: Effect of Sample Size on t Critical Value
Statement II says: "For a given level of significance ($\alpha$), the critical value of Student's t increases as n increases".
As discussed above, the shape of the t-distribution depends on the degrees of freedom (df), which are usually calculated as sample size minus one ($df = n-1$). As the sample size (n) increases, the degrees of freedom ($n-1$) also increase. A larger degree of freedom makes the t-distribution less spread out and its tails become thinner, resembling more closely the standard normal (z) distribution.
Consider a fixed level of significance $\alpha$. As the t-distribution becomes less spread out (closer to the z-distribution) with increasing sample size, the value you need to reach in the tail to capture the area $\alpha$ (or $\alpha/2$) gets closer to the z-critical value. Since the t-critical value is always greater than the z-critical value for finite n, and the t-distribution approaches the z-distribution from having fatter tails to thinner tails as n increases, the critical value of Student's t actually decreases as n increases.
Let's look at an example (approximate critical values for $\alpha = 0.05$ two-tailed):
| Sample Size (n) | Degrees of Freedom (df = n-1) | Approx. t-critical value ($\alpha=0.05$, two-tailed) |
|---|---|---|
| 2 | 1 | 12.71 |
| 10 | 9 | 2.26 |
| 30 | 29 | 2.045 |
| ∞ (approaches z) | ∞ | 1.96 (z-critical value) |
As you can see from the table, the t-critical value decreases as the sample size (n) increases, approaching the z-critical value of 1.96.
Therefore, Statement II is false.
Conclusion
Based on the analysis:
- Statement I: For a given sample size and level of significance ($\alpha$), the critical value of Student's t always exceeds that of z - True.
- Statement II: For a given level of significance ($\alpha$), the critical value of Student's t increases as n increases - False.
Thus, Statement I is true but Statement II is false.
Revision Table: Critical Values Summary
| Concept | Student's t Critical Value | Z Critical Value |
|---|---|---|
| Comparison (given $\alpha$, finite n) | Greater than Z critical value | Less than t critical value |
| Effect of Sample Size (n) | Decreases as n increases (approaches Z) | Unaffected by sample size (constant for given $\alpha$) |
| Distribution Shape | Heavier tails (especially for small n) | Standard shape, tails not as heavy as t |
| Approach with large n | Approaches Z critical value | $-$ |
Additional Information: T and Z Distributions in Hypothesis Testing
Both the t-distribution and the z-distribution are used in hypothesis testing to determine if a sample statistic is significantly different from a population parameter. The choice between using a t-test (and t-distribution) and a z-test (and z-distribution) depends on whether the population standard deviation is known and the sample size.
- Z-test: Used when the population standard deviation ($\sigma$) is known and the sample size is large (n > 30), or when the population is normally distributed and $\sigma$ is known (regardless of sample size). The test statistic follows a z-distribution.
- T-test: Used when the population standard deviation ($\sigma$) is unknown and must be estimated from the sample standard deviation (s). It is typically used for small sample sizes (n < 30), although it is applicable for any sample size when $\sigma$ is unknown. The test statistic follows a t-distribution with $n-1$ degrees of freedom.
The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from a sample, which is why its tails are heavier and critical values are larger than the z-distribution for finite sample sizes.
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