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.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
