Q.(i) Find the critical $t$ value for $\alpha = 0.01$ with d.f. = 22 for a left-tailed test.

Critical T Values: The Intuition First

Imagine you're a quality control inspector at a factory that makes 500-gram packets of rice. You weigh a random sample of 30 packets and find the average is 498 grams. Is that just random chance, or is the machine under-filling? You need a rule to decide.

The problem is that even if the machine is perfectly calibrated, your sample average will almost never be exactly 500 grams — it'll bounce around due to random sampling. So how far from 500 is "too far"? That's where the critical t value comes in.

Think of it as a threshold or a cutoff line. If your sample result falls beyond this line, you say "this is too unlikely to be just chance" and conclude something real is happening. If it falls inside the line, you say "this could easily be random fluctuation."

The Precise Definition

A critical t value is the boundary point on the t-distribution that separates the region where we reject a null hypothesis from the region where we fail to reject it. It depends on three things:

1. Significance level ($\alpha$) — how much risk of being wrong you're willing to accept (commonly 0.05, meaning 5% chance of a false alarm)
2. Degrees of freedom ($df$) — related to your sample size ($n - 1$ for a one-sample test)
3. Tail type — one-tailed or two-tailed test

In plain English: the critical t value is the number on the horizontal axis of the t-distribution such that the total area in the tail(s) equals your chosen $\alpha$.

How You Actually Use It

You calculate a test statistic from your data:

Then you compare it to the critical t value:

• If $|t| > t_{\text{critical}}$ → reject the null hypothesis (the result is statistically significant)
• If $|t| \leq t_{\text{critical}}$ → fail to reject the null hypothesis (not enough evidence)

Why "t" and Not "z"?

The t-distribution is used when you don't know the population standard deviation and have to estimate it from your sample (using $s$). It's wider and has heavier tails than the normal distribution, especially for small samples. As your sample size grows, the t-distribution approaches the normal distribution.

A Concrete Example

Suppose you test whether a new teaching method changes test scores. You have 16 students, so $df = 15$. You choose $\alpha = 0.05$ for a two-tailed test.

From a t-table (or calculator), the critical t value is:

This means: if your calculated t-statistic is greater than 2.131 or less than -2.131, you reject the null hypothesis. If it's between -2.131 and 2.131, you don't have enough evidence.

The Big Picture

Critical t values are just standardized decision boundaries. They translate "how much risk am I willing to take?" into a concrete number you can compare against your data. The t-distribution itself accounts for the extra uncertainty that comes from estimating the population standard deviation from a small sample.

That's it. No magic. Just a carefully chosen cutoff line.