ISS 2016 Statistics Paper-1 Solution: Question 38 (Confidence Interval Using t-Distribution)

Continuing our question-by-question walk through the ISS Statistics Paper-1 archive, today we pick up right where we left off in the 2016 paper and tackle a classic estimation problem: building a confidence interval for a population mean when we only know the sample mean and sample standard deviation. This type of question shows up almost every year in some form, so getting comfortable with every step here pays off well beyond this one paper.

Quick Summary

• Topic:Confidence interval for the population mean using the t-distribution

• Question Reference:ISS 2016, Statistics Paper-1, Question 38

• Correct Answer:Option (a) — (9.5, 10.5)

The Question

To find out the mean yield per plant, 64 plants were selected at random from a field. The mean yield and standard deviation were found to be 10 and 2, respectively. (For α = 0.05 and 63 degrees of freedom, the critical value of t is 2.) What is the 95% confidence interval for the population mean?

• (a) (9.5, 10.5)

• (b) (9.75, 10.25)

• (c) (6.0, 4.0)

• (d) (8.0, 12.0)

Step 1: Identify What We're Given

Before doing any calculation, it helps to simply list out every piece of information the question hands us:

• Sample size, n = 64

• Sample mean, x̄ = 10 (this is our best point estimate of the true population mean)

• Sample standard deviation, s = 2

• Degrees of freedom = n − 1 = 63

• Critical t-value at α = 0.05 (two-tailed, 95% confidence) with 63 d.f. = 2 (given directly, so we don't need t-tables)

Since the population standard deviation is not known, and we only have the sample standard deviation s, the correct sampling distribution to use here is the t-distribution rather than the normal (Z) distribution. This is exactly why the question hands us a t critical value instead of a Z critical value like 1.96.

Step 2: Recall the Confidence Interval Formula

For a population mean, when the population standard deviation is unknown and estimated by the sample standard deviation, the (1 − α) confidence interval is:

CI = x̄ ± t(α/2, n−1) × (s / √n)

Here, the quantity s/√n is called the standard error of the mean — it tells us how much the sample mean is expected to vary from sample to sample. The critical t-value then tells us how many standard errors wide our interval needs to be to capture the true mean with 95% confidence.

Step 3: Compute the Standard Error

We first need √n. Since n = 64, and 64 is a perfect square:

√64 = 8

So the standard error is:

Standard Error = s / √n = 2 / 8 = 0.25

This tiny number makes sense: with as many as 64 plants sampled, our estimate of the mean yield should be fairly precise, so the spread of possible sample means around the true mean is small.

Step 4: Compute the Margin of Error

The margin of error is simply the critical t-value multiplied by the standard error:

Margin of Error = t × Standard Error = 2 × 0.25 = 0.5

Pro Tip: The single most common slip on this type of question is multiplying the critical value by the raw standard deviation (s = 2) instead of the standard error (s/√n = 0.25). That mistake alone is what produces the tempting-but-wrong option (d), (8.0, 12.0) — which comes from 10 ± (2 × 2)/1, essentially forgetting to divide by √n at all. The way this step is usually taught is to write the formula out in full symbolic form first, plug in numbers only at the very last step, and always double-check that dividing by √n has actually happened before you multiply by t — that habit alone eliminates most careless errors on interval-estimation questions.

Step 5: Build the Confidence Interval

Now we simply add and subtract the margin of error from the sample mean:

Lower limit = x̄ − Margin of Error = 10 − 0.5 = 9.5

Upper limit = x̄ + Margin of Error = 10 + 0.5 = 10.5

So the 95% confidence interval for the population mean yield per plant is (9.5, 10.5), which corresponds to option (a).

Step 6: Confirm Against the Official Answer Key

Checking the official ISS 2016 Statistics Paper-1 answer key, Question 38 is indeed marked as option (a). Our independently derived answer matches the official key exactly, so we can be fully confident in this working.

Why This Question Matters

Confidence interval construction is one of the most frequently tested ideas across ISS, IIT JAM, and similar statistics entrance exams, precisely because it blends several core concepts into one short problem: sampling distributions, standard error, and critical values from a named distribution. Questions like this one are also deliberately designed with distractor options that correspond to specific, predictable mistakes — using s instead of s/√n, using the wrong sign convention, or forgetting to take the square root of n at all. Recognizing the shape of each wrong option is often just as useful for exam preparation as knowing the right formula, since it lets you sanity-check your own work by asking "which mistake would I need to make to land on each wrong answer?" Students who drill this kind of check until it becomes automatic — a habit commonly emphasized in coaching programs such as Sunrise Classes — tend to catch these errors well before submitting an answer.

Frequently Asked Questions

Why do we use the t-distribution here instead of the normal distribution?

The normal (Z) distribution is appropriate only when the population standard deviation σ is known. Here we are only given the sample standard deviation s as an estimate of σ, which introduces additional uncertainty. The t-distribution accounts for this extra uncertainty, especially important information when working with a single sample rather than the true population parameters.

Would the answer change much if we used the normal distribution instead of the t-distribution?

With 63 degrees of freedom, the t-distribution is already very close to the standard normal distribution, so a Z critical value of 1.96 would give a nearly identical interval to a t critical value of 2. The question deliberately simplifies the t-value to exactly 2 to keep the arithmetic clean, but the underlying logic of using t rather than Z remains the technically correct choice whenever s replaces σ.

What does "95% confidence" actually mean in this context?

It means that if we were to repeat this sampling process many times, drawing a new random sample of 64 plants each time and constructing a new interval the same way, approximately 95% of those constructed intervals would contain the true, unknown population mean yield. It is a statement about the reliability of the method, not a probability statement about this one specific interval.

Why is option (b), (9.75, 10.25), a tempting wrong answer?

Option (b) results from correctly computing the standard error as 0.25 but then forgetting to multiply it by the critical t-value of 2, effectively using a margin of error of 0.25 instead of 0.5. It is a reminder to always complete every multiplication in the formula rather than stopping partway through.

How would this calculation change if the sample size had been smaller, say n = 16?

With n = 16, the standard error would become 2/√16 = 2/4 = 0.5, doubling the standard error compared to this question. Additionally, with only 15 degrees of freedom, the critical t-value itself would be noticeably larger than 2 (closer to 2.131 for a 95% interval), since smaller samples carry more uncertainty. Both effects combine to produce a visibly wider confidence interval, illustrating why larger samples generally yield more precise estimates.

Is it a problem that the confidence interval is symmetric around the sample mean?

No — this symmetry is expected and correct whenever we are estimating a population mean using the t-distribution (or the normal distribution), because both distributions are symmetric about zero. Confidence intervals only become asymmetric when working with parameters that follow skewed sampling distributions, such as a population variance based on the chi-square distribution.

If any step above felt unclear, or if you worked through this problem differently, drop a comment below and share this post with a fellow ISS aspirant who might be revising the same topic — working through these one question at a time is exactly how this series is meant to be used.