Consider two independent samples. One sample has size 100 and comes from a distribution with mean
𝜇
1
and standard deviation
𝜎
1
, and we observe a sample mean of 5 and a sample standard deviation of 2. The other sample has size 50 and comes from a distribution with mean
𝜇
2
and standard deviation
𝜎
2
, and we observe a sample mean of 10 and a sample standard deviation of 3.

Assuming
𝜎
1
=
𝜎
2
, what critical value should we use to form an 80% confidence interval for
𝜇
1
−
𝜇
2
?

Question

Consider two independent samples.  One sample has size 100 and comes from a distribution with mean 
𝜇
1
and standard deviation 
𝜎
1
, and we observe a sample mean of 5 and a sample standard deviation of 2.  The other sample has size 50 and comes from a distribution with mean 
𝜇
2
and standard deviation 
𝜎
2
, and we observe a sample mean of 10 and a sample standard deviation of 3.

Assuming 
𝜎
1
=
𝜎
2
, what critical value should we use to form an 80% confidence interval for 
𝜇
1
−
𝜇
2
?

BlackTom AI · Accepted Answer

Since equal variances are assumed, use the pooled t-distribution with degrees of freedom df = n1 + n2 − 2 = 100 + 50 − 2 = 148. For an 80% two-sided confidence interval the appropriate critical value is the t-quantile with tail area α/2 = 0.10, i.e. t0.1,148.

1) Relevant knowledge points and concepts
- When population standard deviations are unknown but assumed equal, the difference of sample means uses the pooled t-distribution.
- A two-sided confidence interval with confidence level C uses α = 1 − C and the critical quantile tα/2,df.
- Degrees of freedom for pooled two-sample t: df = n1 + n2 − 2.

2) Step-by-step reasoning
- Confidence level C = 0.80 ⇒ α = 0.20.
- Two-sided interval ⇒ split α into two tails: α/2 = 0.10.
- With equal-variance assumption, df = 100 + 50 − 2 = 148.
- Therefore the critical value is tα/2,df = t0.10,148.

3) Intermediate calculations
- α = 1 − 0.80 = 0.20
- α/2 = 0.10
- df = 100 + 50 − 2 = 148
- Critical value notation: t0.10,148 (numerically ≈ 1.28, since t approaches the z-value 1.2816 for large df).

4) Why this answer is correct
- The t0.10,148 value correctly reflects the two-sided 80% CI (α/2 = 0.10) and the correct pooled degrees of freedom (148) under the equal-variance assumption.

5) Why the other options are incorrect
- Any option using t0.2 corresponds to α/2 = 0.20 (i.e., a 60% CI), so it’s the wrong tail area.
- Any option using 150 degrees of freedom miscomputes df (150 = n1 + n2, but pooled df is n1 + n2 − 2 = 148), so df = 150 is incorrect.

6) Conclusion
The correct critical value to use is t0.1,148, which matches the provided answer.

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