The population standard deviation
𝜎
is not known, and the sample size is 30. If we wish to estimate the population mean
𝜇
with 90% confidence, what is the appropriate critical value t to use?

Round to THREE DECIMAL PLACES.

Example: If the critical value
𝑡
𝛼
2
=
9.123
, enter 9.123 as your answer.

Question

The population standard deviation 
𝜎
 is not known, and the sample size is 30.  If we wish to estimate the population mean 
𝜇
 with 90% confidence, what is the appropriate critical value t to use?

Round to THREE DECIMAL PLACES.

Example: If the critical value 
𝑡
𝛼
2
=
9.123
 , enter 9.123 as your answer.

BlackTom AI · Accepted Answer

The problem asks for the appropriate t critical value to estimate a population mean with 90% confidence when the population standard deviation is unknown and the sample size is 30.
First, recognize that with unknown sigma and a small-to-moderate sample size, we should use the t-distribution rather than the z-distribution. The degrees of freedom are df = n - 1 = 30 - 1 = 29.
To construct a 90% confidence interval, the two-tailed alpha is 0.10, so each tail has 0.05. Thus we need t_{0.05, 29}, the critical value with 29 degrees of freedom that leaves 5% in the right tail (and 5% in the left tail).
Consulting a t-table (or using a calculator) for df = 29 at the 0.05 significance level in the upper tail yields t ≈ 1.699. When rounded to three decimals, this is 1.699.
Why this value makes sense: as degrees of freedom increase, the t-quantile approaches the standard normal quantile, but with df = 29 it remains slightly larger than the z-value for 90% CI, reflecting additional uncertainty due to estimating sigma from the sample.
Common pitfalls to avoid: using z_{0.05} = 1.960 would be incorrect here because it assumes known sigma (z distribution) and a large-sample normal approximation; using a df too small (e.g., df = 28 or df = 50) would yield slightly different t_{0.05, df}, but with df = 29 the value is 1.699.
If you were to round differently, ensure you preserve three decimal places as required; 1.699 is already at three decimals, matching the instruction.

BU-STAT-2103-750-49509-202503 Quiz 4

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