Find the critical $$ t $$-value for a $$ 90 \% $$ confidence interval using a $$ t $$-distribution with 16 degrees of freedom. Round your answer to three decimal places, if necessary.

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1. The confidence level is $$90\%$$, so the total area in the two tails is $$ \alpha = 1 - 0.90 = 0.10 $$. Since the t-distribution is symmetric, each tail has an area of

$$
\frac{\alpha}{2} = \frac{0.10}{2} = 0.05.
$$

2. We need to find the critical t-value, $$ t^* $$, such that the cumulative probability is

$$
P(T \leq t^* ) = 1 - 0.05 = 0.95,
$$

with $$ 16 $$ degrees of freedom.

3. Consulting a t-distribution table or using a calculator shows that for $$ 16 $$ degrees of freedom, the $$ 95^{\text{th}} $$ percentile is approximately

$$
t^* \approx 1.746.
$$

Thus, the critical t-value for a $$90\%$$ confidence interval with $$16$$ degrees of freedom is

$$
t^* \approx 1.746.
$$
The critical t-value for a 90% confidence interval with 16 degrees of freedom is approximately 1.746.

Find the critical \( t \)-value for a \( 90 \% \) confidence interval using a \( t \)-distribution with 16 degrees of freedom. Round your answer to three decimal places, if necessary.

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