The critical value t
𝛼
2
,k is the (1-
𝛼
/2) - quantile for a t distribution with k degrees of freedom.

Compute t
𝛼
2
,k when
𝛼
= .02 and k = 17

Question

The critical value t
𝛼
2
,k is the (1-
𝛼
/2) - quantile for a t distribution with k degrees of freedom.

Compute  t
𝛼
2
,k when  
𝛼
 = .02 and k = 17

BlackTom AI · Accepted Answer

Begin by identifying what the problem asks: we need the critical t value at a two-sided significance level alpha = 0.02 with k = 17 degrees of freedom. This corresponds to the (1 - alpha/2) quantile of the t distribution, i.e., the 0.99 quantile, since alpha/2 = 0.01 for each tail.

Step 1: Compute the tail probability. Because it is a two-sided test with alpha = 0.02, each tail beyond the critical value must have probability 0.01. In notation, we want t such that P(T > t) = 0.01 for T ~ t_df, where df = 17.

Step 2: Locate the quantile. Look up the t value for df = 17 at the cumulative probability 0.99 (since 1 - 0.01 = 0.99) or equivalently find t so that P(T ≤ t) = 0.99. This is the 0.99 quantile of t with 17 degrees of freedom.

Step 3: Interpret and report. Using a t-distribution table, a t value around 2.57 is typical for df near 17 at the 0.99 quantile. More precise calculation (from software or a precise table) yields t_{0.01,17} ≈ 2.566934.

Step 4: Verify consistency. If you computed using a statistics calculator or software (e.g., t.inv(0.99, 17) or equivalent), you should obtain a value very close to 2.566934, since rounding differences may occur depending on the tool.

Notes on potential pitfalls:
- Ensure you are using the correct tail probability: for a two-sided test with alpha = 0.02, each tail is 0.01, not 0.02.
- The degrees of freedom must be exactly k = 17 in this problem; using a different df changes the critical value noticeably.
- If you only have a t-table that lists common quantiles, you may need to interpolate to approximate the 0.99 quantile; software provides the most exact value.

In summary, the critical t value t_{alpha/2, k} for alpha = 0.02 and k = 17 is approximately 2.566934, which aligns with the provided numerical answer.

The critical value t 𝛼 2 ,k is the (1- 𝛼 /2) - quantile for a t distribution with k degrees of freedom. Compute t 𝛼 2 ,k when 𝛼 = .02 and k = 17数值题

类似问题

Which of the following is true regarding the t-distribution?

This time, let's construct a 95% confidence interval for a true mean price difference between a 0.99 carat and a 1 carat diamond. What is the correct t-score for the margin of error?

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