Q: What confidence levels are commonly used in hypothesis testing?

Common confidence levels are 90% (α = 0.10), 95% (α = 0.05) - most common, 99% (α = 0.01) - very strict, and 99.9% (α = 0.001) - extremely strict. Lower α means higher confidence but harder to reject the null hypothesis.

Q: How do you interpret the critical t-value?

If your calculated test statistic (t-statistic) is greater than the critical t-value (in absolute value), you reject the null hypothesis. If it's less, you fail to reject. For two-tailed tests, compare with ±critical value.

Q: What is the relationship between t-value and confidence interval?

The critical t-value is used to construct confidence intervals: CI = x̄ ± t(α/2, df) × (s/√n). The t-value determines the margin of error. Higher confidence (lower α) means larger t-values and wider confidence intervals.

Q: How do you find critical t-value for different sample sizes?

Degrees of freedom = n - 1 for one sample mean test. For paired samples: df = n - 1 where n is number of pairs. For two independent samples: df = n₁ + n₂ - 2. For regression: df = n - 2. Use these df values to look up critical t-values.

Q: What are practical applications of critical t-values?

Critical t-values are used in: 1) t-tests for comparing sample means, 2) Constructing confidence intervals for means, 3) Regression analysis for testing coefficients, 4) Quality control and process monitoring, 5) Medical research and clinical trials, 6) Market research and consumer studies.

Question 1

What is a critical t-value?

Accepted Answer

A critical t-value is the boundary value that separates the rejection region from the acceptance region in hypothesis testing using the t-distribution. It depends on the degrees of freedom, significance level (α), and whether the test is one-tailed or two-tailed.

Question 2

How do you calculate critical t-value?

Accepted Answer

To calculate a critical t-value: 1) Determine degrees of freedom (df = n - 1 for one sample). 2) Choose confidence level (e.g., 95% → α = 0.05). 3) Select one-tailed or two-tailed test. 4) Look up the value in a t-distribution table using df and α/2 for two-tailed tests.

Question 3

What is the difference between one-tailed and two-tailed tests?

Accepted Answer

A one-tailed test (directional) tests for an effect in one direction (greater or less than), splitting α in one tail. A two-tailed test (nondirectional) tests for any difference without direction, splitting α/2 in each tail. Two-tailed tests are more conservative.

Question 4

How do degrees of freedom affect the critical t-value?

Accepted Answer

Degrees of freedom (df) determine the shape of the t-distribution. Lower df means fatter tails and higher critical values. As df increases, the t-distribution approaches the normal distribution. Common df = n - 1 for one sample, (n₁ + n₂ - 2) for two samples.

Question 5

When do you use the t-distribution instead of the z-distribution?

Accepted Answer

Use the t-distribution when: 1) Sample size is small (n < 30 typically), 2) Population standard deviation is unknown, 3) You're estimating population parameters from sample data. Use z-distribution when population SD is known and sample size is large.

Question 6

What confidence levels are commonly used in hypothesis testing?

Accepted Answer

Common confidence levels are 90% (α = 0.10), 95% (α = 0.05) - most common, 99% (α = 0.01) - very strict, and 99.9% (α = 0.001) - extremely strict. Lower α means higher confidence but harder to reject the null hypothesis.

Question 7

How do you interpret the critical t-value?

Accepted Answer

If your calculated test statistic (t-statistic) is greater than the critical t-value (in absolute value), you reject the null hypothesis. If it's less, you fail to reject. For two-tailed tests, compare with ±critical value.

Question 8

What is the relationship between t-value and confidence interval?

Accepted Answer

The critical t-value is used to construct confidence intervals: CI = x̄ ± t(α/2, df) × (s/√n). The t-value determines the margin of error. Higher confidence (lower α) means larger t-values and wider confidence intervals.

Question 9

How do you find critical t-value for different sample sizes?

Accepted Answer

Degrees of freedom = n - 1 for one sample mean test. For paired samples: df = n - 1 where n is number of pairs. For two independent samples: df = n₁ + n₂ - 2. For regression: df = n - 2. Use these df values to look up critical t-values.

Question 10

What are practical applications of critical t-values?

Accepted Answer

Critical t-values are used in: 1) t-tests for comparing sample means, 2) Constructing confidence intervals for means, 3) Regression analysis for testing coefficients, 4) Quality control and process monitoring, 5) Medical research and clinical trials, 6) Market research and consumer studies.

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