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Solve t-distribution Probability and Critical Value Problems (24 and 16 Degrees of Freedom)
Statistics
College (University Level)
Question Content
(a) Consider a t distribution with 24 degrees of freedom. Compute P(t ≤ 1.16). Round your answer to at least three decimal places. (b) Consider a t distribution with 16 degrees of freedom. Find the value of c such that P(-c < t < c) = 0.95. Round your answer to at least three decimal places.
Correct Answer
(a) 0.868; (b) 2.120
Detailed Solution Steps
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Part (a):
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1. Identify the parameters: t-distribution with degrees of freedom (df) = 24, and we need to find the cumulative probability for t = 1.16.
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2. Use a t-distribution table, calculator (like ALEKS calculator), or statistical software. Input df=24 and the t-score 1.16 to find the cumulative probability.
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3. The cumulative probability P(t ≤ 1.16) is calculated to be approximately 0.868.
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Part (b):
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1. Identify the parameters: t-distribution with df=16, and the middle probability P(-c < t < c)=0.95. This means the total tail probability is 1 - 0.95 = 0.05, so each tail has a probability of 0.05/2 = 0.025.
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2. We need to find the t-score c where the cumulative probability P(t ≤ c) = 0.95 + 0.025 = 0.975 (or find the critical t-value for a two-tailed test with α=0.05 and df=16).
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3. Use a t-distribution table, inverse t-calculator (like ALEKS calculator), or statistical software. Input df=16 and the cumulative probability 0.975 to find the critical value c.
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4. The value of c is calculated to be approximately 2.120.
Knowledge Points Involved
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t-distribution Cumulative Probability
The t-distribution is a continuous probability distribution used when the population standard deviation is unknown and sample size is small. Cumulative probability P(t ≤ t₀) gives the area under the t-distribution curve to the left of a given t-score t₀, calculated using degrees of freedom (df = n-1, where n is sample size) and t-score tables or calculators.
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Degrees of Freedom for t-distribution
Degrees of freedom represent the number of independent values in a statistical calculation that can vary. For a one-sample t-test, df = sample size - 1. It determines the shape of the t-distribution, with higher df making the t-distribution closer to the standard normal distribution.
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Critical t-values for Confidence Intervals
Critical t-values (like c in part (b)) define the bounds of a middle probability in a t-distribution. For a 95% middle probability, the area in each tail is (1 - 0.95)/2 = 0.025. The critical value corresponds to the t-score where the cumulative probability is 1 - tail probability, used to construct confidence intervals or conduct hypothesis tests for population means with unknown variance.
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