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Solve t-Distribution Probability and Critical Value Problems (17 and 14 Degrees of Freedom)
Statistics
College (First-Year Statistics)
Question Content
(a) Consider a t distribution with 17 degrees of freedom. Compute $P(-1.23 < t < 1.23)$. Round your answer to at least three decimal places. (b) Consider a t distribution with 14 degrees of freedom. Find the value of $c$ such that $P(t \\leq c) = 0.01$. Round your answer to at least three decimal places.
Correct Answer
(a) 0.770; (b) -2.624
Detailed Solution Steps
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Step 1: Solve part (a):
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1. Recognize that for a t-distribution, which is symmetric about 0, $P(-1.23 < t < 1.23) = 2 \\times P(0 < t < 1.23) = 1 - 2 \\times P(t \\geq 1.23)$.
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2. Use a t-distribution table or calculator with degrees of freedom (df) = 17. Look up the one-tailed probability for t = 1.23, which is approximately 0.115.
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3. Calculate the desired probability: $1 - 2 \\times 0.115 = 1 - 0.230 = 0.770$.
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Step 2: Solve part (b):
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1. Identify that $P(t \\leq c) = 0.01$ is a left-tailed probability with df = 14. Due to the symmetry of the t-distribution, this is equivalent to finding the negative of the value where $P(t \\geq |c|) = 0.01$.
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2. Use a t-distribution table or calculator with df = 14 to find the critical t-value for a one-tailed probability of 0.01, which is 2.624.
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3. Since the probability is for the left tail, the value of $c$ is the negative of this critical value: $c = -2.624$.
Knowledge Points Involved
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t-Distribution Symmetry
The t-distribution is a symmetric, bell-shaped probability distribution centered at 0. This symmetry means $P(t < -a) = P(t > a)$ for any positive value $a$, which simplifies calculations for two-tailed or left-tailed probabilities.
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Degrees of Freedom in t-Distribution
Degrees of freedom (df) for a t-distribution represent the number of independent observations minus the number of estimated parameters. It determines the shape of the t-distribution: as df increases, the t-distribution approaches the standard normal distribution.
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Calculating t-Distribution Probabilities
To find probabilities for a t-distribution, use a t-table, statistical calculator, or software. For two-tailed probabilities, use the symmetry of the distribution to simplify the calculation. For left-tailed lower-tail probabilities, use the negative of the corresponding right-tailed critical value.
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Critical t-Values
A critical t-value is a threshold value that defines the boundary of a specific probability region in the t-distribution. For a given significance level and degrees of freedom, it is used to find cutoff points for hypothesis testing or confidence interval construction.
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