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Solve t-distribution Probability and Critical Value Problems (df=17, df=14)
Statistics
Grade 11 of Senior High School / University Introductory 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) 0.771; (b) -2.624
Detailed Solution Steps
1
Part (a) Step 1: Recall that for a symmetric t-distribution, $P(-a < t < a) = 2*P(0 < t < a) = 1 - 2*P(t \\geq a)$. We use a t-table or t-distribution calculator for df=17, t=1.23.
2
Part (a) Step 2: Look up the one-tailed probability for t=1.23, df=17: $P(t \\geq 1.23) \\approx 0.1145$.
3
Part (a) Step 3: Calculate the desired probability: $1 - 2*0.1145 = 1 - 0.229 = 0.771$.
4
Part (b) Step 1: Recognize $P(t \\leq c) = 0.01$ is a left-tailed probability for df=14. Due to symmetry, this is equivalent to the negative of the critical value for $P(t \\geq |c|) = 0.01$.
5
Part (b) Step 2: Look up the one-tailed critical t-value for df=14, $\\alpha=0.01$, which is 2.624.
6
Part (b) Step 3: Since we need the left tail, the value of c is the negative of this critical value: $c = -2.624$.
Knowledge Points Involved
1
t-distribution Symmetry
The t-distribution is a symmetric, bell-shaped distribution centered at 0. This means $P(t < -a) = P(t > a)$ and $P(-a < t < a) = 1 - 2P(t > a)$. This symmetry simplifies calculating two-tailed or symmetric interval probabilities.
2
Degrees of Freedom (df) for t-distribution
Degrees of freedom represent the number of independent observations minus the number of estimated parameters, calculated as $n-1$ for one-sample t-problems. The value of df determines the shape of the t-distribution: higher df makes it closer to the standard normal distribution.
3
Tailed Probabilities for t-distribution
One-tailed probabilities ($P(t \\geq a)$ or $P(t \\leq a)$) correspond to the area in one tail of the t-distribution, while two-tailed probabilities cover both tails. These are used for hypothesis testing and confidence interval construction for population means when the population standard deviation is unknown.
4
Critical t-values
Critical t-values are threshold values that correspond to a specific tail probability and degrees of freedom. They are used to define rejection regions in t-tests or to find bounds for confidence intervals.
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