Math Problem Statement
Solution
The image contains two problems related to the t-distribution. Here is the breakdown of each problem and how to solve them:
(a) Problem:
- Consider a t-distribution with 20 degrees of freedom. Compute the probability:
The goal is to find the probability that the t-statistic lies between -1.58 and 1.58, with 20 degrees of freedom. You can use a t-distribution calculator or look up the cumulative probability from a t-distribution table.
Steps to solve:
- Find the cumulative probability for t = 1.58 for 20 degrees of freedom (df = 20).
- Because the t-distribution is symmetric, the probability for P(-1.58 < t < 1.58) can be found by doubling the cumulative probability for t = 1.58 and subtracting from 1.
The formula is:
(b) Problem:
- Consider a t-distribution with 6 degrees of freedom. Find the value of c such that:
Here, you are looking for the critical value of c for which the cumulative probability is 0.01 for a t-distribution with 6 degrees of freedom (df = 6). This is a lower-tail problem, so you would look up the inverse cumulative probability for 0.01 in a t-distribution table with 6 degrees of freedom or use a calculator.
Let me know if you would like further help solving these problems, or if you need more details!
Would you like the detailed step-by-step solution for these problems?
Here are some related questions for further practice:
- How do you compute the cumulative probability from a t-distribution?
- What is the difference between a t-distribution and a normal distribution?
- How can you use a t-table to find probabilities?
- What is the effect of degrees of freedom on the t-distribution?
- How do you interpret a p-value in the context of hypothesis testing with t-distribution?
Tip: In hypothesis testing, using a t-distribution is crucial when working with small sample sizes (typically n < 30) or when the population standard deviation is unknown.
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Math Problem Analysis
Mathematical Concepts
t-distribution
probability
degrees of freedom
Formulas
P(-1.58 < t < 1.58) = 2 * P(t < 1.58) - 1
P(t ≤ c) = 0.01 (inverse cumulative probability for t-distribution)
Theorems
t-distribution properties
Cumulative distribution function (CDF)
Inverse cumulative distribution
Suitable Grade Level
Undergraduate Statistics
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