Math Problem Statement
if a sample has 20 observations and a 95% confidence est for a pop is needed. tell me the t- multiple
Solution
To find the t-multiple for a 95% confidence interval when the sample has 20 observations, you need to use the Student's t-distribution.
Here's the breakdown:
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Degrees of freedom (df): For a sample of 20 observations, the degrees of freedom are:
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Confidence level: 95% confidence means that the remaining 5% is split between both tails of the distribution (2.5% in each tail). Hence, you are looking for the critical value that corresponds to the 97.5th percentile of the t-distribution (to cover 95% in the center).
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t-multiple: Looking up the t-distribution table or using a calculator for and a 95% confidence level gives a t-multiple of approximately:
Thus, the t-multiple is approximately 2.093.
Would you like further details on the computation, or how to use this value?
Related Questions
- How do you construct a confidence interval using the t-multiple?
- What if the sample size was larger—how does it affect the t-multiple?
- How does a 99% confidence interval change the t-multiple?
- Why do we use the t-distribution instead of the normal distribution for small samples?
- What is the difference between a one-tailed and two-tailed confidence interval?
Tip:
For samples larger than 30, the t-distribution begins to approximate the normal distribution, and the t-multiple converges to the z-score (like 1.96 for 95% confidence).
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Interval
Student's t-distribution
Formulas
Degrees of freedom: df = n - 1
Critical value from the t-distribution table: t_{α/2, df}
t-multiple for confidence interval
Theorems
Student's t-distribution
Suitable Grade Level
College-Level Statistics
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