Chapter 10: Problem 43
Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations: a. Two-tailed test, \(\mathrm{df}=9, t=0.73\) b. Upper-tailed test, \(\mathrm{df}=10, t=-0.5\) c. Lower-tailed test, \(n=20, t=-2.1\) d. Lower-tailed test, \(n=20, t=-5.1\) e. Two-tailed test, \(n=40, t=1.7\)
Short Answer
Step by step solution
Determine P-value for Two-tailed test, df=9, t=0.73
Determine P-value for Upper-tailed test, df=10, t=-0.5
Determine P-value for Lower-tailed test, n=20, t=-2.1
Determine P-value for Lower-tailed test, n=20, t=-5.1
Determine P-value for Two-tailed test, n=40, t=1.7
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
t-distribution
This means the t-distribution is more forgiving to outliers in data, making it more appropriate for smaller datasets where sample sizes are limited. As the sample size increases, the t-distribution becomes more like the normal distribution.
- Origin of the t-distribution: It was introduced by William Sealy Gosset for small sample datasets, who published under the pseudonym "Student," hence the name Student's t-distribution.
- Application: Commonly used for estimating the mean in smaller sample sizes or when the standard deviation is unknown.
degrees of freedom
To calculate the degrees of freedom for a t-test, use the formula: \[ df = n - 1 \] where "n" is the sample size.
- Importance: Degrees of freedom impact the reliability of statistical tests and influence the decision threshold for statistical significance.
- Effect: More degrees of freedom mean the t-distribution will have thinner tails and be closer to a normal distribution.
two-tailed test
Here's how it works:
- Symmetric interest: If the null hypothesis states that a parameter has a specific value, a two-tailed test checks for any deviation, whether higher or lower.
- Example: If we are testing if a mean is `10`, the two-tailed test will check if it's significantly different from `10`, whether it is larger or smaller.
- P-value calculation: The p-value in a two-tailed test is calculated by doubling the one-tailed p-value because it considers the probability of deviation in both directions.
lower-tailed test
Key aspects of a lower-tailed test include:
- Direction: It focuses on statistically significant lower values. We are testing if the observed statistic falls significantly below the hypothesized parameter.
- P-value: The p-value is calculated as the area under the curve to the left of the calculated test statistic in a t-distribution.
- Application: Useful for testing hypotheses where the interest is whether the effect might be less rather than greater than expected, such as testing if a new drug decreases blood pressure.
upper-tailed test
Here are some key features:
- Focus: The test aims at identifying cases where the observed statistic is likely higher than the hypothesized statistic.
- P-value calculation: P-values are obtained by looking at the right-hand tail of the distribution, indicating higher values.
- Common Usage: If you are testing whether a new training method increases employees' productivity over a standard method, the interest is in effects on the upper end of the distribution.