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Magazine 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
Expert verified
The P-values for the given situations are approximately: a. 1, b. 0.69, c. 0.025, d. 0.001, e. 0.096
Step by step solution
01
Determine P-value for Two-tailed test, df=9, t=0.73
To find the p-value, we must look at a t-distribution table for the given degrees of freedom (df = 9). In this case, we find that t=0.73 corresponds to a probability in the t-distribution table of about 0.77, which is the one-tailed p-value. Because the test is two-tailed, we must double this to get the two-tailed P-value: 0.77*2 = 1.54. Since a probability cannot exceed 1, we interpret this as a P-value of 1, indicating weak evidence against the null hypothesis.
02
Determine P-value for Upper-tailed test, df=10, t=-0.5
Looking at the t-distribution table for df = 10, we find that t = -0.5 corresponds to a probability in the table of about 0.69, which is the two-tailed p-value. Because the test is upper-tailed (or one-tailed), the P-value is equal to this value: P = 0.69, suggesting weak evidence against the null hypothesis.
03
Determine P-value for Lower-tailed test, n=20, t=-2.1
When n=20, this implies df=20-1=19. Given df = 19 and t = -2.1, we have a one-tailed p-value of around 0.025 according to the t-distribution table. Since this is a lower-tailed (or one-tailed) test, the P-value is the same as the one-tailed p-value: P = 0.025, suggesting moderate evidence against the null hypothesis.
04
Determine P-value for Lower-tailed test, n=20, t=-5.1
Given df = 19 (from n=20) and t = -5.1, we find a one-tailed p-value close to 0.001 (or possibly even less) in the t-distribution table. Again, because it's a lower-tailed test, the P-value equals the one-tailed p-value: P 鈮� 0.001, suggesting strong evidence against the null hypothesis.
05
Determine P-value for Two-tailed test, n=40, t=1.7
Given df = 40-1=39 and t = 1.7, we find a one-tailed p-value of about 0.048 in the t-distribution table. Because this is a two-tailed test, we must double this value to get the P-value: 0.048*2 = 0.096, indicating weak to moderate evidence against the null hypothesis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
t-distribution
The t-distribution is a key statistical distribution used in hypothesis testing, especially when dealing with small sample sizes. It's a kind of bell-shaped curve, much like the normal distribution, but it differs in that it has heavier tails.
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.
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
Degrees of freedom (df) are a crucial consideration in statistical analyses, especially when using the t-distribution. In the context of the t-test, degrees of freedom determine the exact shape of the t-distribution.
To calculate the degrees of freedom for a t-test, use the formula: \[ df = n - 1 \] where "n" is the sample size.
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
A two-tailed test is used in hypothesis testing when we are interested in checking for effects in both directions, either an increase or a decrease.
Here's how it works:
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
A lower-tailed test, also known as a left-tailed test, investigates if a sample statistic is significantly less than a stated value in the null hypothesis. This is useful when the primary concern is a reduction or minimization effect.
Key aspects of a lower-tailed test include:
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
An upper-tailed test is designed to determine if a sample statistic is significantly higher than a value specified in the null hypothesis. It is particularly useful in cases where a positive increase is the focus.
Here are some key features:
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.
