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Magazine Chapter 10: Problem 71
Use Table 4 in Appendix I to bound the following \(p\) -values: a. \(P(t>1.2)\) with \(5 d f\). b. \(P(t>2)+P(t<-2)\) with \(10 d f\) c. \(P(t<-3.3)\) with \(8 d f\) d. \(P(t>0.6)\) with \(12 d f\)
Short Answer
Expert verified
Answer:
a. 0.10 < \(P(t>1.2)\) < 0.20 with 5 df.
b. \(P(t>2)+P(t<-2) = 0.05\) with 10 df.
c. \(P(t<-3.3) < 0.005\) with 8 df.
d. 0.25 < \(P(t>0.6)\) < 0.30 with 12 df.
Step by step solution
01
a. \(P(t>1.2)\) with \(5 d f\) #
First, we need to locate the appropriate row corresponding to our given degrees of freedom (df = 5) in the table. Then, we find the closest value to our test statistic (1.2) in that row. Since there isn't an exact match, we look for the two values that the test statistic falls in between to bound the \(p\)-value. Finally, we observe the column headers to find the \(p\)-values associated with these bounds.
Using Table 4 and df = 5, the test statistic 1.2 falls in between 1.476 and 0.989, which are associated with \(p\)-values 0.10 and 0.20, respectively. Thus, our answer is: 0.10 < \(P(t>1.2)\) < 0.20.
02
b. \(P(t>2)+P(t
First, we need to locate the appropriate row corresponding to our given degrees of freedom (df = 10) in the table. Then, we find the closest value to our test statistic (2) in that row. Since we are considering a two-tailed test, we need to add the \(p\)-values of both tails (i.e., \(P(t>2)+P(t<-2)\)).
Using Table 4 and df = 10, the test statistic 2 is the exact value in the table, which is associated with a \(p\)-value of 0.025 in the right tail. Since we need the sum of the \(p\)-values in both tails, our answer is: \(P(t>2)+P(t<-2) = 0.025 + 0.025 = 0.05\).
03
c. \(P(t
First, we need to locate the appropriate row corresponding to our given degrees of freedom (df = 8) in the table. Then, we find the closest value to our test statistic (3.3) in that row. Since there isn't an exact match, we look for the two values that the test statistic falls in between to bound the \(p\)-value. Finally, we would observe the column headers to find the \(p\)-values associated with these bounds.
Using Table 4 and df = 8, the test statistic 3.3 is beyond the largest value in the row. Thus, our answer is: \(P(t<-3.3) < 0.005\) (closer to 0).
04
d. \(P(t>0.6)\) with \(12 d f\) #
First, we need to locate the appropriate row corresponding to our given degrees of freedom (df = 12) in the table. Then, we find the closest value to our test statistic (0.6) in that row. Since there isn't an exact match, we look for the two values that the test statistic falls in between to bound the \(p\)-value. Finally, we observe the column headers to find the \(p\)-values associated with these bounds.
Using Table 4 and df = 12, the test statistic 0.6 falls in between 0.725 and 0.565, which are associated with \(p\)-values 0.25 and 0.30, respectively. Thus, our answer is: 0.25 < \(P(t>0.6)\) < 0.30.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Degrees of Freedom
In statistics, the term 'degrees of freedom' refers to the number of values in a calculation that are free to vary. It's a crucial concept when working with statistical measures such as the t-distribution. Degrees of freedom are used to describe the extent of freedom in selecting independent variables, and they often determine the shape of the probability distribution being used.
- The degrees of freedom (df) are generally calculated as the sample size minus the number of parameters being estimated.
- The formula for degrees of freedom in a dataset can be roughly described as: \[ df = n - k \] where \( n \) is the number of observations and \( k \) is the number of parameters or groups being estimated.
- In t-distributions, greater degrees of freedom signify that the estimates are more stable and the distribution approaches a normal distribution.
T-Distribution
The t-distribution, also known as Student鈥檚 t-distribution, is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is employed primarily in small sample sizes or when the population standard deviation is unknown. Here are some key points to understand about the t-distribution:
- The shape of the t-distribution is determined by the degrees of freedom. Lower degrees of freedom result in wider and flatter distributions, while higher degrees of freedom make the distribution more similar to a normal distribution.
- It is particularly useful for small sample sizes, typically less than 30.
- As the sample size increases, the t-distribution converges to the normal distribution, making it a versatile tool in statistics.
Statistical Tables
Statistical tables, like the t-distribution table mentioned in the original exercise, are crucial tools in statistical analysis. They provide critical values for different distributions and reveal the probability of obtaining certain results under the null hypothesis. Here鈥檚 why statistical tables are important:
- They allow statisticians to derive probability values for variable data. For example, you can determine a p-value based on a specific t-score and degrees of freedom.
- These tables save time by offering pre-calculated results. This means you don't need to run detailed calculations for standard tests.
- They help to interpret test statistics by giving boundaries or ranges for specific values when calculating probabilities.
