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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
Without the actual t-distribution table or software, we can't specify exact p-values for scenarios a, c, and e. However, we can say that for scenario b, the p-value is nearly 1, and for scenario d, it is approximately 0 due to the respective nature of their t-statistics.

Step by step solution

01

Analyze Scenario a

In scenario a, the t-test is two-tailed with 9 degrees of freedom and a t-statistic of 0.73. The p-value is the probability of observing a t-statistic as extreme, in either positive or negative direction, under the null hypothesis. Using a t-distribution table or software, find the cumulative probability associated with t=0.73 for df=9, and subtract it from 1 to find the one-tailed p-value. Double this to find the p-value for the two-tailed test.
02

Analyze Scenario b

In scenario b, the test is upper-tailed with 10 degrees of freedom and t=-0.5. The p-value is the probability of observing a t-statistic greater than or equal to the observed one under the null hypothesis. Because the observed t-statistic is negative, the p-value is essentially 1, as nearly all possible t-values under the null hypothesis are greater than this.
03

Analyze Scenario c

In scenario c, the test is lower-tailed with 19 degrees of freedom (since \(n-1=20-1\) ) and t=-2.1. The p-value is the probability of observing a t-statistic lower than or equal to -2.1 under the null hypothesis. Using a t-distribution table or software, find the cumulative probability associated with t=-2.1 for df=19.
04

Analyze Scenario d

In scenario d, the test is lower-tailed with 19 degrees of freedom (since \(n-1=20-1\)) and t=-5.1. The p-value is the probability of observing a t-statistic lower than or equal to -5.1 under the null hypothesis. Given how extreme this t-statistic is, the p-value will be very close to 0.
05

Analyze Scenario e

In scenario e, the test is two-tailed with 39 degrees of freedom (since \(n-1=40-1\)) and t=1.7. The p-value is the probability of observing a t-statistic as extreme, in either positive or negative direction, under the null hypothesis. Using a t-distribution table or software, find the cumulative probability associated with t=1.7 for df=39, subtract it from 1 to find the one-tailed p-value, and double this to find the p-value for the two-tailed test.

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Most popular questions from this chapter

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