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Ch. 7 - Hypothesis Testing with One Sample
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 7 - Hypothesis Testing with One SampleProblema 7.2.24
Capitolo 7, Problema 7.2.24

Finding Critical Values and Rejection Regions In Exercises 23–28, find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer.


Left-tailed test, α = 0.09

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Step 1: Understand the problem. This is a left-tailed z-test, which means the rejection region is located in the left tail of the standard normal distribution. The level of significance (α) is given as 0.09, which represents the area in the left tail where we would reject the null hypothesis.
Step 2: Recall the relationship between the level of significance (α) and the critical value. The critical value is the z-score that corresponds to the cumulative probability equal to α. For a left-tailed test, this means finding the z-score where the cumulative probability is 0.09.
Step 3: Use a z-table or statistical software to find the z-score corresponding to a cumulative probability of 0.09. This z-score will be negative because it is in the left tail of the standard normal distribution.
Step 4: Define the rejection region. For a left-tailed test, the rejection region includes all z-scores less than the critical value found in Step 3. Mathematically, the rejection region is expressed as z < critical value.
Step 5: Visualize the rejection region on a standard normal distribution graph. Draw a bell curve, shade the left tail corresponding to α = 0.09, and mark the critical value on the horizontal axis. This visual representation helps to clearly identify the rejection region.

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Critical Value

A critical value is a point on the scale of the test statistic that separates the region where the null hypothesis is rejected from the region where it is not rejected. In hypothesis testing, critical values are determined based on the significance level (α) and the type of test being conducted, such as one-tailed or two-tailed tests.
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Critical Values: t-Distribution

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. For a left-tailed test, this region is located to the left of the critical value, indicating that if the test statistic falls within this region, the null hypothesis can be rejected in favor of the alternative hypothesis.
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Step 4: State Conclusion

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). It represents the threshold for determining whether the observed data is statistically significant. In this case, with α = 0.09, it indicates a 9% risk of incorrectly rejecting the null hypothesis.
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Step 4: State Conclusion Example 4
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Left-tailed test


z=-1.68

alpha=0.05

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In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.

Right-tailed test, α=0.05, n=23

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Ha: μ < 3


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Identifying a Test In Exercises 21–24, determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.


Ha: p = 0.25

H0: p ≠ 0.25

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