Textbook Question
In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.
Left-tailed test, α=0.02
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9. Hypothesis Testing for One Sample
Steps in Hypothesis Testing
1:56 minutesProblem 7.5.4
Textbook Question
Describe the difference between calculating the standardized test statistic, Z^2, for a chi-square test for variance and a chi-square test for standard deviation.
Verified step by step guidance1
Understand the purpose of the chi-square test: Both tests are used to assess whether the variance or standard deviation of a population matches a hypothesized value. The difference lies in the parameter being tested (variance vs. standard deviation).
Recall the formula for the chi-square test statistic: The test statistic is calculated as \( \chi^2 = \frac{(n-1) \cdot s^2}{\sigma^2} \), where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \sigma^2 \) is the hypothesized population variance.
For a chi-square test for variance: Directly use the sample variance \( s^2 \) and hypothesized population variance \( \sigma^2 \) in the formula. The test statistic \( \chi^2 \) is calculated based on these values.
For a chi-square test for standard deviation: Convert the hypothesized standard deviation \( \sigma \) into variance by squaring it (\( \sigma^2 = \sigma \cdot \sigma \)). Similarly, ensure the sample standard deviation \( s \) is squared to obtain \( s^2 \). Then proceed with the same formula for \( \chi^2 \).
Interpretation: While the calculation process is similar, the key difference is whether the hypothesized and sample values are expressed in terms of variance or standard deviation. Always ensure consistency in units (variance or standard deviation) before applying the formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chi-Square Test for Variance
The chi-square test for variance assesses whether the variance of a population is equal to a specified value. It uses the chi-square statistic, which is calculated by comparing the observed variance from a sample to the expected variance under the null hypothesis. This test is particularly useful in quality control and research to determine if variability in data is consistent with a theoretical model.
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Chi-Square Test for Standard Deviation
The chi-square test for standard deviation is similar to the test for variance but focuses specifically on the standard deviation. Since the standard deviation is the square root of the variance, the test evaluates whether the sample standard deviation significantly differs from a hypothesized population standard deviation. This distinction is important in contexts where standard deviation is a more intuitive measure of spread than variance.
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08:45Calculating Standard Deviation
Standardized Test Statistic (Z^2)
The standardized test statistic, often denoted as Z^2 in the context of chi-square tests, is a measure that allows for the comparison of the observed data to the expected data under the null hypothesis. It is calculated by taking the difference between observed and expected values, squaring it, and normalizing by the expected values. This statistic helps determine the significance of the results, indicating whether to reject the null hypothesis based on the calculated p-value.
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06:34Step 2: Calculate Test Statistic
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