Textbook Question
Graphical Analysis In Exercises 9–12, state whether each standardized test statistic t allows you to reject the null hypothesis. Explain.
b. t = 1.42
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9. Hypothesis Testing for One Sample
Steps in Hypothesis Testing
1:51 minutesProblem 7.3.1
Textbook Question
Explain how to find critical values for a t-distribution.
Verified step by step guidance1
Step 1: Understand the t-distribution. The t-distribution is a probability distribution used when estimating population parameters when the sample size is small, and the population standard deviation is unknown. It is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails.
Step 2: Identify the degrees of freedom (df). The degrees of freedom for a t-distribution are calculated as the sample size (n) minus 1, i.e., df = n - 1. This value determines the shape of the t-distribution.
Step 3: Determine the significance level (α). The significance level represents the probability of rejecting the null hypothesis when it is true. Common values for α are 0.05 (5%) or 0.01 (1%). For a two-tailed test, divide α by 2 to account for both tails of the distribution.
Step 4: Use a t-distribution table or statistical software. Locate the critical value by finding the intersection of the degrees of freedom (df) and the column corresponding to the desired significance level (α or α/2 for two-tailed tests) in a t-distribution table. Alternatively, use statistical software or a calculator to find the critical value.
Step 5: Interpret the critical value. The critical value(s) represent the cutoff point(s) on the t-distribution. For a one-tailed test, there will be one critical value. For a two-tailed test, there will be two critical values (positive and negative). These values are used to determine whether the test statistic falls in the rejection region.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
t-distribution
The 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 used primarily in statistics for hypothesis testing and confidence intervals when the sample size is small and the population standard deviation is unknown. The shape of the t-distribution changes based on the degrees of freedom, which are determined by the sample size.
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critical values
Critical values are the threshold points that define the boundaries of the acceptance region in hypothesis testing. They are determined based on the significance level (alpha) and the distribution being used, such as the t-distribution. For a given alpha level, critical values help to decide whether to reject the null hypothesis by indicating the cutoff points for the test statistic.
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degrees of freedom
Degrees of freedom (df) refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of the t-distribution, degrees of freedom are typically calculated as the sample size minus one (n-1). This concept is crucial for determining the appropriate t-distribution to use when finding critical values, as it affects the shape and spread of the distribution.
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