Textbook Question
What happens to the shape of the chi-square distribution as the degrees of freedom increase?
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9. Hypothesis Testing for One Sample
Steps in Hypothesis Testing
2:20 minutesProblem 6.R.30
Textbook Question
In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.99, n = 10
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the critical values for a given confidence level (c = 0.99) and sample size (n = 10). Critical values are used in hypothesis testing and confidence intervals to determine the range of values that are likely to contain the population parameter.
Step 2: Determine the degrees of freedom (df). For a t-distribution, the degrees of freedom are calculated as df = n - 1. Since n = 10, calculate df = 10 - 1.
Step 3: Identify the confidence level and the corresponding significance level (α). The confidence level is c = 0.99, so the significance level is α = 1 - c = 1 - 0.99.
Step 4: Divide the significance level (α) by 2 to find the area in each tail of the t-distribution. This is because the t-distribution is symmetric, and the critical values are located at the tails. Calculate α/2.
Step 5: Use a t-distribution table or statistical software to find the critical t-values for the given degrees of freedom (df) and the area in each tail (α/2). These critical values will correspond to the boundaries of the confidence interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Values
Critical values are the points on the scale of the test statistic that define the boundaries for rejecting the null hypothesis. They are determined based on the desired level of confidence and the distribution of the test statistic. For example, in a normal distribution, critical values correspond to specific z-scores that capture the central area of the distribution, reflecting the confidence level.
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Level of Confidence
The level of confidence, denoted as 'c', represents the probability that the confidence interval will contain the true population parameter. A higher confidence level, such as 0.99, indicates a greater certainty that the interval includes the parameter, but it also results in a wider interval. This concept is crucial for understanding how confident we can be in our estimates based on sample data.
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Sample Size
Sample size, denoted as 'n', refers to the number of observations or data points collected in a study. It plays a significant role in statistical analysis, as larger sample sizes generally lead to more reliable estimates and narrower confidence intervals. In this context, a sample size of 10 may limit the precision of the confidence interval, affecting the critical values derived from it.
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