Chapter 7: Problem 7
Finding Critical Values and Confidence Intervals. In Exercises \(5-8,\) use thegiven information to find the number of degrees of freedom, the criticalvalues \(\mathcal{X}_{L}^{2}\) and \(\mathcal{X}_{R}^{2},\) and the confidenceinterval estimate of \(\boldsymbol{\sigma} .\) The samples are from Appendix\(\boldsymbol{B}\) and it is reasonable to assume that a simple random samplehas been selected from a population with a normal distribution. Platelet Counts of Women \(99 \%\) confidence; \(n=147, s=65.4\)
Short Answer
Expert verified
df = 146. Locate critical values based on 99% CI. CI for \(\sigma\) using provided values.
Step by step solution
01
- Determine Degrees of Freedom
The degrees of freedom (df) for the sample can be calculated using the formula: \[\text{df} = n - 1\] where \(n\) is the sample size.
02
- Find the Critical Values
Using the degrees of freedom, look up the critical values \(\chi_{L}^{2}\) and \(\chi_{R}^{2}\) from the chi-squared distribution table for a 99% confidence interval. The critical values correspond to the \(1 - \alpha/2\) and \(\alpha/2\) percentiles of the chi-squared distribution with \(\text{df}\) degrees of freedom. For a 99% confidence interval, \(\alpha = 0.01\).
03
- Calculate the Confidence Interval for \(\sigma^2\)
The confidence interval for the variance \(\sigma^2\) is given by: \[ \left( \frac{(n-1) s^2}{\chi_{R}^{2}}, \frac{(n-1) s^2}{\chi_{L}^{2}} \right) \] where \(s\) is the sample standard deviation, and \(n\) is the sample size.
04
- Determine the Confidence Interval for \(\sigma\)
To find the confidence interval for \(\sigma\), take the square root of the lower and upper bounds of the confidence interval for \(\sigma^2\): \[ \text{CI} = \left( \sqrt{\frac{(n-1) s^2}{\chi_{R}^{2}}}, \sqrt{\frac{(n-1) s^2}{\chi_{L}^{2}}} \right) \]
05
- Substitute Values
Substitute the known values into the equations: \(n = 147\), \(s = 65.4\), and the critical values obtained from the chi-squared distribution table for 99% confidence with \(146\) degrees of freedom.
06
- Final Calculation
Calculate the numerical values for the confidence interval.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
degrees of freedom
Degrees of freedom (df) is a critical concept in statistics, especially when working with sample data. It represents the number of independent values that can vary in an analysis without violating any given constraints. For a sample of size \(n\), the degrees of freedom is calculated as \( df = n - 1 \). This is because, in many statistical calculations, one data point is used to estimate the sample mean, reducing the number of independent pieces of information by one.
For example: In the given exercise, if the sample size \( n = 147 \), the degrees of freedom would be \( df = 147 - 1 = 146 \). This value is essential for identifying the correct critical values from the chi-squared distribution, which are needed to calculate the confidence interval.
critical values
Critical values are the points at which the tails of the probability distribution are defined for a given confidence level. They help in bounding the confidence interval. To find the critical values for a specific confidence level and degrees of freedom, you would reference a chi-squared distribution table.
Critical values for \( \chi_{L}^{2} \) and \( \chi_{R}^{2} \) correspond to the \( 1 - \alpha/2 \) and \( \alpha/2 \) percentiles, respectively, where \( \alpha \) is the significance level. For a 99% confidence interval, \( \alpha = 0.01 \), meaning you would look up the 0.005 and 0.995 percentiles in the chi-squared table for 146 degrees of freedom.
Example Calculation: Suppose the chi-squared values are found to be \( \chi_{L}^{2} \) = 99.635 and \( \chi_{R}^{2} \) = 206.874 for a 99% confidence interval with 146 degrees of freedom.
chi-squared distribution
The chi-squared distribution is used extensively in hypothesis testing and confidence interval estimation for variance and standard deviation. It is a continuous distribution defined for non-negative values (greater than or equal to zero). Each chi-squared distribution is defined by its degrees of freedom.
The chi-squared distribution is skewed to the right, and this skewness decreases as the degrees of freedom increase, making the distribution more symmetric. This distribution helps in determining critical values, which in turn are used to calculate confidence intervals for variance and standard deviation.
In Practice: Using our exercise data with 146 degrees of freedom, critical values from the chi-squared distribution are crucial for determining the range within which we are confident the true population variance or standard deviation lies.
variance and standard deviation
Variance \( \sigma^2 \) measures the spread of data points in a dataset around the mean. It provides a numerical value for this dispersion. Standard deviation \( \sigma \) is simply the square root of variance, offering a measure of spread in the same units as the data.
Confidence Interval for Variance: To find the confidence interval for the variance, use the formula: \[ \frac{(n-1) s^2}{\chi_{R}^{2}}, \frac{(n-1) s^2}{\chi_{L}^{2}} \]
Confidence Interval for Standard Deviation: For standard deviation, take the square root of the upper and lower bounds of the variance confidence interval: \[ \text{CI} = \left( \sqrt{\frac{(n-1) s^2}{\chi_{R}^{2}}}, \sqrt{\frac{(n-1) s^2}{\chi_{L}^{2}}} \right) \]
For the given exercise, plugging in the values \( n = 147 \), \( s = 65.4 \), and the obtained critical values, you'll calculate the numerical confidence intervals for both variance and standard deviation.
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