use the given information to find the number of degrees of freedom, the critical values X2L and X2R, and the confidence interval estimate of σ. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution:


Heights of Men 99% confidence; n=153, s=7.10 cm.

Verified step by step guidance

1. Step 1: Identify the degrees of freedom (df). The degrees of freedom for a chi-square distribution is calculated as df = n - 1, where n is the sample size. In this case, n = 153, so df = 153 - 1.
2. Step 2: Determine the critical values X²L and X²R. For a 99% confidence level, the significance level (α) is 1 - 0.99 = 0.01. Divide α into two tails: α/2 = 0.005 for each tail. Use a chi-square distribution table or statistical software to find the critical values for df = 152 at α/2 = 0.005 (right tail) and 1 - α/2 = 0.995 (left tail).
3. Step 3: Use the formula for the confidence interval of the population standard deviation (σ). The formula is: CI for σ = [sqrt((df * s²) / X²R), sqrt((df * s²) / X²L)], where s is the sample standard deviation, df is the degrees of freedom, and X²L and X²R are the critical values from Step 2.
4. Step 4: Substitute the known values into the formula. Use df = 152, s = 7.10 cm, and the critical values X²L and X²R obtained in Step 2. Compute the lower and upper bounds of the confidence interval for σ.
5. Step 5: Interpret the result. The confidence interval provides a range of plausible values for the population standard deviation (σ) at the 99% confidence level. Ensure the units (cm) are included in the final interval.

Key Concepts

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of a sample, df is typically calculated as the sample size minus one (n-1). This concept is crucial for determining the appropriate statistical tests and critical values, especially in t-distributions and chi-squared tests.

Critical Values

Critical values are the threshold points that define the boundaries of a confidence interval or the rejection region in hypothesis testing. For a chi-squared distribution, critical values (X2L and X2R) are determined based on the degrees of freedom and the desired confidence level. These values help in assessing whether the sample statistic falls within a specified range, thereby influencing conclusions about the population parameter.

Confidence Interval for Standard Deviation

A confidence interval for the standard deviation (σ) provides a range of values within which the true population standard deviation is likely to fall, given a certain level of confidence (e.g., 99%). This interval is calculated using the sample standard deviation and the chi-squared distribution, which accounts for the sample size and variability. Understanding this concept is essential for making inferences about population parameters based on sample data.