8.6
Finding Critical Values for Chi-Square
Consider a curve representing sample data drawn randomly from a normally distributed population. One must construct confidence intervals to estimate o…
Consider sample data on the fuel economy of some car brands. To obtain a 95% confidence interval for the population standard deviation, one must calculate the critical values that separate the likely results from the unlikely ones.
A 95% confidence level covers 95% of the area under the curve, while the remaining 5% area distributed equally on either side.
As the chi-square distribution is asymmetrical, the right and left critical values separating an area of 2.5% or 0.025 on both sides are individually determined.
To determine the right-tailed critical value, locate nine on the left column for degrees of freedom in the chi-square table and find 0.025 across the top row, yielding a value of 19.023.
Since the table provides cumulative areas to the right of the critical value, subtract the remaining 0.025 area from the total area under the curve to obtain 0.975. Now, using the chi-square table, the left-tailed critical value is calculated.
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Q1: Why are left and right critical values determined separately in chi-square distributions?
The chi-square distribution is asymmetrical, unlike symmetric distributions. For a 95% confidence interval, the remaining 5% is split equally as 2.5% on each side. Since the distribution's shape differs on each side, the left and right critical values that separate these 2.5% areas must be calculated independently using the chi-square table.
Q2: How do you find the right-tailed critical value using a chi-square table?
Locate the degrees of freedom in the left column, then find the significance level (0.025 for a 95% confidence interval) across the top row. The intersection gives the right-tailed critical value. For example, with nine degrees of freedom and 0.025 significance level, the value is 19.023.
Q3: What calculation is needed to find the left-tailed critical value?
Since chi-square tables provide cumulative areas from the right, subtract the significance level from 1 to find the left-tailed area. For a 0.025 right-side area, calculate 1 minus 0.025 equals 0.975. Then locate 0.975 in the table's column to obtain the left-tailed critical value.
Q4: How does the confidence level relate to the area under the chi-square curve?
A 95% confidence level covers 95% of the area under the chi-square curve, with the remaining 5% distributed equally on both sides as 2.5% each. These tail areas represent unlikely values, while the central 95% represents likely results for estimating population standard deviation.
Q5: What role does degrees of freedom play in locating chi-square critical values?
Degrees of freedom, calculated as sample size minus one (df = n - 1), determines which row to use in the chi-square table. Each row corresponds to a different degrees of freedom value, and the correct row must be selected before finding the critical value at the intersection with the appropriate significance level column.
Q6: Why is the chi-square table based on cumulative values from the right?
Chi-square tables use right-cumulative values because the distribution is asymmetrical and naturally skewed. This format allows direct reading of right-tailed critical values. For left-tailed values, you must convert by subtracting the desired tail area from 1 to find the cumulative area from the right.
Q7: How do critical values help construct confidence intervals for population standard deviation?
Critical values separate likely sample results from unlikely ones at a specified confidence level. They define the boundaries of the confidence interval, ensuring that 95% of possible sample standard deviations fall within the interval for normally distributed populations, making them essential for reliable population parameter estimation.