http://www.ck12.org Chapter 10. Chi-Square
X^2 =Chi-Square statistical value
d f=degrees of freedom=N−1, whereN=size of the sample
s^2 =sample variance
σ^2 =population variance
Similar to thez−test, we want to test a hypothesis that the sample comes from a population with a variance greater
than the obseved variance. Let’s take a look at an example to help clarify.
Example:Suppose we have a sample of 41 female gymnasts from Mission High School. We want to know if their
heights are truly a random sample of the general high school population, with respect to variance. We know from a
previous study that the standard deviation for height of high school women is 2.2.
To test this question, we first need to generate null and alternative hypotheses. Our null hypothesis states that the
sample comes from the population that has a variance of 4.84 (σ^2 =the standard deviation of the overall population
squared or 4.84). Therefore:
Null Hypothesis H 0 :σ^2 ≤ 4. 84 ( the variance of the sample is greater than or equal to that of the population)
Alternative Hypothesis Ha:σ^2 > 4. 84 (the variance of the sample is less than that of the population)
Using the sample of the 41 gymnasts, we compute the standard deviation and find it to be 1. 2 (s= 1. 2 ). Using the
information from above, we can calculate our Chi-Square value and find that:
=X^2 =
d f s^2
σ^2= ( 40 · 1. 22 )/ 4. 84 = 11. 90
Therefore, since 11.90 is less than 55.76, we fail to reject the null hypothesis and therefore cannot conclude this
sample female gymnasts has significantly higher variance in height when compared to the general female high
school population.
Calculating a Confidence Interval for a Population Variance
Once we know how to test a hypothesis about a single variance, calculating a confidence interval for a population
variance is relatively easy. Again, it is important to remember that this test is dependent on the normality of the
population. For non-normal populations, it is best to use the ANOVA test which we will cover in greater detail in
another lesson.
Similar to constructing confidence intervals in other types of tests, we construct a confidence interval when testing
a population variance to identify a range that we think will encompasses the variance. To construct a confidence
interval for the population variance, we need three pieces of information: the number of observations in a sample,
the variance of the sample, and the desired confidence interval. With the desired confidence interval (most often this
is set at 90 or 95%), we can construct the upper and lower limits around the significance level.
To construct the upper limit of the confidence interval, we set the value equal toα/2 (alpha is the Greek letter “a”)
whereα=probability that the variance isnot inthe interval) and the lower limit to( 1 −(α/ 2 )). Therefore, when
constructing a 90% confidence interval(α= 0. 1 )we would find that the two limits of the confidence interval would
be at 0. 05 (α/ 2 )and 0. 95 ( 1 −(α/ 2 )). Similarly, a 98% confidence interval(α= 0. 02 )would have limits set at
0 .01 and 0.99. Using these limits and the number of degrees of freedom from the sample, we can use the standard
Chi-Square distribution table to look up actual values to construct our confidence interval for population variance.
Let’s look at an example to help clarify.
Example:We randomly select 30 samples of Coca Cola and measure the amount of sugar in each sample. Using the
formula that we learned earlier, we calculate that the variance of the sample is 5.20. What would be the population