chi

Examples of chi in the following topics:

• Chi-Square Probability Table

  • Areas in the chi-square table always refer to the right tail.

• Student Learning Outcomes

• Chi Square Distribution

  • Define the Chi Square distribution in terms of squared normal deviates
  • A Chi Square calculator can be used to find that the probability of a Chi Square (with 2 df) being six or higher is 0.050.
  • The mean of a Chi Square distribution is its degrees of freedom.
  • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.
  • Chi Square distributions with 2, 4, and 6 degrees of freedom

• Structure of the Chi-Squared Test

  • The chi-square ($\chi^2$) test is a nonparametric statistical technique used to determine if a distribution of observed frequencies differs from the theoretical expected frequencies.
  • Chi-square statistics use nominal (categorical) or ordinal level data.
  • If a chi squared test is conducted on a sample with a smaller size, then the chi squared test will yield an inaccurate inference.
  • First, we calculate a chi-square test statistic.
  • Second, we use the chi-square distribution.

• Notation

  • where df = degrees of freedom depend on how chi-square is being used.
  • (If you want to practice calculating chi-square probabilities then use df = n−1.
  • The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.

• Randomization for two-way tables and chi-square

  • In short, we create a randomized contingency table, then compute a chi-square test statistic.
  • When the minimum threshold is met, the simulated null distribution will very closely resemble the chi-square distribution.

• Estimating a Population Variance

  • The chi-square distribution is used to construct confidence intervals for a population variance.
  • A chi-square distribution can be used to construct a confidence interval for this variance.
  • In fact, the chi-square distribution enters all analyses of variance problems via its role in the $F$-distribution, which is the distribution of the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom.
  • The chi-square distribution is a family of curves, each determined by the degrees of freedom.
  • Note that these critical values are found on the chi-square critical value table, similar to the table used to find $z$-scores.

• Finding a p-value for a chi-square distribution

  • This is where the chi-square distribution becomes useful.
  • How many degrees of freedom should be associated with the chi-square distribution used for ?
  • The chi-square distribution and p-value are shown in Figure 6.10.
  • There are three conditions that must be checked before performing a chi-square test:
  • Degrees of freedom: We only apply the chi-square technique when the table is associated with a chi-square distribution with 2 or more degrees of freedom.

• Facts About the Chi-Square Distribution

  • When df > 90, the chi-square curve approximates the normal.
  • In the next sections, you will learn about four different applications of the Chi-Square Distribution.
  • Think about the implications of right-tailed versus left-tailed hypothesis tests as you learn the applications of the Chi-Square Distribution.

• The chi-square distribution and finding areas

  • To do so, a new table is needed: the chi-square table, partially shown in Table 6.8.
  • Figure 6.9(d) shows a cutoff of 11.7 on a chi-square distribution with 7 degrees of freedom.
  • Figure 6.9(e) shows a cutoff of 10 on a chi-square distribution with 4 degrees of freedom.
  • Figure 6.9(f) shows a cutoff of 9.21 with a chi-square distribution with 3 df.
  • (a) Chi-square distribution with 3 degrees of freedom, area above 6.25 shaded.