Sec 53-55 Intro to Chi-Square, Computation for 1-Way Chi-Square

Question 1

Chi-Square

Answer 1

CHI-SQUARE is the test done to determine if the results of NAMING DATA (eg. political parties, or flavors of ice cream.) are able to reject the NULL HYPOTHESIS.

NOMINAL (NAMING) DATA does NOT directly permit the computation of means and standard deviations. Instead, researchers report the number of subjects who named each category (i.e., the frequency) and the corresponding proportions or percentages.

• It is NOT possible to use a t -Test or ANOVA to test this null hypothesis because the result does NOT consist of means and standard deviations.
• The appropriate test for data that includes frequencies or numbers of cases is CHI-SQUARE (Symbol = X²)

There are two types of CHI-SQUARE (ONE-WAY and TWO-WAY):

ONE-WAY CHI-SQUARE – also known as a GOODNESS-OF-FIT chi-square. The subjects are classified in only one way (Ex: whom they plan to vote for).

• Ex: of ONE-WAY CHI-SQUARE: A random sample of 200 registered voters was drawn, and voters were asked which of two candidates running for an elected office they planned to vote for.
  
  • Candidate Smith n = 110 (55.0%)
  • Candidate Doe n = 90 (45.0%)
• This suggests Smith is leading. HOWEVER, because only a random SAMPLE of voters was surveyed. It is POSSIBLE, for instance, that the population of voters is evenly split and that a DIFFERENCE of 10 percentage points was OBTAINED BECAUSE OF SAMPLING ERRORS associated with random sampling (i.e. the NULL HYPOTHESIS).
  
  • More specifically, this NULL HYPOTHESIS says that THERE IS NO TRUE DIFFERENCE (in candidate preference) in the POPULATION and that, in the population, the voters are evenly split.
• COMPUTATION of a ONE-WAY CHI-SQUARE – (Please refer to the “CHI-SQUARE 1-WAY” Tab in the “Quant Psych Tool” Spreadsheet.
  
  • OBSERVED FREQUENCIES (Symbol = O ) – are the frequencies OBTAINED from the SAMPLE.
  • EXPECTED FREQUENCIES (Symbol = E ) – are the frequencies EXPECTED based on the NULL HYPOTHESIS.
  • OBSERVED VALUE of CHI-SQUARE (Symbol = X² ) – Calculated using the Chi-square formula.
  • CRITICAL VALUE of CHI-SQUARE – Gotten using the formula for Chi-Square Degrees of Freedom (df = number of categories - 1) and Table 11 at the end of the book.
• DECISION RULE: “If the observed value of chi-square is greater than the critical value, then reject the null hypothesis. Otherwise, do not reject it.”
• In the example above, the Observed Value of Chi-Square = 1.241 and the Critical Value of Chi-square = 5.991.
  
  • So, because the observed value (1.241) is NOT greater than the critical value (5.991), do NOT reject the null hypothesis at the .05 level. The conclusion is that the differences are NOT statistically significant.

TWO-WAY CHI-SQUARE – The subjects are classified in TWO ways (Ex: Samples from two populations of voters (i.e., males and females) were classified in terms of whom they plan to vote for (Candidate #1 or #2)​.

• Ex: of TWO-WAY CHI-SQUARE: A random sample of 200 male registered voters and a random sample of 200 female registered voters were drawn (Gender is the first Classification), and voters were asked which of two candidates running for elected office they planned to vote for (Candidate selection the second Classification).
  
  • Males – Candidate #1 n = 80, Candidate #2 n = 120
  • Females – Candidate #1 n = 120, Candidate #2 n = 80
• This suggests that Candidate #1 is a stronger candidate among females, and Candidate #2 is a stronger candidate among males.
• However, only a random sample was surveyed. So we must consider how likely it is that the observed differences between the two groups (males and females) were created by random sampling errors.
• CHI-SQUARE is the appropriate test because the data are at the NOMINAL (NAMING) level.

THE DECISION RULE FOR 2-WAY CHI SQUARE – Same as that for the 1-Way. The idea is that you can REJECT the NULL HYPOTHESIS because the differences in the findings were STATISTICALLY SIGNIFICANT – or they weren’t.

CAUTION: The probabilities obtained with chi-square will NOT be accurate if the EXPECTED FREQUENCY in any cell is small (i.e., less than about 10).

• In most cases, this can be prevented by using a reasonably large sample size.
• Another solution is to COLLAPSE adjoining cells when there is a logical basis for doing so.
  
  • Ex: Pretend you had the followoing NOMINAL Data: (See image below)