Stats without Tears
Solutions for Chapter 12

(If you have MATH200A V6, you’ll see the p-value, degrees of freedom, and χ² test statistic on the same screen as the graph.)

Common mistake: When a model is given in percentages, some students like to convert the observed numbers to percentages. Never do this! The observed numbers are always actual counts and their total is always the actual sample size.

Remark: You could give the model as decimals, .25, .20, .15 and so on. But for the model, all that matters is the relative size of each category to the others, so it’s simpler to use whole-number ratios.

Common mistake: If you do convert the percentages to decimals, remember that 8% and 7% are 0.08 and 0.07, not 0.8 and 0.7.

• L3 shows the expected counts, and the lowest is 70, so all are ≥5.
• The problem says that the 1000 people were a random sample.
• There are millions of ice cream lovers, so the sample of 1000 is less than 10% of population.

Remark: For Case 6 only, you could write your non-conclusion as something like “the model is not inconsistent with the data” or “the data don’t disprove the model.”

Remark: The χ² test keeps you from jumping to false conclusions. Eyeballing the observed and expected numbers (L2 and L3), you might think they’re fairly far off and the model must be wrong. Yet the test gives a largish p-value.

Remark: If it had gone the other way — if p was less than α — you would say something like “At the .05 level of significance, the model is inconsistent with the data” or “the data disprove the model” or simply “the model is wrong”.

• The problem states that the sample was random.
• With millions of party members, the samples are under 10% of the population.
• Check the B matrix and find that the lowest expected count is 106.45. Therefore, all expected counts are above the minimum of 5.

Alternative: use MATH200A part 7 for steps 3–4 and RC.

Remark: “Depends on” does not mean that’s the only factor. But if you don’t like “depends on”, you could say “is not independent of”. Or you could say, “party affiliation is a factor in a person’s opinion on gun control.”

• Random sample: given.
• There are many, many first graders, far more than 10×425 = 4250.
• All L3’s (expected counts) are 85, so all are ≥5.

• Random sample: given.
• At a glance, it looks like the sample size is around 100. But it’s obviously less than 10% of the number of women.
• Expected values (Matrix B) show one value 4.8148, which is below 5. You can say that it’s just barely below 5, and it’s the only one, so the requirement is effectively met. That’s true, but it’s also a moot point because of the high p-value.

Remark: The large p-value makes it really tempting to declare that the two variables are independent. But that would be accepting H₀, which we must never do. It’s always possible that there is a connection and we were just unlucky enough that this particular sample didn’t show it.

Some researchers would say “There is insufficient evidence to reject the hypothesis of independence.” Strictly speaking, that’s the same error. However, when the audience is researchers, rather than the non-technical public, it may be understood that they’re not really accepting H₀, only failing to reject it pending the outcome of a further study.

Remark: There are a lot of reasons for this. Judges tend to be older and tend to prefer jurors closer to their own age. Also, older candidates are more likely to be retired, which means they are less likely to be exempt by reason of their occupation.

• Simple random sample: given.
• 500 men is obviously far below 10% of the total number.
• All expected counts (Matrix B) are 14.364 or greater, ≥5.

Results: χ² = 4.74, df = 6, p-value = 0.5769

Common mistake: Never enter the totals in a two-way test.

• Random sample? Yes, randomized experimental design. ✔
• Sample less than 10% of population? Yes, the population of people exposed to the common cold is indefinitely large. ✔
• All expected values ≥5? Yes, matrix B shows all values at least 5.6. ✔

Remark: Researchers might write something like “Echinacea made no significant difference to infection rates in our study” with the p-value or significance level. It’s understood that this does not prove Echinacea ineffective — this particular study fails to reach a conclusion. But as additional studies continue to find p > α, our confidence in the null hypothesis increases.