20 Mar Statistics short answer questions

Two of the main goals of statistical inference are

• estimating unknown parameters, using both point and interval estimates;
• evaluating particular claims about the values of unknown parameters, using null hypothesis significance testing.

Chapters 5 and 6 of your textbook introduce these two goals of statistical inference in the context of data analysis involving categorical variables. When we make inferences from categorical variables, we usually focus on unknown population proportions π. These parameters quantify the proportions of the target population that belong to the different levels constituting a categorical variable or the different possible combinations of levels between two categorical variables. In the simple case of an indicator variable, π refers to the proportion of "successes" present in the population. Conversely, for multilevel categorical variables, π refers to the proportions of a particular level or group out of the overall population.

In Chapters 5-6 and in class, you explored several confidence intervals and hypothesis tests for unknown values of π:

1. confidence intervals for one unknown population proportion;
2. hypothesis tests for one unknown population proportion;
3. confidence intervals for the difference between two unknown population proportions;
4. hypothesis tests for differences between two unknown population proportions;
5. hypothesis tests for differences between multiple unknown population proportions.

The first two of these methods rely on a normal approximation of the sampling distribution of proportions. The third and fourth similarly rely on a normal approximation of the sampling distribution of differences in proportions. The fifth relies on a chi-square approximation of the sampling distribution of X2 statistics.

In your quiz section discussions, you explored how each of these methods can be applied to realistic social science research, using an example from archaeological demography. This example is founded on the idea that the proportion of a population (π) who were juveniles (less than 15 years old) when they died is associated with the population growth rate: as the population growth rate increases, π also increases. More specifically, when π equals approximately 0.16, the population growth rate is 0, holding steady in abundance over time. Alternatively, if π is less than 0.16, this is an indirect indicator of a decreasing population, while a value of π greater than 0.16 indicates a growing population. Archaeologists and demographers also hypothesize that population growth rates of past populations responded to important economic and other cultural changes over time. By implication, π for a study population are expected to differ at different points in time.

The problem archaeological demographers confront is that we do not actually know π for past populations represented by individual cemetery samples. Consequently, we cannot compare values of π between populations represented by different cemeteries. Instead, many archaeologists have treated each cemetery’s sample proportion p as a point estimate of the value of π for the population it represents, comparing p rather than π between samples. However, as you now know, relying on point estimates alone is ill-advised because they are almost always erroneous estimates of unknown parameters.

For this participation report, respond to all of the following questions:

1. When you calculated confidence intervals and performed null hypothesis tests using a z-test for one sample proportion (tentatively Tuesday 15 February and Thursday 17 February), what was the important value of π that you were evaluating, and what does this mean for archaeological demographers?

2. When you performed null hypothesis tests for two samples using a z-test (tentatively Tuesday 22 February), what would the archaeological and demographic implication be if you rejected the null hypothesis? (Think about what π means for population structure and growth.)

3. When you performed null hypothesis tests for multiple samples using a chi-square test (tentatively Thursday 24 February), what would the archaeological and demographic implication be if you rejected the null hypothesis? How is this insight limited compared to the two-sample test?

4. Did you notice that the success-failure condition was more restrictive in some cases than others? If so, why?