Question

In Exercises 7.1 to \(7.4,\) find the expected counts in each category using the given sample size and null hypothesis. \(H_{0}:\) All three categories \(A, B, C\) are equally likely; \(\quad n=1200\)

Short answer

The expected counts for each category \(A, B, C\) under the null hypothesis is \(400\).

Step-by-step solution

Identify the Total Sample Size and Categories

The total sample size provided is \(n=1200\) and the categories are identified as \(A, B, C\). According to the null hypothesis, these categories are equally likely.

Divide the Total Sample Size by the Number of Categories

Since the null hypothesis asserts that all categories are equally likely, we simply need to divide the total sample size by the number of categories to find out the expected counts in each category. So, divide 1200 by 3.

Compute the Expected Counts

On dividing the total sample size by the number of categories, we find that the expected count for each category \(A, B, C\) is \(400\). So, under the null hypothesis, we would expect 400 occurrences in each category if all are equally likely.

Key concepts

Null Hypothesis

The null hypothesis, often denoted as H₀, is a fundamental concept in statistics that refers to a general statement or default position asserting that there is no significant difference or effect. In the context of expected counts, the null hypothesis takes on a specific claim about the population that is being tested. For example, the idea that 'All three categories A, B, C are equally likely' is a form of the null hypothesis. This assumption is critical as it provides the baseline that the actual data is compared to in order to determine if there is a statistically significant difference from what is expected.

When working with the null hypothesis and expected counts, you are essentially asking, 'If there truly were no preference or difference between categories, what would we expect the data to look like?' Thus, the null hypothesis acts as a benchmark for comparison. Testing the null hypothesis involves comparing observed data from an experiment or survey to what we would expect to find if the null hypothesis were true.

Under the null hypothesis of equal likelihood for categories A, B, and C, we assume that each outcome is just as probable as the others. Therefore, in a sample of 1200 instances, we would anticipate each category to represent a third of the sample, given no other influencing factors.

Sample Size

The sample size, denoted as n, refers to the total number of observations or data points that are collected and analyzed in a study. It is a crucial element of any statistical analysis as it affects the power of a test and therefore the reliability of the conclusions that can be drawn from the study. In statistical terms, a larger sample size can provide more accurate estimates of population parameters and increase the likelihood of detecting a true effect.

When examining expected counts in the context of the null hypothesis, the sample size becomes important as it dictates the precision of the expected counts and the robustness of the subsequent statistical test. Taking our exercise as an example, with a sample size of n=1200, we gain enough data points to distribute among the categories and determine if the observed counts significantly deviate from what was expected under the null hypothesis.

In the given exercise, a substantial sample size allows for an effective illustration of the concept of expected counts under the assumption of equally likely categories. If the sample size were smaller, the expected counts might be less reliable, and it could be harder to draw meaningful conclusions or detect statistically significant differences.

Category Probability

Category probability is a term that describes the likelihood of an observation falling into a particular category out of a set of possible categories. In relation to expected counts, the category probability is used to determine what those counts should be under specific assumptions, such as those stated in the null hypothesis.

For instance, if the null hypothesis claims that three categories are equally likely, as in our textbook exercise, then the probability for each category A, B, and C is \( \frac{1}{3} \), considering there are three categories. This uniform probability distribution is applied to calculate the expected count for each category. With the sample size of n=1200, the expected count for each category is found by multiplying the category probability by the total sample size: \(1200 \times \frac{1}{3} = 400\).

The probability assigned to each category directly impacts the expected count; a change in the probability would alter the expected counts accordingly. Understanding category probability is essential for interpreting the results of the hypothesis test and for determining the distribution of the expected counts amongst the categories.