Question

A response can fall into one of \(k=4\) categories with hypothesized cell probabilities given If 250 responses are recorded, what are the four expected cell counts for the chi-square test? $$ p_{1}=.25, p_{2}=.15, p_{3}=.10, p_{4}=? $$

Short answer

Answer: The expected counts for the four categories are as follows: - Category 1: 62.5 - Category 2: 37.5 - Category 3: 25 - Category 4: 125

Step-by-step solution

Find the missing cell probability (p4)

To find the probability of p4, we need to use the fact that the sum of all the probabilities should be equal to 1. We were given the probabilities for p1, p2, and p3, so we can find p4 as follows: $$ p_{4} = 1 - (p_{1} + p_{2} + p_{3}) $$ Plug in the given probabilities into the formula and calculate p4: $$ p_{4} = 1 - (0.25 + 0.15 + 0.10) = 1 - 0.50 = 0.50 $$ So, the missing cell probability (p4) is 0.50.

Calculate the expected counts for each category

Now that we have all the cell probabilities (p1, p2, p3, and p4), we can calculate the expected counts for each category. To do this, we will multiply the probability of each category by the total number of responses (250): $$ \text{Expected Count}_{1} = 250 \times p_{1} = 250 \times 0.25 = 62.5 \\ \text{Expected Count}_{2} = 250 \times p_{2} = 250 \times 0.15 = 37.5 \\ \text{Expected Count}_{3} = 250 \times p_{3} = 250 \times 0.10 = 25 \\ \text{Expected Count}_{4} = 250 \times p_{4} = 250 \times 0.50 = 125 $$ The expected counts for the four categories for the chi-square test are as follows: Category 1: 62.5 Category 2: 37.5 Category 3: 25 Category 4: 125

Key concepts

Expected Cell Counts

When you perform a chi-square test, it's crucial to understand the concept of 'expected cell counts'. These are the theoretical frequencies that we would expect for each category if the null hypothesis is true.

In our exercise, to calculate expected cell counts, we multiply each hypothesized probability by the total number of responses. For instance, if we have a probability of 0.25 and 250 responses, the expected count is calculated as: \( 250 \times 0.25 = 62.5 \). It's similar to predicting how many times you will get heads if you flip a fair coin four times. On average, you'd expect heads to come up twice, because the probability is 0.5 for each flip.

Getting the expected cell counts accurate is essential for the next steps in the chi-square test, as they are used to measure how far off the observed counts are from what we would expect to see by chance alone.

Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty. It plays a fundamental role in statistics and dictates how we calculate expected cell counts.

In the case of the chi-square test, we deal with categorical data spread across different categories, each having a certain probability of occurrence. As in our exercise, the probabilities for each cell must add up to 1 (100%), as it's certain that every response will fall into one of the categories.

Understanding the basics of probability, such as how to calculate cumulative probabilities and the concept of independent events, helps in not only solving such problems but in predicting and interpreting different outcomes in natural and social phenomena.

Null Hypothesis

The null hypothesis, symbolized as \( H_0 \), is a statement in statistics that there is no effect or no difference, and it serves as a starting point for any statistical significance testing. In a chi-square test, the null hypothesis often claims that the observed frequencies will fit the expected frequencies if the underlying assumptions are correct.

For example, if a die is fair, you'd expect each number to appear with an equal probability. The null hypothesis would state that the die rolls do not favor any specific number. Rejecting the null hypothesis implies that the observed data provides enough evidence to suggest an effect or a difference that cannot be attributed to random chance.

Statistical Significance

Statistical significance is a determination that a relationship between variables or differences within a variable is caused by something other than chance. In the context of the chi-square test, it's the measure of whether the observed distribution of data differs significantly from what we would expect under the null hypothesis.

If we have a low probability (p-value) associated with the chi-square statistic, lower than a predetermined significance level such as 0.05, we say the result is statistically significant. This means that it's unlikely these results occurred by chance, leading us to reject the null hypothesis and consider alternative explanations.