Suppose we have a multinomial population with four categories: A, B, C, and D . The null hypothe

step 1 So we will be assuming that all four of those proportions, A, B, C, and D are all equal to .25,

Suppose we have a multinomial population with four...

Suppose we have a multinomial population with four categories: $\mathrm{A}, \mathrm{B}, \mathrm{C},$ and $\mathrm{D} .$ The null hypothesis is that the proportion of items is the same in every category. The null hypothesis is $$ H_{0}: p_{\mathrm{A}}=p_{\mathrm{B}}=p_{\mathrm{C}}=p_{\mathrm{D}}=.25 $$ A sample of size 300 yielded the following results. $$ \begin{array}{llll} \text { A: } 85 & \text { B: } 95 & \text { C: } 50 & \text { D: } 70 \end{array} $$ Use $\alpha=.05$ to determine whether $H_{0}$ should be rejected. What is the $p$ -value?

Suppose we have a multinomial population with four categories: $\mathrm{A}, \mathrm{B}, \mathrm{C},$ and $\mathrm{D} .$ The null hypothesis is that the proportion of items is the same in every category. The null hypothesis is
$$
H_{0}: p_{\mathrm{A}}=p_{\mathrm{B}}=p_{\mathrm{C}}=p_{\mathrm{D}}=.25
$$
A sample of size 300 yielded the following results.
$$
\begin{array}{llll}
\text { A: } 85 & \text { B: } 95 & \text { C: } 50 & \text { D: } 70
\end{array}
$$
Use $\alpha=.05$ to determine whether $H_{0}$ should be rejected. What is the $p$ -value?

Key Concepts

Multinomial Distribution

A multinomial distribution generalizes the binomial distribution to more than two outcomes. It describes the probabilities of counts for each category when there are multiple possible outcomes in independent trials, given fixed probabilities for each outcome and a fixed number of trials.

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement of no difference or no effect. In the context of a multinomial test for equal proportions, it asserts that the probabilities associated with each category are equal, reflecting the assumption that the observed differences are only due to random variation.

Chi-square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a statistical test used to determine whether the observed frequencies in categorical data significantly deviate from the expected frequencies under the null hypothesis. It compares the squared differences between observed and expected counts, scaled by the expected counts, summing these differences across all categories to form the test statistic.

Degrees of Freedom

Degrees of freedom, in the context of the chi-square goodness-of-fit test, are typically calculated as the number of categories minus one. This value is used in determining the critical value or p-value from the chi-square distribution to assess how unusual the observed test statistic is under the null hypothesis.

p-value

A p-value is the probability, under the assumption that the null hypothesis is true, of observing a test statistic as extreme as, or more extreme than, the value calculated from the sample data. It provides an indication of the strength of the evidence against the null hypothesis; a small p-value suggests that the observed data are unlikely under the null hypothesis, leading to its potential rejection.

Significance Level

The significance level, often denoted by alpha, is a pre-specified threshold for judging whether an observed effect is statistically significant. In hypothesis testing, if the p-value is less than the significance level, the null hypothesis is rejected, implying that the observed data provide sufficient evidence to conclude that the effect under study is real.

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