The following hypotheses are given: H0: Forty percent of the...

The following hypotheses are given: $H_{0}:$ Forty percent of the observations are in category $A, 40$ percent are in $B,$ and 20 percent are in $\mathrm{C}$ $H_{1}:$ The observations are not as described in $H_{0}$. We took a sample of $60,$ with the following results. $$ \begin{array}{|cc|} \hline \text { Category } & f_{o} \\ \hline \mathrm{A} & 30 \\ \mathrm{~B} & 20 \\ \mathrm{C} & 10 \\ \hline \end{array} $$ a. State the decision rule using the .01 significance level. b. Compute the value of chi-square. c. What is your decision regarding $H_{0} ?$

The following hypotheses are given:
$H_{0}:$ Forty percent of the observations are in category $A, 40$ percent are in $B,$ and 20 percent are in $\mathrm{C}$ $H_{1}:$ The observations are not as described in $H_{0}$.
We took a sample of $60,$ with the following results.
$$
\begin{array}{|cc|}
\hline \text { Category } & f_{o} \\
\hline \mathrm{A} & 30 \\
\mathrm{~B} & 20 \\
\mathrm{C} & 10 \\
\hline
\end{array}
$$
a. State the decision rule using the .01 significance level.
b. Compute the value of chi-square.
c. What is your decision regarding $H_{0} ?$

Key Concepts

Expected Frequencies

Expected frequencies represent the counts anticipated under the null hypothesis for each category. They are computed by multiplying the total sample size by the hypothesized proportions for each category. These values are essential for determining the chi-square statistic, as they provide the baseline against which the observed frequencies are compared.

Null and Alternative Hypotheses

These are the fundamental propositions in hypothesis testing. The null hypothesis (H0) represents a statement of no effect or a specific expected distribution in the population, while the alternative hypothesis (H1) indicates that the data deviate from this expected pattern. They form the basis for deciding whether observed differences are due to chance or reflect a true underlying effect.

Chi-Square Goodness-of-Fit Test

This test is used to compare the observed frequencies in categorical data with the expected frequencies derived from a theoretical distribution as stated by the null hypothesis. It involves calculating a chi-square statistic that measures the discrepancy between observed and expected counts, which is then used to assess the adequacy of the assumed distribution.

Decision Rule and Significance Level

In hypothesis testing, the decision rule specifies when to reject the null hypothesis based on a predetermined significance level (such as 0.01). For the chi-square goodness-of-fit test, if the computed chi-square statistic exceeds the critical value from the chi-square distribution (with appropriate degrees of freedom), the null hypothesis is rejected, indicating that the observed data significantly deviate from the expected distribution.

Degrees of Freedom

Degrees of freedom for the chi-square goodness-of-fit test are calculated as the number of categories minus one. This value is crucial for determining the correct critical value from the chi-square distribution table, ensuring the test accurately accounts for the number of parameters estimated from the data.

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