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Chapter 7: Problem 9

Give a null hypothesis for a goodness-of-fit test and a frequency table from a sample. For each table, find: (a) The expected count for the category labeled \(\mathrm{B}\) (b) The contribution to the sum of the chi-square statistic for the category labeled \(\mathrm{B}\) (c) The degrees of freedom for the chi-square distribution for that table $$ \begin{aligned} &H_{0}: p_{a}=p_{b}=p_{c}=p_{d}=0.25\\\ &H_{a}:\\\ &\text { Some } p_{i} \neq 0.25\\\ &\begin{array}{lccc|c} \hline \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \text { Total } \\ 40 & 36 & 49 & 35 & 160 \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified
The expected count for category B is 40. The contribution to the sum of the chi-square statistic for category B is 0.4. The degrees of freedom for the chi-square distribution for the table is 3.

Step by step solution

01

Calculating Expected Count

Since the null hypothesis assumes equal probability for each category, the expected frequency (count) for any of these categories can be obtained by equally dividing the total sample size. For category B:\n\nThe total sample size = 160\n\nSo, Expected count for B, \( E_B = \frac{Sample \,Size}{Number\, of\, Categories} = \frac{160}{4} = 40.\)
02

Calculating Chi-square statistic

The chi-square statistic, denoted by 蠂2, is calculated using the formula:\n\n\( 蠂2 = \sum \frac{(O-E)^2}{E} \)\n\nHere, 'O' is the Observed frequency and 'E' is the Expected frequency.\n\nFor category B, Observed count 'O' = 36 and Expected count 'E' = 40 from Step 1.\n\nSo, for category B: \( 蠂2_B = \frac{(O_B - E_B)^2}{E_B} = \frac{(36-40)^2}{40} = 0.4 \)
03

Calculating Degree of Freedom

In a Chi-square Goodness of Fit test, the degrees of freedom is given by the formula:\n\n\( df = number\, of\, categories - 1 \)\n\nHere, number of categories = 4 (A, B, C, and D)\n\nSo, \( df = 4 - 1 = 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
In statistics, the null hypothesis is a fundamental concept used to test if there is enough evidence to reject a certain assumption about a data set. For a goodness-of-fit test, the null hypothesis is typically a statement that assumes no significant difference between the observed sample data and the expected data predicted by a particular model.

In our example, the null hypothesis is expressed as:
  • \( H_0: p_a = p_b = p_c = p_d = 0.25 \)
This suggests that each category (A, B, C, D) has an equal probability of occurrence, specifically 0.25 or 25% each. Essentially, this null hypothesis assumes that the distribution of the observed sample matches the theoretical distribution perfectly. It's crucial to test this assumption to determine whether the observed data significantly deviates from what is expected, indicating a possible difference that may require further investigation.
Chi-square Statistic
The chi-square statistic is a measure used to compare the observed and expected frequencies in categorical data to test hypotheses about distributions. This statistic helps determine how well the observed data fits the expected distribution under the null hypothesis.

To compute the chi-square statistic for a particular category, like B in this exercise, the formula is:
  • \( \chi^2 = \sum \frac{(O - E)^2}{E} \)
Where:
  • \( O \) is the observed frequency
  • \( E \) is the expected frequency
For our exercise, with the observed count for B being 36 and the expected count 40, the contribution to the chi-square statistic from category B is:
  • \( \chi^2_B = \frac{(36 - 40)^2}{40} = 0.4 \)
A low chi-square value indicates that the observed data fits well with the expected distribution. Conversely, a high value suggests a poor fit, indicating potential rejection of the null hypothesis.
Degrees of Freedom
Degrees of freedom (df) represent the number of independent values or quantities that can vary in a statistical calculation while still conforming to a set constraint. In the context of a chi-square test, degrees of freedom are essential as they influence the shape of the chi-square distribution used to interpret results.

For the goodness-of-fit test, the degrees of freedom can be calculated as:
  • \( df = \text{number of categories} - 1 \)
This calculation considers that, if you know the total sum and all but one of the individual category counts, the last one is fixed. In our example, with four categories (A, B, C, D), it results in:
  • \( df = 4 - 1 = 3 \)
These degrees of freedom help determine the critical value when evaluating the chi-square statistic to decide whether to accept or reject the null hypothesis. Degrees of freedom thus play a crucial role in statistical testing, influencing the conclusions drawn from data analysis.

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Most popular questions from this chapter

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