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Chapter 19: Problem 767

What is the relationship between the \(\mathrm{X}^{2}\) statistic and sample size given that everything else remains constant?

Short Answer

Expert verified
The relationship between the chi-square (\(\mathrm{X}^{2}\)) statistic and sample size, with all else remaining constant, is that the chi-square statistic tends to increase as the sample size increases. This is due to the increase in degrees of freedom with a larger sample size, making the chi-square test more sensitive to deviations from the null hypothesis. Additionally, an increased sample size improves the power and accuracy of the test, allowing for more reliable conclusions based on the chi-square statistic.

Step by step solution

01

Understanding the chi-square distribution and its properties.

The chi-square distribution represents the distribution of the sum of squared independent standard normal random variables. The chi-square (\(\mathrm{X}^{2}\)) statistic is used in various statistical tests, such as goodness-of-fit tests and tests for independence in contingency tables. An important property of the chi-square distribution is its degrees of freedom (df), which is affected by the number of categories in the analysis or the sample size. The degrees of freedom is defined as the number of independent pieces of information that are used to estimate a parameter. In the context of the chi-square test, the degrees of freedom usually equal (number of categories - 1) or (number of rows - 1) 脳 (number of columns - 1) for a contingency table.
02

Understanding how sample size affects the chi-square statistic

When the sample size increases, assuming all other factors remain constant, the degrees of freedom increase. This is because more information is available to estimate the parameters, making the chi-square statistic more sensitive to deviations from the null hypothesis. As a result, the chi-square statistic will tend to increase as sample size increases. In particular, with a larger sample size, even small deviations from the null hypothesis can yield a large chi-square statistic, leading to a higher chance of rejecting the null hypothesis. With a small sample size, the chi-square test may fail to detect deviations from the null hypothesis, leading to a lower chi-square statistic and a higher chance of failing to reject the null hypothesis when it is actually false.
03

Analyzing the effect of sample size on power and accuracy

The power of a statistical test is defined as the probability of correctly rejecting the null hypothesis when it is false. The power is affected by factors such as the sample size, effect size, and significance level. As mentioned earlier, an increased sample size results in a higher power for the chi-square test, meaning a higher probability of detecting deviations from the null hypothesis when they exist. Accuracy in statistical tests is also related to sample size. When the sample size is large, estimates of parameters are generally more accurate, resulting in more reliable test statistics and conclusions. With a small sample size, there may be more uncertainty in the estimates, which can lead to less accurate test statistics and conclusions.
04

Conclusion

The relationship between the chi-square (\(\mathrm{X}^{2}\)) statistic and sample size, with all else remaining constant, is that the chi-square statistic tends to increase as the sample size increases. This occurs because the degrees of freedom increase with sample size, making the chi-square test more sensitive to deviations from the null hypothesis. Increased sample size also improves the power and accuracy of the test, allowing for more reliable conclusions based on the chi-square statistic.

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Key Concepts

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

Sample Size
Sample size is one of the key factors that significantly impacts the results of statistical tests, including the chi-square (\( \chi^2 \)) test. In simple terms, it refers to the number of observations or data points collected in a study. A larger sample size provides more information and a richer dataset. This often leads to a more accurate estimate of the population parameters. In the context of chi-square tests, a larger sample size increases the sensitivity of the test. This means even small differences from the null hypothesis can result in a significant chi-square statistic.On the flip side, a small sample size might lead the chi-square test to miss these differences. This can result in a failure to identify the true relationship between variables.
Degrees of Freedom
Degrees of freedom play an essential role in statistical tests, including those utilizing the chi-square distribution. They represent the number of independent values or quantities that have the freedom to vary. For example, in a chi-square goodness-of-fit test, the degrees of freedom are calculated as the number of categories minus one.Increasing the sample size typically increases the degrees of freedom, assuming the number of categories remains constant. More degrees of freedom imply the \( \chi^2 \) statistic can capture more variation in the data. The relationship between degrees of freedom and chi-square statistics means that as degrees of freedom increase, the test becomes more robust and sensitive. This is crucial for drawing more reliable conclusions.
Null Hypothesis
The null hypothesis is a fundamental concept in statistics. It is a statement that there is no effect or no difference between groups or variables in the study. When conducting a chi-square test, the null hypothesis might state that two distributions are independent or that observed frequencies match expected frequencies. The chi-square test aims to assess whether the observed data deviate significantly from the null hypothesis. If the chi-square statistic is high and the p-value is below a chosen significance level, we might reject the null hypothesis. A larger sample size increases the likelihood of detecting any deviations from the null hypothesis, providing a better chance for accurate detection.
Statistical Power
Statistical power is the probability that a test will correctly reject a false null hypothesis. It is a measure of a test's ability to detect an effect when there is one. Several factors affect power, including sample size, effect size, and significance level. For chi-square tests, a larger sample size enhances statistical power. This increase in power means there is a higher likelihood of identifying true relationships within the data. With improved power, conclusions drawn from the test are more reliable, and the risk of a Type II error (failing to reject a false null hypothesis) is reduced. Ensuring adequate statistical power is vital for the validity of your study's findings.

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Most popular questions from this chapter

Consider two independent multinomial distributions with parameters \(\mathrm{n}_{1}, \mathrm{P}_{11}, \mathrm{P}_{21}, \ldots \mathrm{P}_{\mathrm{k} 1}\) and \(\mathrm{n}_{2}, \mathrm{P}_{12}, \mathrm{P}_{22}, \ldots \mathrm{P}_{\mathrm{k} 2}\) Devise a test for the hypothesis that the parameters satisfy, \(\mathrm{P}_{11}=\mathrm{P}_{12}, \ldots \mathrm{P}_{\mathrm{k} 1}=\mathrm{P}_{\mathrm{k} 2}\) where all probabilities are unspecified.

The results of a survey show that the 518 respondents may 168 be categorized as ie \(\begin{aligned}&\text { be categorized as } \\\&\text { Protestant - Republicans } & 126 \\\&\text { Protestant - Democrats } & 71 \\\&\text { Protestant - Independents } & 19 \\\&\text { Catholic - Republicans } & 61 \\\&\text { Catholic - Democrats } & 93 \\\&\text { Catholic - Independents } & 14 \\\&\text { Jewish - Republicans } & 38 \\\&\text { Jewish - Democrats } & 69 \\\&\text { Jewish - Independents } & 27 \\\&\text { Given this data construct a } & \text { contingency table }\end{aligned}\) table.

Is there a significant relationship between the hair color of a husband and wife? A researcher observes 500 pairs and collects the following data. \begin{tabular}{|c|c|c|c|c|c|} \hline \multirow{2}{*} { WIFE } & \multicolumn{4}{|c|} { H U S B A N D } & Row \\\ \cline { 2 - 5 } & Red & Blonde & Black & Brown & Total \\ \hline Red & 10 & 10 & 10 & 20 & 50 \\ \hline Blonde & 10 & 40 & 50 & 50 & 150 \\ \hline Black & 13 & 25 & 60 & 52 & 150 \\ \hline Brown & 17 & 25 & 30 & 78 & 150 \\ \hline Column total & 50 & 100 & 150 & 200 & 500 \\ \hline \end{tabular} The researcher wishes to test the theory that people tend to select mates with the same hair color. Test this theory at the level of significance \(\alpha=.01\).

Experience in a certain apple-growing region indicates that about 20 per cent of the total apple crop is classified in the best grade, about 40 per cent in the next highest rating, about 30 per cent in the next highest rating, and the remaining 10 per cent in the fourth and lowest rating. A random sample of 50 trees in a certain orchard yielded 600 bushels of apples, classified as follows: 150 bushels of the highest quality apples 290 bushels of the next highest, 140 bushels of the next highest, and 20 bushels of the lowest quality apples Can we conclude that the applet in this orchard are typical?

To find out whether there is a significant association between the sensitivity of the skin to sunlight and the color of a person's eyes, we could take a random sample of, say, 100 people, and test them all with a standard dose of ultra violet rays, noting both their reaction to this test and their eye color. Both of these features are noted for each person in the trial, so the observations are matched. The results could then be tabulated in an association table like this \begin{tabular}{|l|l|l|l|l|l|} \hline \multicolumn{2}{|l|} {} & \multicolumn{3}{|c|} { Ultra-violet reaction } & \multirow{2}{*} {\(\mathrm{R}\)} \\ \cline { 3 - 5 } \multicolumn{2}{|l|} {} & \(++\) & \(+\) & \(-\) & \\ \hline \multirow{2}{*} { Eye Color } & Blue & 19 & 27 & 4 & 50 \\ \cline { 2 - 6 } & Grey or green & 7 & 8 & 5 & 20 \\ \cline { 2 - 6 } & Brown & 1 & 13 & 16 & 30 \\ \hline C & & 27 & 48 & 25 & \(100=\mathrm{N}\) \\ \hline \end{tabular} Is this relationship significant? What are some measures of association which describe the strength of association? How strongly associated are these variables?
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