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Chi-Square Goodness-of-Fit Test: Using a Chi-Square Goodness of Fit Test
Chi-Square Goodness-of-Fit Test: Using a Chi-Square Goodness of Fit Test with a significance level of 0.05, test the hypothesis that Set 1 is sampled from a Normal Distribution with a population mean equal to the sample mean and a population standard deviation equal to the sample standard deviation. Similarly, test the hypothesis with a significance level of 0.05 that Set 2 is sampled from an Exponential Distribution with a population mean equal to the sample mean. For each test, start with the data classes from your histogram and merge them to ensure each class has a sufficient number of observations. Then, for each data class, calculate the following:
Numbers of observations in the data. Class probability. Class expected value. Chi-square component values. Finally, for each test, calculate the chi-square value, describe the degrees of freedom, and explain your conclusion.
EXAMPLE SETUP
Class
Observed Frequency (oi)
Class Probability
Expected Frequency (ei)
2 Class Component
X ≤ 2
Count observations based on your collected data.
Calculate using the assumed probability distribution.
For each class, take its probability and multiply by n.
2 < X ≤ 7
7 < X ≤ 12
X > 12
Total
n
1.0
n
Full Answer Section
H₁: The data does not follow a Normal distribution.
Set 2 (Exponential):
H₀: The data follows an Exponential distribution with the specified mean.
H₁: The data does not follow an Exponential distribution.
Calculate Expected Frequencies:
For each class, use the assumed distribution (Normal or Exponential) and its parameters (mean and standard deviation) to calculate the probability of an observation falling into that class.
Multiply the probability by the total number of observations to get the expected frequency.
Calculate the Chi-Square Test Statistic: χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ] where:
Oᵢ is the observed frequency for the i-th class
Eᵢ is the expected frequency for the i-th class
Determine Degrees of Freedom:
Degrees of freedom (df) = number of classes - number of parameters estimated from the data - 1
Find the Critical Value:
Use a chi-square distribution table or statistical software to find the critical value for the given significance level (α = 0.05) and degrees of freedom.
Compare the Test Statistic to the Critical Value:
If the calculated chi-square value is greater than the critical value, reject the null hypothesis.
Otherwise, fail to reject the null hypothesis.
Example:
Let's assume we have the following data for Set 1 (Normal) and Set 2 (Exponential):
Class
Observed Frequency (Oᵢ)
Class Probability
Expected Frequency (Eᵢ)
(Oᵢ-Eᵢ)²/Eᵢ
...
...
...
...
...
Total
n
1
n
χ²
After calculating the chi-square value, we would compare it to the critical value from the chi-square distribution table with the appropriate degrees of freedom. If the calculated value is larger, we would reject the null hypothesis and conclude that the data does not follow the assumed distribution.
Note:
It's crucial to ensure that the expected frequencies are large enough to use the chi-square test. A common rule of thumb is that all expected frequencies should be at least 5.
For more complex distributions or large datasets, statistical software can be used to perform the calculations and generate the p-value.
By following these steps and using appropriate statistical software, you can effectively test the goodness-of-fit of your data to specific distributions.
Sample Answer
A Chi-Square Goodness-of-Fit Test is a statistical test used to determine whether a sample data fits a specific distribution. In this case, we'll test if Set 1 follows a Normal Distribution and Set 2 follows an Exponential Distribution.
Steps Involved:
Data Preparation:
Create a histogram for each dataset to visualize the distribution.
Merge classes if necessary to ensure each class has an expected frequency of at least 5.
Hypotheses:
Set 1 (Normal):
H₀: The data follows a Normal distribution with the specified mean and standard deviation.