can you please try to help me understand this?
Q:The table below shows a distribution and the observed frequencies of the values of a variable from a simple random sample of a population. Perform a chi-square goodness-of-fit test, at the specified significance level, to decide whether the distribution of the variable differs from the given distribution.
Distribution: 0.2 0.2 0.1 0.2 0.3
Observed freq: 37 15 12 23 43
significance level=0.05
20
Certainly! To perform a chi-square goodness-of-fit test, you need to follow these steps:
Step 1: Set up hypotheses
- Null hypothesis (H0): The distribution of the variable does not differ from the given distribution.
- Alternative hypothesis (Ha): The distribution of the variable differs from the given distribution.
Step 2: Calculate Expected Frequencies
- Calculate the expected frequencies for each category under the null hypothesis. This is done by multiplying the distribution probabilities by the total observed frequency.
- Expected freq for category 1 = 0.2 * (Total Observed Frequency) = 0.2 * (37 + 15 + 12 + 23 + 43) = 26
- Expected freq for category 2 = 0.2 * (Total Observed Frequency) = 0.2 * (37 + 15 + 12 + 23 + 43) = 26
- Expected freq for category 3 = 0.1 * (Total Observed Frequency) = 0.1 * (37 + 15 + 12 + 23 + 43) = 13
- Expected freq for category 4 = 0.2 * (Total Observed Frequency) = 0.2 * (37 + 15 + 12 + 23 + 43) = 26
- Expected freq for category 5 = 0.3 * (Total Observed Frequency) = 0.3 * (37 + 15 + 12 + 23 + 43) = 39
Step 3: Calculate the Test Statistic
- The chi-square test statistic is given by the formula:
- Chi-square = ∑ [(Observed freq - Expected freq)^2 / Expected freq]
Step 4: Determine the Critical Value
- Consult the chi-square distribution table with degrees of freedom equal to k - 1 (k is the number of categories) and the given significance level (0.05). Identify the critical value.
Step 5: Compare the Test Statistic with the Critical Value
- If the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Now let's perform the calculations:
Expected freq: 26 26 13 26 39
Observed freq: 37 15 12 23 43
Step 3: Calculate the Test Statistic
- Calculate the chi-square test statistic using the formula mentioned above.
Step 4: Determine the Critical Value
- Consult the chi-square distribution table with degrees of freedom = 5 - 1 = 4 and the given significance level (0.05). Identify the critical value.
Step 5: Compare the Test Statistic with the Critical Value
- If the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
I hope this explanation helps you understand the steps involved in performing a chi-square goodness-of-fit test.