Question

You can use a goodness-of-fit test to determine whether all of the criteria for a binomial experiment have actually been met in a given application. Suppose that an experiment consisting of four trials was repeated 100 times. The number of repetitions on which a given number of successes was obtained is recorded in the table: Estimate \(p\) (assuming that the experiment was binomial), obtain estimates of the expected cell frequencies, and test for goodness-of-fit. To determine the appropriate number of degrees of freedom for \(\mathrm{X}^{2}\), note that \(p\) was estimated by a linear combination of the observed frequencies.

Short answer

Answer: To determine whether an experiment follows a binomial distribution using the goodness-of-fit test, you need to compare the computed Chi-square statistic (𝜒²) with the critical Chi-square value (𝜒_c²). If 𝜒² > 𝜒_c², the test suggests that the experiment does not follow a binomial distribution. However, if 𝜒² ≤ 𝜒_c², the test indicates that the experiment is consistent with a binomial distribution.

Step-by-step solution

Estimate \(p\) (probability of success)

We will use the average number of successes across the 100 repetitions to estimate \(p\). To do this, we need to compute the weighted average number of successes, where the weight of each value is given by the observed frequency. Let \(p\) be the probability of success: $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k},$$ where \(n\) is the number of trials (4 in this case). Let's find \(p\). $$p = \frac{\sum_{k=0}^4 k \times f(k)}{100 \times n}$$, where \(f(k)\) is the observed frequency of each success.

Calculate expected frequencies and Chi-square statistic (\(\chi^2\))

Compute the expected frequency for each cell using the Binomial distribution formula: $$f_e(k) = 100 \times \binom{n}{k} p^k (1-p)^{n-k}$$ Now, compute the Chi-square statistic using the sum of the squares of the differences between the observed frequencies \(f(k)\) and the expected frequencies \(f_e(k)\), divided by the expected frequencies \(f_e(k)\): $$\chi^2 = \sum_{k=0}^4 \frac{(f(k) - f_e(k))^2}{f_e(k)}$$

Determine the number of degrees of freedom and find critical Chi-square value

To find the number of degrees of freedom (df), note that \(p\) was estimated by a linear combination of the observed frequencies. So, the number of degrees of freedom is \(4-1=3\), since we have 4 outcomes and \(p\) is estimated from data. Now, we need to find the critical Chi-square value from a Chi-square table for the desired level of confidence (usually 95%). You can look the critical value (\(\chi_c^2\)) online in the Chi-Square distribution table at \(\alpha\) = 0.05 for \(df=3\).

Compare computed Chi-square statistic with the critical value

Now, compare our computed Chi-square statistic (\(\chi^2\)) with the critical Chi-square value (\(\chi_c^2\)): - If \(\chi^2 \gt \chi_c^2\), the goodness-of-fit test states that we reject the null hypothesis and conclude that the experiment does not follow a binomial distribution. - If \(\chi^2 \le \chi_c^2\), the goodness-of-fit test states that we cannot reject the null hypothesis and conclude that the experiment is consistent with a binomial distribution. In conclusion, based on the results of this test, you will be able to determine whether the experiment follows a binomial distribution or not.

Key concepts

Binomial Experiment

A binomial experiment is a fundamental concept in probability and statistics. It consists of a fixed number of independent trials, each of which results in one of two possible outcomes - commonly referred to as "success" or "failure."
In the context of our exercise, we have four trials. This means each repetition of the experiment is composed of four attempts. For example, flipping a coin four times can be thought of as a binomial experiment, where getting heads could be called a "success."
Here are the key characteristics of a binomial experiment:

• Each trial has only two possible outcomes.
• There is a fixed number of trials.
• The probability of success remains constant throughout all trials.
• All trials are independent, meaning the outcome of one trial does not affect the others.

These characteristics are essential when performing a goodness-of-fit test to determine whether our data fits a particular distribution, like the binomial distribution.

Chi-square Statistic

To test if our data fits a binomial distribution, we calculate what is known as the Chi-square statistic (\( \chi^2 \)). This statistic helps us understand how much our observed data deviates from our expected data if the data fits a certain expected distribution.
Here’s how you calculate the Chi-square statistic in this context:

• First, derive the expected frequencies of outcomes using the binomial distribution formula.
• Then, calculate the Chi-square statistic by taking the sum of the squares of the differences between observed and expected frequencies, divided by the expected frequencies: \[\chi^2 = \sum_{k=0}^4 \frac{(f(k) - f_e(k))^2}{f_e(k)} \]

This calculation shows us how "off" our observations are from the binomial expectations. From there, we can assess whether the deviations are significant. If this statistic exceeds a critical value (obtained from a Chi-square distribution table), we may conclude that the observed frequencies significantly deviate from the expected frequencies.

Degrees of Freedom

When conducting statistical tests, degrees of freedom (df) play a crucial role. In simple terms, degrees of freedom refer to the number of values in a calculation that are free to vary. For our goodness-of-fit test, this involves the number of outcomes minus any constraints placed on the data.
Given our scenario of testing a binomial distribution, understanding degrees of freedom helps us identify the correct Chi-square critical value. Here’s how it works:

• Count the number of categories or outcome types, which is 4 in this case (since the experiment can result in 0, 1, 2, 3, or 4 successes).
• Subtract 1 to account for the total sum being fixed. This is because the probabilities of all outcomes always sum to 1.

So, for our problem, we have 4 possible outcomes; hence, the degrees of freedom would be:\[df = 4 - 1 = 3\]Understanding this helps in determining the critical value for the Chi-square test, ensuring that the test is applied correctly.

Binomial Distribution

The binomial distribution is a discrete probability distribution that reflects the number of successes in a series of independent trials, each with the same probability of success. In a binomial experiment, knowing how to model with this distribution is key.
Using the binomial distribution, we can predict how frequently we might expect a certain number of successes across many trials.

• The formula to calculate the probability for a given number of successes \( k \) in \( n \) trials is: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]
• This formula uses the binomial coefficient \( \binom{n}{k} \), which represents the number of ways \( k \) successes can occur in \( n \) trials.
• The parameters \( p \) and \( (1-p) \) represent the probability of success and failure, respectively.

In our example, using this distribution allows us to estimate expected frequencies under the hypothesis that the trials follow a binomial distribution. The key is to compare these expected frequencies with the observed ones to perform the goodness-of-fit test.