7 coins are tossed at a time 128 times, the numbers of heads observed at each toss are recorded. Fit the binomial distribution to the data assuming that the coins are 1 biased 2 un... 7 coins are tossed at a time 128 times, the numbers of heads observed at each toss are recorded. Fit the binomial distribution to the data assuming that the coins are 1 biased 2 unbiased.
Understand the Problem
The question is asking us to analyze the results of tossing 7 coins 128 times and fit the observed heads to a binomial distribution, considering both biased and unbiased coins. We will approach the solution by determining the parameters of the binomial distribution for both scenarios and comparing the fits.
Answer
Fit the observed heads to binomial distributions using the calculations provided for both biased and unbiased scenarios, leading to the appropriate model selection based on the fit quality.
Answer for screen readers
To find the fit of the observed heads to the binomial distribution for both biased and unbiased coins, perform the calculations and tests as detailed above. The final output will be the fitting results, potentially noting which parameterization yields a better fit.
Steps to Solve
- Determine the number of trials and successes In this problem, we are tossing 7 coins 128 times, which implies:
- Number of trials ($n$) = 7 (the number of coins)
- Total number of experimental trials = 128
- Identify the parameters for unbiased coins For unbiased coins, the probability of getting heads ($p$) is 0.5. We can use the binomial distribution formula: $$ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} $$ where:
- $k$ is the number of heads.
- $n$ is the number of coins, which is 7.
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Identify the parameters for biased coins For biased coins, you may have a different probability $p$, which needs to be defined or estimated from data. The same binomial distribution formula applies, but we substitute the value of $p$ accordingly.
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Calculate the expected counts for both scenarios For each number of heads ($k = 0$ to $k = 7$), compute the expected counts of heads using the formula mentioned above, multiplied by the number of trials (128). Therefore, for unbiased coins: $$ \text{Expected Count} = 128 \cdot P(X = k) $$ Repeat this for the biased coins with the assumed $p$ value.
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Fit observed data to binomial distribution Once you have the expected counts for both unbiased and biased scenarios, you can fit them against the observed counts from your experiments using statistical methods such as Chi-Squared goodness-of-fit tests to determine which model fits better.
To find the fit of the observed heads to the binomial distribution for both biased and unbiased coins, perform the calculations and tests as detailed above. The final output will be the fitting results, potentially noting which parameterization yields a better fit.
More Information
Fitting a binomial distribution to experimental data allows for comparison between different models of randomness (biased vs unbiased), which can reveal insights about the nature of the coin tosses. This method is commonly used in probability and statistics to validate assumptions about random processes.
Tips
- Incorrectly calculating the probabilities for different numbers of heads in the binomial distribution.
- Confusing the expected counts with observed counts—ensure that you are formally distinguishing between these two.
- Not adjusting the parameters as needed when switching from unbiased to biased coin scenarios.
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