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Magazine Chapter 25: Problem 14
Find a \(99 \%\) confidence interval for \(p\) in the binomial distribution frorn a classical result by K. Pearson, who In 24000 trials of tossing a coin obtained 12012 Heads. Do you think that the coin was fair?
Short Answer
Expert verified
The 99% confidence interval can be computed, and if it includes 0.5, the coin is likely fair.
Step by step solution
01
Identify Parameters
Identify the key parameters given in the problem: The number of trials is \(n = 24000\), and the number of successes (Heads) is \(X = 12012\). The sample proportion \(\hat{p}\) is given by \(\hat{p} = \frac{X}{n} = \frac{12012}{24000}\).
02
Calculate Sample Proportion
Calculate the sample proportion \(\hat{p}\): \[ \hat{p} = \frac{12012}{24000} = 0.5005 \]
03
Determine Z-value for 99% Confidence Level
For a 99% confidence interval, find the Z-value corresponding to \(\frac{1 - 0.99}{2} = 0.005\). The Z-value for a 99% confidence interval is approximately \(Z = 2.576\).
04
Calculate Standard Error
Calculate the standard error of the sample proportion using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.5005 \times 0.4995}{24000}} \]
05
Compute the Margin of Error
Calculate the margin of error \(ME\) using the Z-value and standard error: \[ ME = Z \times SE = 2.576 \times \sqrt{\frac{0.5005 \times 0.4995}{24000}} \]
06
Construct Confidence Interval
The 99% confidence interval for the true proportion \(p\) is: \[ (\hat{p} - ME, \hat{p} + ME) \] Calculate this interval using the values obtained from previous steps.
07
Interpret the Confidence Interval
The confidence interval provides a range in which we expect the true proportion \(p\) of heads would fall if the coin were fair (p = 0.5). If 0.5 is within this interval, the coin may be considered fair.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Binomial Distribution
When dealing with the concept of binomial distribution, it's important to know that it describes the outcomes of a series of experiments or trials. In this case, we refer to a situation where you can have only two possible results, such as a coin toss resulting in heads or tails. Here, the number of trials is set, which in our original problem is 24,000 coin tosses.
- Defining Success: One outcome is defined as "success". For example, getting a head in a coin toss can be a success.
- Probability of Success (p): This is the probability of getting a success on any given trial.
Identifying Sample Proportion
In statistics, the sample proportion is represented by the symbol \(\hat{p}\), and it gives us a measure of how often an outcome occurs within a sample size. It is calculated by dividing the number of successes by the total number of trials. In Pearson's case, he observed 12,012 heads out of 24,000 tosses.
To calculate \(\hat{p}\), use the formula: \[\hat{p} = \frac{X}{n} = \frac{12012}{24000} = 0.5005\]
To calculate \(\hat{p}\), use the formula: \[\hat{p} = \frac{X}{n} = \frac{12012}{24000} = 0.5005\]
- Practical Interpretation: The sample proportion means that about 50.05% of the tosses resulted in heads. This is incredibly close to what you'd expect from a fair coin (which would have 50% heads).
- Accuracy of Sample Proportion: It gives us an initial estimate of the true population proportion, but can vary because it is based on a limited sample size.
Calculating the Standard Error
Standard error (SE) is a key concept when assessing how much sample proportions fluctuate due to random sampling. It helps to understand the variability of these sample proportions and can be a tool to gauge the reliability of our estimate of the true parameter. The formula to calculate the standard error of a sample proportion is:
\[SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\]
For Pearson's coin toss experiment, the calculation is:\[SE = \sqrt{\frac{0.5005 \times 0.4995}{24000}}\]
\[SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\]
For Pearson's coin toss experiment, the calculation is:\[SE = \sqrt{\frac{0.5005 \times 0.4995}{24000}}\]
- Function: The standard error gives an idea of how much the sample proportion would vary if different samples were taken from the same population.
- Importance: It indicates how precise \(\hat{p}\) is as an estimate of the true proportion \(p\).
The Role of Z-value in Confidence Intervals
In statistics, the Z-value is a number representing the number of standard deviations a data point is from the mean. It plays a crucial role when we build confidence intervals, which are used to estimate the true parameter of a population based on sample data.
- Use in Confidence Intervals: The Z-value helps to determine the margin of error for a confidence interval. For a 99% confidence level, we use a Z-value of approximately 2.576.
- Understanding Confidence Levels: A 99% confidence interval means that if you repeat your experiment many times, about 99% of the intervals calculated will contain the true population proportion.
