A coin is tossed 1000 times,out of which we observe 560 heads,the question of interest is whether the coin is biased or not?
To determine whether the coin is biased or not, we can use a statistical test called the binomial test. The binomial test is used to assess whether an observed proportion significantly differs from a specified population proportion.
In this case, our null hypothesis (H0) is that the coin is fair, with an equal chance of getting heads or tails, i.e., 50% heads and 50% tails. The alternative hypothesis (H1) is that the coin is biased.
Now, let's perform the binomial test using the observed data:
Step 1: Define the parameters
n = 1000 (number of coin tosses)
x = 560 (number of observed heads)
p = 0.5 (hypothesized probability of getting heads in a fair coin)
Step 2: Calculate the p-value
The p-value is the probability of obtaining a result as extreme as the observed result (or more extreme), assuming the null hypothesis is true.
We can use statistical software or online calculators to calculate the p-value. For example, using the binom.test() function in R, the p-value for this test is determined as follows:
p-value = binom.test(560, 1000, p = 0.5)$p.value
Step 3: Determine the significance level
The significance level (α) is a pre-defined threshold below which we reject the null hypothesis. Common significance levels are 0.05 and 0.01, indicating that we are willing to falsely reject the null hypothesis 5% or 1% of the time, respectively.
Step 4: Compare the p-value with the significance level
If the p-value is less than the significance level, we reject the null hypothesis and conclude that the coin is biased. Otherwise, we fail to reject the null hypothesis and conclude that there is no evidence of bias in the coin.
Remember, statistical tests can only provide evidence for or against a hypothesis but cannot prove or disprove it definitively. They provide a measure of confidence in our conclusions based on the available data.
So, by following these steps and performing the binomial test, you can determine whether the coin is biased or not based on the given data.