in 1000 tosses of a coin, 560 heads & 440 tails appear. is it reasonable to assume that coin is fair.
biased.. its beyond 3 standard deviations
To determine whether it is reasonable to assume that the coin is fair based on the given data, we can use statistical analysis.
Step 1: Calculate the expected number of heads and tails for a fair coin.
For a fair coin, the probability of getting heads is 0.5, and tails is also 0.5.
So, for 1000 tosses, we would expect the number of heads to be (0.5 * 1000) = 500, and the number of tails to be (0.5 * 1000) = 500.
Step 2: Compare the observed data to the expected data.
In this case, we observed 560 heads and 440 tails. We can calculate the difference between the observed and expected values for heads and tails respectively.
Difference in heads: 560 - 500 = 60
Difference in tails: 440 - 500 = -60 (Negative because the observed value is less than the expected value)
Step 3: Assess the reasonability of assuming the coin is fair.
To evaluate the reasonableness, we need to consider the deviation from the expected values. If the differences between the observed and expected values are significant, it could indicate that the coin is biased.
In this scenario, the difference in heads is 60 and the difference in tails is -60. Both differences indicate some level of deviation from the expected values of a fair coin. However, the differences are not excessively large, and they could potentially occur due to random chance.
Therefore, based solely on the given data, it would be reasonable to assume that the coin is approximately fair, although there may be a slight bias. However, to make a more definitive conclusion, additional statistical tests or more data would be needed.
To determine whether the coin is fair, we can perform a hypothesis test using statistical analysis. The null hypothesis in this case would state that the coin is fair (i.e., the probability of getting heads or tails is 0.5 for each toss), while the alternative hypothesis would suggest that the coin is biased.
To conduct the hypothesis test, we can use the chi-square test of independence for categorical data. In this case, we would compare the observed frequencies (560 heads, 440 tails) with the expected frequencies (based on the assumption of a fair coin).
To calculate the expected frequencies, we can assume that the probability of getting heads or tails on each toss is 0.5. In 1000 tosses, we would expect approximately 500 heads and 500 tails. Therefore, for a fair coin, the expected frequencies would be 500 heads and 500 tails.
Next, we can calculate the chi-square statistic using the formula:
χ² = Σ((O - E)² / E),
where Σ indicates the sum of the calculations for each category, O is the observed frequency, and E is the expected frequency.
Substituting in the values:
χ² = ((560 - 500)² / 500) + ((440 - 500)² / 500).
Evaluating this equation will give us the chi-square statistic. We can then compare this statistic with the critical value from the chi-square distribution table to determine the p-value.
If the p-value is less than a chosen significance level (e.g., 0.05), we can reject the null hypothesis and conclude that the coin is likely biased. On the other hand, if the p-value is greater than the significance level, we fail to reject the null hypothesis and cannot conclude that the coin is biased.
In summary, we can determine whether the coin is fair by conducting a chi-square test of independence using the observed and expected frequencies, and comparing the calculated chi-square statistic with the critical value from the chi-square distribution table.