Final Answer:
The answer is b. Reject the null hypothesis. With 5080 heads out of 10000 tosses, the probability of obtaining this extreme result or more with a fair coin is very low, leading to the rejection of the hypothesis that the coin is fair. Thus the correct option is b. Reject the null hypothesis.
Step-by-step explanation:
In hypothesis testing for the fairness of a coin, the null hypothesis (H0) assumes that the coin is fair (probability of heads = 0.5), while the alternative hypothesis (H1) suggests the coin is not fair. With 10000 tosses and 5080 heads, we calculate the probability of getting 5080 or more heads with a fair coin. This can be done using statistical methods like the binomial distribution or normal approximation.
Using the binomial distribution, the expected number of heads for a fair coin is 5000 (0.5 * 10000). We can then compute the standard deviation (√(np(1-p)), where n = number of trials, p = probability of success) and perform a z-test to see if 5080 heads significantly deviates from this expected value.
After calculating the z-score and finding the corresponding p-value, we compare it to the chosen significance level (usually 0.05). If the p-value is less than 0.05, we reject the null hypothesis in favor of the alternative hypothesis, suggesting that the coin is likely not fair.
In this scenario, with a large enough sample size and the proportion of heads significantly higher than expected, we would reject the null hypothesis and conclude that the coin is probably not fair. Hence, the correct choice is to "Reject the null hypothesis" (b).
Thus the correct option is b. Reject the null hypothesis.