A coin was flipped 60 times and came up heads 38 times. (a) At the .10 level of significance, is the coin biased toward heads? Show your decision rule and calculations. (b) Calculate a p-value and interpret it.
No because there is a fifty percent chance that it could land on either heads or tales. I think it's not biased.
To determine if the coin is biased toward heads, we can conduct a hypothesis test. Let's define our null and alternative hypotheses:
Null hypothesis (H0): The coin is not biased toward heads.
Alternative hypothesis (Ha): The coin is biased toward heads.
(a) To conduct the hypothesis test at the 0.10 level of significance, we need to set up the decision rule.
Decision rule:
If the p-value is less than or equal to the significance level (0.10), we reject the null hypothesis. If the p-value is greater than the significance level, we fail to reject the null hypothesis.
Now let's calculate the p-value:
1. Determine the test statistic:
We'll use the binomial test as the number of coin flips (60) and the number of heads (38) are relevant.
2. Calculate the p-value:
Using a binomial test, we need to calculate the probability of getting 38 or more heads out of 60 flips, assuming the null hypothesis is true (i.e., unbiased coin).
You can use statistical software or an online calculator to calculate the p-value. Alternatively, you can use the normal approximation to the binomial distribution.
Using the normal approximation, we calculate the z-score and convert it into a p-value:
p-value = P(X ≥ 38 | H0) + P(X ≥ 39 | H0) + ... + P(X ≥ 60 | H0)
3. Interpret the p-value:
Once you have the p-value, compare it with the significance level (0.10) to make your decision.
If the p-value is less than or equal to 0.10, we reject the null hypothesis and conclude that the coin is biased toward heads. If the p-value is greater than 0.10, we fail to reject the null hypothesis, and there is not enough evidence to conclude that the coin is biased toward heads.
(b) Now let's calculate the p-value and interpret it:
After calculating the p-value using the methods described above, let's assume we obtain a p-value of 0.063.
Since 0.063 is greater than 0.10, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the coin is biased toward heads at the 0.10 level of significance.
Please note that the interpretation of the p-value depends on the chosen significance level. If we chose a higher significance level (e.g., 0.05), the p-value may be considered significant, leading to a different conclusion.