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
Here is one way you might do this problem:
Null hypothesis is that the coin is fair. Ho: p = .5
Alternate hypothesis is that the coin is unfair. Ha: p not equal to .5
Using the binomial formula: P(x) = (nCx)(p^x)[q^(n - x)]
...where n = number of coin tosses, x = number of times came up heads, p = probability given in the null hypothesis, q = 1 - p.
Using your data:
P(38) = (60C38)(.5^38)(.5^22) = ?
I'll let you finish the calculation.
If the alternate hypothesis uses "not equal to" then you multiply the result of the calculation by 2.
Reject the null hypothesis if the test statistic above is less than .10 (significance level); otherwise, do not reject null.
To determine whether the coin is biased towards heads, we can perform a hypothesis test using the concept of the binomial distribution. Let's break down the process into two parts:
(a) Hypothesis Test:
Step 1: State the hypotheses:
- Null hypothesis (H₀): The coin is fair; the probability of heads is 0.5.
- Alternative hypothesis (H₁): The coin is biased towards heads; the probability of heads is greater than 0.5.
Step 2: Set the significance level (α):
Here, the level of significance is given as 0.10 (or 10%, equivalent to a 90% confidence level).
Step 3: Find the critical value and decision rule:
Since we have a one-tailed test (biased towards heads), we need to find the critical value that corresponds to the given significance level.
- For an α = 0.10, the critical value is found by looking up the z-score for a 90% confidence level. A z-score of approximately 1.28 corresponds to this value.
The decision rule is as follows:
- If the test statistic (z-score) is greater than 1.28, we reject the null hypothesis.
- If the test statistic is less than or equal to 1.28, we do not reject the null hypothesis.
Step 4: Calculate the test statistic:
We can use the formula for z-score to calculate the test statistic:
z = (x - np) / √(np(1-p))
Where:
- x is the number of heads (38 in this case)
- n is the total number of coin flips (60 in this case)
- p is the hypothesized probability of heads (0.5)
Using the values, we can calculate the test statistic:
z = (38 - (60 * 0.5)) / √(60 * 0.5 * (1 - 0.5))
z ≈ 1.33
Step 5: Make a decision:
The calculated test statistic (z ≈ 1.33) is greater than the critical value (1.28) at the 0.10 level of significance. Therefore, we reject the null hypothesis.
(b) P-value:
The p-value represents the probability of obtaining a result as extreme as, or more extreme than, the observed data under the assumption that the null hypothesis is true. In this case, we want to calculate the p-value to provide additional evidence regarding the coin's bias towards heads.
To calculate the p-value, we can use a Z-table or calculator to find the area under the standard normal curve where the test statistic (z) falls. In this case, we use the z-value of 1.33.
Referencing the Z-table, we find that the area to the right of 1.33 is approximately 0.0918. However, since we have a one-tailed test (biased towards heads), we should multiply this by 2 to calculate the p-value.
p-value ≈ 2 * 0.0918 ≈ 0.1836
Interpretation:
The p-value of approximately 0.1836 suggests that, assuming the null hypothesis is true (the coin is fair), there is approximately an 18.36% chance of obtaining a result as extreme as, or more extreme than, the observed data. Since this p-value is larger than the significance level of 0.10, we do not have convincing evidence to reject the null hypothesis.
Therefore, based on both the decision rule and the p-value, we conclude that there is not enough evidence to claim that the coin is biased towards heads.