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. I need more than the answer. I'm looking for help on the steps to get to the answer, please. Thanks
To determine if the coin is biased toward heads, we can conduct a hypothesis test using the binomial distribution. Here are the steps to calculate and interpret the results:
Step 1: State the hypotheses:
The null hypothesis (H0): The coin is fair and not biased toward heads.
The alternative hypothesis (H1): The coin is biased toward heads.
Step 2: Set the significance level:
The significance level is given as 0.10 (10%).
Step 3: Calculate the test statistic:
We will calculate the test statistic using the formula for the z-score of a proportion:
z = (p - P) / sqrt(P(1-P)/n)
Where:
- p is the sample proportion of heads (38/60 = 0.6333)
- P is the hypothesized proportion under the null hypothesis (0.5 for a fair coin)
- n is the sample size (60)
Substituting the values, we get:
z = (0.6333 - 0.5) / sqrt(0.5 * (1 - 0.5) / 60)
Calculating this expression, we get:
z ≈ 2.32
Step 4: Determine the critical value:
To determine the critical value, we need to look it up in the z-table for a one-tailed test at the 0.10 significance level.
The critical value for a one-tailed test at α = 0.10 is approximately 1.28.
Step 5: Make a decision:
If the calculated test statistic (z) is greater than the critical value (1.28), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, since z = 2.32 is greater than 1.28, we reject the null hypothesis.
Step 6: Interpret the results:
Based on the statistical analysis, we have sufficient evidence to conclude that the coin is biased toward heads at the 0.10 level of significance.
Note: The steps outlined above assume a one-tailed test since we are testing for a bias toward heads. If you are conducting a two-tailed test, the critical value will be different, and the decision rule would be based on comparing the absolute value of z to the critical value.
To determine whether the coin is biased toward heads, we can conduct a hypothesis test using the concept of statistical significance. Here are the steps to get to the answer:
Step 1: State the hypotheses.
The null hypothesis (H0) is that the coin is not biased toward heads, meaning it is fair. The alternative hypothesis (Ha) is that the coin is biased toward heads.
H0: The coin is fair (p = 0.5)
Ha: The coin is biased toward heads (p > 0.5)
Step 2: Set the significance level (α).
The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is true. In this case, α = 0.10, or 10%.
Step 3: Calculate the test statistic.
We will use the binomial test statistic formula to calculate the test statistic:
𝑧 = (𝑝̂ - 𝑝) / √(𝑝(1 - 𝑝) / 𝑛)
where:
- 𝑝̂ is the sample proportion of successes (flips resulting in heads)
- 𝑝 is the hypothesized population proportion (0.5 for a fair coin)
- 𝑛 is the sample size (60 flips)
In this case, 𝑝̂ = 38/60 = 0.6333, 𝑛 = 60, and 𝑝 = 0.5.
𝑧 = (0.6333 - 0.5) / √(0.5 * (1 - 0.5) / 60)
Step 4: Calculate the critical value or p-value.
The critical value is the value that the test statistic must exceed to reject the null hypothesis. In this case, we will calculate the critical value using a one-tailed z-test.
To obtain the critical value, we need to find the z-score associated with the significance level (α). In a one-tailed test, the critical value is the z-score that corresponds to the α level in the upper tail of the standard normal distribution.
Using a standard normal distribution table or a z-table, we find the z-score for α = 0.10 to be approximately 1.28.
Step 5: Make a decision.
- If the test statistic is greater than the critical value, we reject the null hypothesis.
- If the p-value is less than α, we reject the null hypothesis.
For this question, because we are using a one-tailed test and the alternative hypothesis suggests that the coin is biased toward heads (p > 0.5), we are interested in the upper tail of the distribution.
If the test statistic is greater than the critical value (1.28), or if the p-value is less than α (0.10), we will reject the null hypothesis and conclude that the coin is biased toward heads. Otherwise, we fail to reject the null hypothesis.
Note: If using a p-value to make a decision, we compare it to α (not the critical value directly) to determine whether to reject or fail to reject the null hypothesis.
I hope this explanation of the steps helps you with your question!