https://file.aiquickdraw.com/mathbot/file/1725546536239ecpfjlj2.jpg

The problem describes a hypothesis test for a potentially biased coin. Let's analyze it step by step.

### Null Hypothesis (\( H_0 \)):
- The coin is fair, meaning the probability of heads \( P(H) = 0.5 \) and the probability of tails \( P(T) = 0.5 \).

### Test Details:
- The test involves flipping the coin 5 times.
- You will reject the null hypothesis (i.e., conclude the coin is unfair) if you observe:
  - 5 heads (all flips come up heads)
  - 5 tails (all flips come up tails)

### Type I Error:
- A **Type I error** occurs when you reject the null hypothesis even though it is true.
- In this case, the Type I error happens if you conclude the coin is unfair, even though it is actually fair.
  
Thus, we need to calculate the probability of getting either 5 heads or 5 tails under the assumption that the coin is fair.

#### Probability of 5 heads:
The probability of getting heads on a single flip is \( P(H) = 0.5 \). For 5 consecutive heads:
\[
P(\text{5 heads}) = (0.5)^5 = \frac{1}{32}
\]

#### Probability of 5 tails:
Similarly, the probability of getting 5 tails is:
\[
P(\text{5 tails}) = (0.5)^5 = \frac{1}{32}
\]

#### Total Probability of Type I Error:
Since these two events (5 heads and 5 tails) are mutually exclusive, we can add their probabilities:
\[
P(\text{Type I error}) = P(\text{5 heads}) + P(\text{5 tails}) = \frac{1}{32} + \frac{1}{32} = \frac{2}{32} = \frac{1}{16}
\]

### Decimal Representation:
\[
P(\text{Type I error}) = \frac{1}{16} = 0.0625
\]

Thus, the Type I error rate is **0.0625**.

Would you like further details or have any questions? Here are 5 follow-up questions to expand on this topic:

1. What is a Type II error and how would it apply to this coin test?
2. How could the Type I error rate be reduced in this experiment?
3. What happens to the error rate if the number of flips is increased?
4. How would the test change if we were to perform a two-tailed hypothesis test?
5. Can we design a test with a higher or lower significance level?

**Tip:** To better understand the balance between Type I and Type II errors, remember that decreasing one often increases the other.

Suppose you have a coin that you suspect may be a trick coin. You decide to perform a hypothesis test: you will flip the coin 5 times, and, if it either comes up heads every time or tails every time, then you will conclude that the coin is not fair. Assuming that the null hypothesis is that the coin is fair, what is the Type I error rate of this test? Enter your answer as a decimal to at least four decimal places.

Determine the Type I error rate for a hypothesis test where you flip a coin 5 times to conclude if it's fair. This problem explores core probability concepts and hypothesis testing suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation