Question 1181661: To test the hypothesis that a coin is fair, the following decision rules are adopted: (1) Accept
the hypothesis if the number of heads in a single sample of 100 tosses is between 40 and 60
inclusive, (2) reject the hypothesis otherwise.
a. Find the probability of rejecting the hypothesis when it is actually correct.
b. Interpret graphically the decision rule and the result of part (a).
c. What conclusions would you draw if the sample of 100 tosses yielded 53 heads? 60
heads
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to address this hypothesis testing problem:
**a. Probability of Rejecting a Correct Hypothesis (Type I Error):**
* **Binomial Distribution:** Since we're dealing with coin tosses, we can model the number of heads using the binomial distribution. If the coin is fair, the probability of heads (p) is 0.5. We have n = 100 trials.
* **Normal Approximation:** Because n is large, we can approximate the binomial distribution with a normal distribution.
* Mean (μ) = np = 100 * 0.5 = 50
* Standard deviation (σ) = sqrt(np(1-p)) = sqrt(100 * 0.5 * 0.5) = 5
* **Rejection Region:** We reject the hypothesis if the number of heads is *less than 40* or *greater than 60*.
* **Calculating Probabilities:** We need to find P(X < 40) + P(X > 60). Using the normal approximation:
* z1 = (40 - 50) / 5 = -2
* z2 = (60 - 50) / 5 = 2
* P(X < 40) = P(z < -2) ≈ 0.0228 (from a z-table or calculator)
* P(X > 60) = P(z > 2) ≈ 0.0228 (due to symmetry)
* P(rejecting correct hypothesis) = 0.0228 + 0.0228 = 0.0456
**b. Graphical Interpretation:**
* Draw a normal distribution curve with a mean of 50 and a standard deviation of 5.
* Shade the areas to the left of 40 and to the right of 60. These shaded areas represent the rejection regions.
* The total area of the shaded regions (0.0456) is the probability of rejecting the hypothesis when it's actually true (Type I error).
**c. Conclusions from Sample Results:**
* **53 Heads:** Since 53 is between 40 and 60 (inclusive), we *accept* the hypothesis that the coin is fair.
* **60 Heads:** Since 60 is *included* in the acceptance region (between 40 and 60 inclusive), we *accept* the hypothesis that the coin is fair. Note: If the rule was *strictly* between 40 and 60, we would reject the hypothesis.
**In summary:**
* The probability of rejecting the hypothesis when it is actually correct (Type I error) is approximately 0.0456 or 4.56%.
* Graphically, this is represented by the areas in the tails of the normal distribution beyond z = ±2.
* A sample of 53 heads leads to accepting the hypothesis.
* A sample of 60 heads leads to accepting the hypothesis.
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