Answer:
Answer is below. Hope this helps!
Explanation:
To determine if we have a balanced coin, we can use a hypothesis test based on the number of heads, Y, obtained from 36 coin tosses. The null hypothesis (H0) is that the coin is balanced, while the alternative hypothesis (Ha) is that the coin is not balanced.
The rejection region is given by |y - 180| ≥ 4, where y is the observed number of heads. If the observed number of heads falls within this rejection region, we reject the null hypothesis.
To find the value of a, we need to determine the critical value(s) that define the rejection region. In this case, the critical value is 4. This means that if the observed number of heads is 4 or more away from the expected number of heads (180 in this case), we would reject the null hypothesis.
The value of β, the probability of making a Type II error, depends on the true value of the proportion of heads, denoted by p. In this case, p = 0.7. To calculate β, we need to determine the probability of falling within the rejection region when the true proportion is 0.7.
Since we know the distribution of Y follows a binomial distribution, we can use the properties of this distribution to calculate β. Specifically, we can calculate the probability of falling within the rejection region when the true proportion is 0.7. This can be done using the binomial distribution formula:
P(|Y - np| ≥ 4) = P(Y ≤ 176) + P(Y ≥ 184)
Using a binomial distribution calculator or software, we can calculate these probabilities. The sum of these probabilities gives us the value of β.
Remember to consult appropriate statistical tables or software to obtain the exact values of the probabilities and make sure to interpret the results in the context of the problem.
In summary:
- The value of a, the critical value defining the rejection region, is 4.
- The value of β depends on the true proportion of heads, p, and can be calculated using the binomial distribution formula. In this case, p = 0.7.