Suppose that the rejection region of a test ψ has the form R={Xn:Xn>c} . Find the smallest c such that ψ has level α .
(If applicable, type abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α quantile of a standard normal variable.)
c≥?
Suppose that the test ψ has level α=0.05 . What is the power of ψ ?
(If applicable, type abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α quantile of a standard normal variable, e.g. enter q(0.01) for q0.01 .)
Power of ψ=?
Answer?
To find the smallest value of c such that ψ has level α, we need to determine the critical value that separates the rejection region from the acceptance region.
The level α represents the maximum probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.
Assuming the test statistic follows a standard normal distribution, we can use the cumulative distribution function (CDF) to find the critical value.
The rejection region R is defined as R={Xn:Xn>c}. This means that if the test statistic Xn is greater than c, we reject the null hypothesis.
To determine the critical value c, we want to find the value for which the probability of the test statistic being greater than c is equal to α.
Mathematically, this can be expressed as P(Xn > c) = α.
Since the test statistic follows a standard normal distribution, we can rewrite this equation using the cumulative distribution function (CDF) Φ(x):
1 - Φ(c) = α.
Now, we need to solve for c.
First, let's subtract α from both sides:
1 - α = Φ(c).
Next, let's subtract 1 from both sides:
-α = Φ(c) - 1.
Now, we need to find the value of c for which Φ(c) - 1 = -α. This value represents the 1 - α quantile of the standard normal distribution, denoted as q(α):
c = q(α).
Therefore, the smallest value of c such that ψ has level α is c = q(α), the 1 - α quantile of a standard normal variable.
To find the smallest value of c such that the rejection region ψ has level α, we need to determine the critical value that corresponds to the specified level of significance.
In this case, the level of the test is denoted as α. Therefore, we need to find the value of c that ensures that the probability of observing a test statistic greater than c, given that the null hypothesis is true, is equal to α.
In other words, we want to find the value of c for which P(Xn > c | H0) = α, where H0 represents the null hypothesis.
To determine this critical value, we can use the standard normal distribution since the test statistic Xn is assumed to follow a normal distribution. We can utilize the cumulative distribution function (CDF) of the standard normal distribution, denoted as Φ(x), to calculate the probability.
The rejection region R is given by R = {Xn: Xn > c}. This means that the probability of observing a test statistic falling in the rejection region is equal to the probability of Xn being greater than c.
We want to find the smallest value of c, so we need to find the smallest value for which P(Xn > c) = α.
Using the properties of the standard normal distribution, we can rewrite this as P(Z > z) = α, where Z follows a standard normal distribution and z is the corresponding quantile for the level α. In other words, z = q(α).
Now, we have P(Z > z) = α, and we want to find the smallest value of z for which this equation holds.
Since the cumulative distribution function Φ(x) gives us the probability of Z being less than or equal to x, we can rewrite the equation as 1 - Φ(z) = α.
Rearranging the equation, we have Φ(z) = 1 - α.
To find the value of z, we can utilize the inverse of the cumulative distribution function, denoted as Φ^(-1)(x) or also known as the quantile function.
So, z = Φ^(-1)(1 - α).
Finally, we substitute the value of z into our original equation to find the critical value c:
c ≥ Φ^(-1)(1 - α).
Therefore, the smallest value of c such that ψ has level α is c ≥ Φ^(-1)(1 - α).
Note: In some statistical software, the inverse of the cumulative distribution function may be denoted as qnorm(). Hence, c ≥ qnorm(1 - α) can also represent the condition for the critical value.