You have an SRS of size n = 16 from a Normal distribution with standard deviation = 1. You wish to test
H_0 :μ = 0
H_a :μ > 0
You decide to reject H_0 if x bar > 0 and to accept H_0 otherwise.
a. Find the probability of a Type I error. That is, find the probability that the test rejects H_0 when in fact μ = 0.
b. Find the probability of a Type II error when μ = 0.2. This is the probability that the test accepts H_0 when in fact μ = 0.2.
c. Find the probability of a Type II error when μ = 0.5.
To calculate the probabilities of Type I and Type II errors, we need to use the concept of the standard normal distribution and the critical value.
a. Probability of a Type I error:
To find the probability of rejecting H_0 when μ = 0 (Type I error), we need to find the critical value. In this case, since the alternative hypothesis is μ > 0, we are conducting a right-tailed test.
The critical value can be found using the standard normal distribution table or through statistical software. The critical value corresponds to the level of significance, denoted as α.
In this case, we want to test H_0: μ = 0 at a certain significance level, let's say α = 0.05 (5%). So, we find the critical value corresponding to the area under the curve at the tail (1 - α) = 0.95.
Using the standard normal distribution table or software, we find the critical value to be approximately 1.645 (rounded to three decimal places). Now we can calculate the probability of Type I error.
Since we have a simple random sample from a Normal distribution with mean μ = 0 and standard deviation σ = 1, the sample mean x bar follows a standard normal distribution.
The probability of Type I error is given by P(x bar > critical value | μ = 0). Plugging in the values, we have:
P(x bar > 1.645 | μ = 0) = 1 - P(x bar ≤ 1.645 | μ = 0)
Using the standard normal distribution table or software, we find P(x bar ≤ 1.645 | μ = 0) ≈ 0.9505.
Therefore, the probability of a Type I error is approximately 1 - 0.9505 = 0.0495, or 4.95%.
b. Probability of a Type II error when μ = 0.2:
To calculate the probability of a Type II error, we need to find the critical value for the given alternative hypothesis and calculate the probability of accepting H_0 when μ = 0.2.
Since the alternative hypothesis is μ > 0, the critical value for a right-tailed test will depend on the level of significance (α). Let's assume α = 0.05 (5%) again.
The critical value can be found as before, by finding the value in the standard normal distribution that corresponds to the area 1 - α = 0.95.
Using the standard normal distribution table or software, the critical value is still approximately 1.645.
Now, we calculate the probability of Type II error P(x bar ≤ critical value | μ = 0.2).
P(x bar ≤ 1.645 | μ = 0.2) can be calculated using the standard normal distribution table or software. This probability depends on the specific values of μ, σ, and the sample size n. Using these values, you can find the required probability.
c. Probability of a Type II error when μ = 0.5:
Just like in the previous case, you need to determine the critical value for a right-tailed test with a given level of significance and calculate the probability of accepting H_0 when μ = 0.5.
The process is the same as in part (b), but this time you need to find P(x bar ≤ critical value | μ = 0.5). Again, you can use the standard normal distribution table or software to calculate this probability.
Please note that the probability of a Type II error varies depending on the specific values of μ, σ, and the sample size n. With these values at hand, you can calculate P(x bar ≤ critical value | μ) accordingly.
To analyze the given hypothesis test, we need to assume that the sample mean, x bar, is normally distributed since we have a large sample size (n = 16). We also know that the population standard deviation is 1.
a. Finding the probability of a Type I error:
A Type I error occurs when we reject the null hypothesis H_0 when it is actually true. In this case, H_0: μ = 0, so rejecting H_0 implies that the sample mean, x bar, is greater than 0.
To find the probability of a Type I error, we need to find the probability that x bar is greater than 0, given that μ is actually 0. We can use the standard normal distribution (Z distribution) to calculate this probability.
Since the sample mean is normally distributed when μ = 0 and σ = 1, we can calculate the z-score corresponding to x bar = 0 and find the area to the right of this value in the standard normal distribution.
The z-score formula is: z = (x - μ)/σ
For x bar = 0 and μ = 0, the z-score is:
z = (0 - 0)/1 = 0
The probability of a Type I error is equivalent to finding the area to the right of z = 0 in the standard normal distribution.
P(Type I error) = P(z > 0)
We can consult a standard normal distribution table or use statistical software to find the probability associated with z = 0. In this case, the probability is 0.5.
Therefore, the probability of a Type I error is 0.5 or 50%.
b. Finding the probability of a Type II error when μ = 0.2:
A Type II error occurs when we accept the null hypothesis H_0 when it is actually false. In this case, H_0: μ = 0 and H_a: μ > 0. So we will accept H_0 if the sample mean, x bar, is less than or equal to 0.
To find the probability of a Type II error, we need to calculate the probability that x bar is less than or equal to 0, given that μ = 0.2.
Again, we can use the standard normal distribution to calculate this probability. Following the same steps as before, we calculate the z-score for x bar = 0 and μ = 0.2:
z = (0 - 0.2)/1 = -0.2
Now we need to find the area to the left of z = -0.2 in the standard normal distribution.
P(Type II error | μ = 0.2) = P(z < -0.2)
Using a standard normal distribution table or statistical software, we find that the probability associated with z = -0.2 is approximately 0.4207.
Therefore, the probability of a Type II error when μ = 0.2 is approximately 0.4207 or 42.07%.
c. Finding the probability of a Type II error when μ = 0.5:
Using the same approach as before, we find the z-score for x bar = 0 and μ = 0.5:
z = (0 - 0.5)/1 = -0.5
Now we need to find the area to the left of z = -0.5 in the standard normal distribution.
P(Type II error | μ = 0.5) = P(z < -0.5)
Using the standard normal distribution table or statistical software, we find that the probability associated with z = -0.5 is approximately 0.3085.
Therefore, the probability of a Type II error when μ = 0.5 is approximately 0.3085 or 30.85%.