Draw the operating characteristic curve for n = 10, 50, and 100 for the following test:

Ho: ì = 400
Hi: ì > 400
á = .05, ó = 50

To draw the Operating Characteristic (OC) curve for a hypothesis test, we need to calculate the probability of rejecting the null hypothesis when it is false for different values of the population mean (ì). In this case, we are given that the null hypothesis (Ho) is that ì = 400 and the alternative hypothesis (Hi) is that ì > 400. We also have the significance level (á) of 0.05 and the standard deviation (ó) of 50.

To calculate the probability of rejecting Ho for different ì values, we need to use the sampling distribution of the sample mean. The sampling distribution of the sample mean follows a normal distribution with a mean of ì and a standard deviation of ó / sqrt(n), where n is the sample size.

For n = 10, 50, and 100, we can calculate the standard deviation of the sample mean as ó / sqrt(n). Given that ó = 50, the standard deviation of the sample mean for each sample size is:

n = 10: Standard deviation = 50 / sqrt(10) ≈ 15.81
n = 50: Standard deviation = 50 / sqrt(50) ≈ 7.07
n = 100: Standard deviation = 50 / sqrt(100) = 5

Next, we need to determine the critical value(s) for the hypothesis test at the chosen significance level (á = 0.05). Since we have a right-tailed test (ì > 400), we need to find the critical value of the z-test corresponding to a cumulative probability of 1 - á = 0.95.

Using a standard normal distribution table or a z-table, we can find the critical value zα. For á = 0.05, the critical value zα is approximately 1.645.

Now, we can calculate the probability of rejecting Ho for different ì values using the formula:

P(reject Ho | ì) = P(sample mean > critical value | ì)

For each sample size, we can use the formula to calculate the probability for a range of ì values (above 400). Plotting these probabilities against the ì values will give us the OC curve.

Let's calculate the probabilities and draw the OC curve for n = 10, 50, and 100:

1. For n = 10:
- Calculate the z-score for the critical value zα = 1.645 and sample size n = 10.
- Use the formula P(reject Ho | ì) = P(z > z-score) = 1 - P(z <= z-score).

Repeat the above steps for different ì values (greater than 400) to calculate the corresponding probabilities.

2. For n = 50 and n = 100:
- Repeat the same steps as above using the appropriate sample sizes and the corresponding standard deviations.

By plotting the probabilities against the ì values for each sample size, the resulting graph will give us the Operating Characteristic curve.