
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
a) From a sample of size \( n \) drawn from a normal population with known standard deviation \( \sigma \), the sample mean \( \bar{x} \) is found. At a significance level \( \alpha \):

1. Find the critical region for testing the null hypothesis \( H_{0}: a = a_{0} \) against the alternative hypothesis \( H_{1}: a > a_{0} \).
2. Find the power function of the test, using the hypothetical population mean \( a = a_{1} \) where \( a_{1} > a_{0} \).
3. Verify that increasing the sample size increases the power of the test.
4. Verify that increasing the significance level increases the power of the test.

b) For a sample of size \( n = 16 \) taken from a normal population with known standard deviation \( \sigma = 4 \), and with a significance level of 0.05, test the null hypothesis \( H_{0}: a = a_{0} = 2 \) against the alternative hypothesis \( H_{1}: a > 2 \).

1. Find the power of the one-sided test for the hypothetical population mean \( a = a_{1} = 3.2 \).
2. Determine the sample size \( n_{1} \) required for the power of the test to be 0.6.

Present the answer in LaTex format: \boxed{Your answer}
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