The provided image contains a statistics problem related to hypothesis testing for a normal distribution. Here's a step-by-step breakdown of the problem and how we approach solving it:

Problem Context:

• Given:

  • Mean ($\mu$) = 400 g
  • Standard deviation ($\sigma$) = 30 g
  • Sample size ($n$) = 20
  • The testing is done at a 10% significance level ($\alpha = 0.10$).

• Tasks:

  1. Determine the limits of the sample mean for no action.
  2. Calculate the probability of a Type I error.
  3. Find the probability that the mean weight of cereal in a sample is ≤ 390 g.
  4. Compute the probability of a Type II error assuming:
     • Actual mean weights of 392 g, 405 g, 410 g, and 420 g.
  5. Calculate probabilities of both Type I and Type II errors if the true mean remains 400 g.

Approach:

1. Find the limits of no action:

   • The critical region for rejecting the null hypothesis is based on $Z$-scores for the given significance level. Use $Z$-tables to determine these bounds.

2. Probability of Type I error ($\alpha$):

   • This is the probability of rejecting the null hypothesis when it is true, which directly corresponds to the significance level.

3. Probability of $\bar{X} \leq 390$:

   • Use the sampling distribution of the sample mean: $\text{Standard error (SE)} = \frac{\sigma}{\sqrt{n}}$
   • Compute the $Z$-score for $\bar{X} = 390$ and use the standard normal distribution to find the probability.

4. Type II error for specified mean weights:

   • Calculate the probability of failing to reject the null hypothesis when the mean weight is 392 g, 405 g, etc. Use the respective alternative hypotheses to compute this.

5. Type I and Type II errors at $\mu = 400$:

   • Determine these probabilities using the actual sampling distribution centered at 400 g.

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