Let's break down the problem step-by-step:

Standard Error (SE):

The Standard Error is given by the formula:

$SE = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma$ is the population standard deviation (which is 30 in this case),
• $n$ is the sample size (which is 70).

Substituting the values:

$SE = \frac{30}{\sqrt{70}} \approx \frac{30}{8.37} \approx 3.58$

So, the Standard Error is approximately 3.58.

Probability that the sample mean is at most 285:

Next, we need to calculate the probability that the sample mean is at most 285. To do this, we can use the Z-score formula:

$Z = \frac{X - \mu}{SE}$

Where:

• $X$ is the sample mean we're interested in (285),
• $\mu$ is the population mean (280),
• $SE$ is the Standard Error (3.58).

Substitute the values into the formula:

$Z = \frac{285 - 280}{3.58} \approx \frac{5}{3.58} \approx 1.397$

Now, we can look up this Z-score in the Z-table to find the probability. A Z-score of 1.397 corresponds to a cumulative probability of approximately 0.917.

So, the probability that the sample mean is at most 285 is 0.917, or 91.7%.

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Summary of Answers:

• The Standard Error is approximately 3.58.
• The probability that the sample mean is at most 285 is approximately 91.7%.

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Would you like a more detailed explanation or have any further questions? Here are some additional related questions:

1. How does the sample size affect the Standard Error?
2. What is the difference between Standard Deviation and Standard Error?
3. How do you calculate the Z-score for other values of $X$?
4. How do you use a Z-table to find probabilities?
5. What assumptions do we make when calculating probabilities using the Z-score?

Tip: Increasing the sample size decreases the Standard Error, which leads to a more accurate estimate of the population mean.