To solve these problems, we will use the sampling distribution of the sample mean. Since the population is normally distributed, we can apply the following steps for both questions:

Key Formulas:

1. Standard error of the mean (SE): $SE = \frac{\sigma}{\sqrt{n}}$ where:

   • $\sigma$ is the population standard deviation,
   • $n$ is the sample size.

2. Z-score for the sample mean: $Z = \frac{\bar{x} - \mu}{SE}$ where:

   • $\bar{x}$ is the sample mean we are interested in,
   • $\mu$ is the population mean,
   • $SE$ is the standard error of the mean.

3. Probability: The probability corresponds to the area under the normal curve to the left (or right) of the Z-score, which can be looked up in the Z-table.

Part (a): Sample size = 16, Find P($\bar{x}$ < 555)

Given:

• Population mean ($\mu$) = 575
• Population standard deviation ($\sigma$) = 40
• Sample size ($n$) = 16
• Desired sample mean ($\bar{x}$) = 555

Step 1: Calculate the standard error (SE)

$SE = \frac{40}{\sqrt{16}} = \frac{40}{4} = 10$

Step 2: Calculate the Z-score

$Z = \frac{555 - 575}{10} = \frac{-20}{10} = -2$

Step 3: Find the probability corresponding to the Z-score

Using the Z-table, a Z-score of -2 corresponds to a probability of 0.0228.

So, the probability that the sample mean is less than 555 is: $P(\bar{x} < 555) = 0.0228$

Final Answer for Part (a):

$P(\bar{x} < 555) = 0.0228 \quad \text{(rounded to four decimal places)}$

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Part (b): Sample size = 25, Find P($\bar{x}$ ≥ 593)

Given:

• Population mean ($\mu$) = 575
• Population standard deviation ($\sigma$) = 40
• Sample size ($n$) = 25
• Desired sample mean ($\bar{x}$) = 593

Step 1: Calculate the standard error (SE)

$SE = \frac{40}{\sqrt{25}} = \frac{40}{5} = 8$

Step 2: Calculate the Z-score

$Z = \frac{593 - 575}{8} = \frac{18}{8} = 2.25$

Step 3: Find the probability corresponding to the Z-score

Using the Z-table, a Z-score of 2.25 corresponds to a cumulative probability of 0.9878.

To find the probability that the sample mean is greater than or equal to 593: $P(\bar{x} \geq 593) = 1 - 0.9878 = 0.0122$

Final Answer for Part (b):

$P(\bar{x} \geq 593) = 0.0122$

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Summary of Answers:

• Part (a): $P(\bar{x} < 555) = 0.0228$
• Part (b): $P(\bar{x} \geq 593) = 0.0122$

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Would you like more details or have any questions about these solutions?

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