Again following on from the previous question, what is the probability that when we take a random sample of 
n
=
78
 adults, the sample mean would be either greater than 7.1187 or less than 6.8813?

To find the probability that the sample mean is either greater than 7.1187 or less than 6.8813, we'll use the concept of the sampling distribution of the sample mean.

### Step 1: Determine the parameters of the sampling distribution
The sampling distribution of the sample mean \(\bar{X}\) follows a normal distribution:
\[
\bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)
\]
where:
- \(\mu\) is the population mean,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.

Given:
- \(n = 78\)
- Population mean \(\mu = 7\) (assumed from context, if not provided)
- Population standard deviation \(\sigma = 0.5\) (assumed from context, if not provided)

The standard error (SE) of the sample mean is:
\[
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{0.5}{\sqrt{78}} \approx 0.0566
\]

### Step 2: Standardize the values
To find the probability, we convert the sample means to \(z\)-scores using the formula:
\[
z = \frac{\bar{X} - \mu}{\text{SE}}
\]

For \(\bar{X} = 7.1187\):
\[
z = \frac{7.1187 - 7}{0.0566} \approx 2.10
\]

For \(\bar{X} = 6.8813\):
\[
z = \frac{6.8813 - 7}{0.0566} \approx -2.10
\]

### Step 3: Find the corresponding probabilities
Using standard normal distribution tables or a calculator:

- \(P(Z > 2.10)\) is the probability that the \(z\)-score is greater than 2.10.
- \(P(Z < -2.10)\) is the probability that the \(z\)-score is less than -2.10.

Since the normal distribution is symmetric, we have:
\[
P(Z > 2.10) = P(Z < -2.10) \approx 0.0179
\]

### Step 4: Calculate the total probability
The total probability that the sample mean is either greater than 7.1187 or less than 6.8813 is the sum of these two probabilities:
\[
P(\bar{X} > 7.1187 \text{ or } \bar{X} < 6.8813) = P(Z > 2.10) + P(Z < -2.10) \approx 0.0179 + 0.0179 = 0.0358
\]

### Final Answer:
The probability that the sample mean is either greater than 7.1187 or less than 6.8813 is approximately **0.0358**.

Would you like more details or have any questions? 

Here are some related questions you might be interested in:

1. How does the sample size affect the probability of the sample mean being within a certain range?
2. What assumptions are made about the population distribution when using the Central Limit Theorem?
3. How would the probability change if the sample size were doubled?
4. What is the difference between the probability of a single observation being within a range and the sample mean being within that range?
5. How can we calculate probabilities for non-normal population distributions?

**Tip:** When dealing with sample means, increasing the sample size decreases the standard error, which narrows the distribution of the sample mean and affects the probabilities of extreme values.

Learn how to calculate the probability that a random sample mean of 78 adults is either greater than 7.1187 or less than 6.8813. Understand the concepts of sampling distribution, normal distribution, and z-scores with step-by-step explanations and examples. Suitable for undergraduate students studying probability and statistics.

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Again following on from the previous question, what is the probability that when we take a random sample of n

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