
 
2. The amounts of time nurses at a hospital work each day are normally distributed, with a mean of 8.2 hours and a standard deviation of 0.35 hour. 

a) If a nurse is randomly selected, what is the probability that she or he works more than 8.5 hours? 

b) If seven nurses are randomly selected, what is the probability that their mean working time is more than 8.5 hours? 

Let's solve both parts of the problem using the properties of the normal distribution.

### Given:
- Mean (\(\mu\)) = 8.2 hours
- Standard deviation (\(\sigma\)) = 0.35 hours

### Part a) Probability that a single nurse works more than 8.5 hours

We need to find \(P(X > 8.5)\), where \(X\) is the time a nurse works.

1. **Calculate the Z-score**:
   \[
   Z = \frac{X - \mu}{\sigma} = \frac{8.5 - 8.2}{0.35}
   \]

2. **Find the probability** corresponding to this Z-score using the standard normal distribution.

### Part b) Probability that the mean working time for seven nurses is more than 8.5 hours

Since we are dealing with the mean of a sample, we'll use the **sampling distribution of the sample mean**.

1. **Standard error of the mean** (\(\sigma_{\bar{X}}\)):
   \[
   \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{0.35}{\sqrt{7}}
   \]

2. **Calculate the Z-score** for the sample mean:
   \[
   Z = \frac{\bar{X} - \mu}{\sigma_{\bar{X}}} = \frac{8.5 - 8.2}{\sigma_{\bar{X}}}
   \]

3. **Find the probability** corresponding to this Z-score using the standard normal distribution.

Let's calculate these values.Here are the results:

### Part a) Probability that a single nurse works more than 8.5 hours

- **Z-score**: \(Z = 0.857\)
- **Probability**: \(P(X > 8.5) \approx 0.196\) (or 19.6%)

### Part b) Probability that the mean working time for seven nurses is more than 8.5 hours

- **Z-score**: \(Z = 2.268\)
- **Probability**: \(P(\bar{X} > 8.5) \approx 0.0117\) (or 1.17%)

### Summary:
- **Part a**: There's a 19.6% chance that a randomly selected nurse works more than 8.5 hours.
- **Part b**: There's a 1.17% chance that the average working time of seven randomly selected nurses exceeds 8.5 hours.

Would you like further details on any part of the solution?

Explore the probabilities in a normal distribution scenario where nurses' working hours are normally distributed with a mean of 8.2 hours and a standard deviation of 0.35 hours. Calculate the probability that a randomly selected nurse works more than 8.5 hours and the probability that the average working time of seven nurses exceeds 8.5 hours.

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