A recent survey by the American Automobile Association showed that a family of two adults and two children on vacation in the United States will pay an average of $247 per day for food and lodging with a standard deviation of $60 per day. Assuming the data are normally distributed, find, to the nearest hundredth, the z-scores for each of the following vacation expense amounts.
a. $197 per day. 

b. $277 per day. 

c. $310 per day. 


To find the z-scores for the given vacation expense amounts, we can use the z-score formula:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- $ x $ is the value for which we are finding the z-score,
- $ \mu = 247 $ is the mean (average expense per day),
- $ \sigma = 60 $ is the standard deviation.

Let's calculate the z-scores for each of the given values:

### a. For $ x = 197 $:
\[
z = \frac{197 - 247}{60} = \frac{-50}{60} = -0.83
\]
The z-score for $197 per day is **-0.83**.

### b. For $ x = 277 $:
\[
z = \frac{277 - 247}{60} = \frac{30}{60} = 0.50
\]
The z-score for $277 per day is **0.50**.

### c. For $ x = 310 $:
\[
z = \frac{310 - 247}{60} = \frac{63}{60} = 1.05
\]
The z-score for $310 per day is **1.05**.

### Summary:
- a. $197 per day: $ z = -0.83 $
- b. $277 per day: $ z = 0.50 $
- c. $310 per day: $ z = 1.05 $

Would you like further details or have any questions about these calculations?

Here are 5 related questions you could explore:
1. What does a z-score of 0 represent in a normal distribution?
2. How can we interpret negative z-scores in real-life scenarios?
3. What is the probability of spending less than $197 per day?
4. How would the z-scores change if the standard deviation was larger or smaller?
5. How can we use z-scores to determine outliers in this data?

**Tip:** A z-score tells you how many standard deviations a data point is from the mean. It helps standardize different scales or data sets for comparison.

A recent survey by the American Automobile Association showed that a family of two adults and two children on vacation in the United States will pay an average of $247 per day for food and lodging with a standard deviation of $60 per day. Assuming the data are normally distributed, find, to the nearest hundredth, the z-scores for each of the following vacation expense amounts. a. $197 per day. b. $277 per day. c. $310 per day.

Discover how to calculate z-scores for different vacation expenses using normal distribution. This example problem includes step-by-step calculations for expenses of $197, $277, and $310, given a mean of $247 and a standard deviation of $60 per day. Ideal for students in Grades 10-12 learning about statistics and normal distribution.

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