https://file.aiquickdraw.com/mathbot/file/1726184614997nig4flr8.jpg

Let's break down the problem step by step based on the provided image.

### Question 2
We are given:
- Mean $\mu = 247$
- Standard deviation $\sigma = 60$
We need to calculate the **z-scores** for each of the following values:
1. $X = 197$
2. $X = 277$
3. $X = 310$

The z-score formula is:
\[
z = \frac{X - \mu}{\sigma}
\]

#### a. For $X = 197$:
\[
z = \frac{197 - 247}{60} = \frac{-50}{60} = -0.83
\]
#### b. For $X = 277$:
\[
z = \frac{277 - 247}{60} = \frac{30}{60} = 0.50
\]
#### c. For $X = 310$:
\[
z = \frac{310 - 247}{60} = \frac{63}{60} = 1.05
\]

### Question 3
We are told that a family has a **z-score of 1.5** and need to find their daily expense, $X$.

Using the z-score formula and solving for $X$:
\[
z = \frac{X - \mu}{\sigma}
\]
\[
1.5 = \frac{X - 247}{60}
\]
Multiplying both sides by 60:
\[
90 = X - 247
\]
\[
X = 247 + 90 = 337
\]

### Question 4
This question asks us to find the percentage of families spending various amounts, assuming the data is normally distributed.

To solve these, we need to find the z-scores for each threshold and use a standard normal distribution table (or a calculator) to get the corresponding percentages.

Would you like to proceed with question 4 or go into further details on any of the steps so far? Let me know if you'd like to see the distribution calculations for question 4.

Here are some related questions for further practice:
1. How do you calculate the percentage of families spending less than a certain amount using the z-score?
2. Can you explain how the z-score formula works in different scenarios?
3. What is the relationship between standard deviation and z-scores?
4. How can you interpret a positive and negative z-score in the context of daily expenses?
5. What are some key properties of a normal distribution?

**Tip:** The area under the normal curve represents probabilities. For example, a z-score of 0.83 corresponds to the probability of spending less than $197 per day.

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. $197 per day. $277 per day. $310 per day. If a family from the data set in question 2 had a z-score of 1.5, what was their daily expense for food and lodging?

Learn how to calculate z-scores for given expenses and determine the actual expenses from a z-score in a normally distributed data set of vacation expenses by families in the US. This example uses a mean of $247 and a standard deviation of $60.

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