According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $2,075. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $500.



Let's break down the information given:

- Mean amount spent: $ \mu = 2,075 $ dollars.
- Standard deviation: $ \sigma = 500 $ dollars.
- The distribution of the amounts spent is normal.

You can use this information to answer several types of questions regarding probabilities or the likelihood of certain spending amounts within this age group. Here are some common questions that can be solved using this information:

1. **What is the probability that a randomly selected adult spends less than a certain amount, say $1,500, on reading and entertainment per year?**
   - This can be calculated using the **z-score** formula and standard normal distribution tables.

2. **What percentage of adults spend between two amounts, for example, between $1,800 and $2,500?**
   - You can find the z-scores for both values and use the standard normal distribution to find the probability between these z-scores.

3. **What is the probability that a randomly selected adult spends more than $3,000 on reading and entertainment?**
   - Again, you would calculate the z-score for $3,000 and then find the corresponding probability.

4. **What is the range of spending within which 95% of adults fall?**
   - For this, you would use the properties of the normal distribution and z-scores corresponding to 95% of the data.

5. **If a sample of 50 adults is taken, what is the probability that the sample mean of their spending exceeds $2,100?**
   - You can apply the **Central Limit Theorem** to calculate this probability.

Would you like to explore one of these questions in more detail, or do you have any other specific questions about this data?

Here are 5 related questions you might consider:
1. What is the z-score for an adult who spends $1,500 on reading and entertainment?
2. What percentage of adults spend more than $2,500 annually?
3. How would the probabilities change if the standard deviation were smaller, say $400?
4. What amount separates the top 10% of spenders from the rest?
5. How would the probability change if we sampled a group of 100 people instead of 50?

**Tip:** The z-score formula is $ z = \frac{x - \mu}{\sigma} $, where $x$ is the value you're comparing to the mean $ \mu $.

According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $2,075. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $500.

Explore the probability and statistics behind the spending habits of adults aged 25-34, using a normal distribution with a mean of $2,075 and a standard deviation of $500. Learn how to calculate probabilities using the Z-Score and understand the application of the Central Limit Theorem. Suitable for college students or advanced high school learners.

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