Math Problem Statement
assume a standard normal distribution for these calculations. Go through calculations in detail
Solution
Great! We'll assume a standard normal distribution for these calculations, which means the mean () is 0 and the standard deviation () is 1. The values given are rates, and we'll treat them as z-scores directly. Here's how we can solve each part:
1. Find the probability that a randomly selected rate is higher than 4.75%.
Since we're using a standard normal distribution:
- The z-score corresponding to 4.75% is directly 4.75.
- We want to find .
Using standard normal distribution tables or a calculator:
- The probability is almost 1 (since 4.75 is far into the tail).
- Therefore, is almost 0.
2. Find the probability that a randomly selected rate is at most 4.52%.
Again, using the standard normal distribution:
- The z-score corresponding to 4.52% is directly 4.52.
- We want to find .
From the standard normal distribution table:
- (since 4.52 is very high).
3. Find the probability that a randomly selected rate is between 4.64% and 5.26%.
Here, we calculate two probabilities:
- and .
- The probability we want is .
Using standard normal distribution tables or a calculator:
- (since 5.26 is very high).
- (since 4.64 is also very high).
4. Find the rate corresponding to a standard score (z-score) of 0.85.
Given:
- The formula to find the corresponding rate (X) is:
Since and :
So, the rate corresponding to a z-score of 0.85 is 0.85 or 85%.
Summary:
- Probability of rate higher than 4.75%: approximately 0.
- Probability of rate at most 4.52%: approximately 1.
- Probability of rate between 4.64% and 5.26%: 0.
- Rate corresponding to a z-score of 0.85: 0.85 or 85%.
Would you like further clarification on any of these points?
Follow-up Questions:
- How would the probabilities change if the rates were closer to the mean?
- What does a z-score of 0.85 signify in a standard normal distribution?
- How would you find probabilities if the mean and standard deviation were different?
- What is the significance of tail probabilities in a normal distribution?
- How do these results apply to real-world scenarios?
Tip:
When dealing with probabilities in a normal distribution, remember that extreme z-scores (far from 0) correspond to events that are very rare.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Standard Normal Distribution
Z-score
Formulas
Z-score formula: \( Z = \frac{X - \mu}{\sigma} \)
Theorems
-
Suitable Grade Level
Advanced High School / College Level
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