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This question refers to a uniform distribution, where numbers are selected between 5 and 69.

Let's address the questions one by one.

### Given:
- The numbers range from 5 to 69, so:
  - Minimum (\(a\)) = 5
  - Maximum (\(b\)) = 69

### a. **The mean of this distribution is:**

The mean (\(\mu\)) for a uniform distribution can be calculated using:
\[
\mu = \frac{a + b}{2}
\]
\[
\mu = \frac{5 + 69}{2} = \frac{74}{2} = 37
\]
**Answer: 37**

### b. **The standard deviation is:**

The standard deviation (\(\sigma\)) for a uniform distribution is:
\[
\sigma = \frac{b - a}{\sqrt{12}}
\]
\[
\sigma = \frac{69 - 5}{\sqrt{12}} = \frac{64}{\sqrt{12}} \approx \frac{64}{3.464} \approx 18.471
\]
**Answer: 18.471**

### c. **The probability that the number will be exactly 19 is:**

For a continuous uniform distribution, the probability of getting any exact value is 0, because the probability of a single point in a continuous distribution is 0.

**Answer: 0**

### d. **The probability that the number will be between 6 and 21:**

To find the probability between two values in a uniform distribution:
\[
P(6 < x < 21) = \frac{21 - 6}{69 - 5}
\]
\[
P(6 < x < 21) = \frac{15}{64} \approx 0.2344
\]
**Answer: 0.2344**

### e. **The probability that the number will be larger than 33:**

\[
P(x > 33) = \frac{69 - 33}{69 - 5}
\]
\[
P(x > 33) = \frac{36}{64} = 0.5625
\]
**Answer: 0.5625**

### f. **Conditional Probability \(P(x > 14 | x < 26)\):**

This is a conditional probability, which is calculated as:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
Here, \(A\) is \(x > 14\) and \(B\) is \(x < 26\). 

1. \(P(14 < x < 26) = \frac{26 - 14}{69 - 5} = \frac{12}{64} = 0.1875\)
2. \(P(B) = P(x < 26) = \frac{26 - 5}{69 - 5} = \frac{21}{64}\)

Thus, the conditional probability is:
\[
P(x > 14 | x < 26) = \frac{0.1875}{0.328125} \approx 0.5714
\]
**Answer: 0.5714**

### g. **Find the 85th percentile:**

The formula for the percentile in a uniform distribution is:
\[
x_p = a + p(b - a)
\]
For the 85th percentile (\(p = 0.85\)):
\[
x_{85} = 5 + 0.85(69 - 5) = 5 + 0.85(64) = 5 + 54.4 = 59.4
\]
**Answer: 59.4**

### h. **Find the maximum for the lower quartile:**

The lower quartile corresponds to the 25th percentile. Using the same formula for percentiles:
\[
x_{25} = 5 + 0.25(69 - 5) = 5 + 0.25(64) = 5 + 16 = 21
\]
**Answer: 21**

---

Let me know if you'd like more details on any of these answers!

Here are five follow-up questions:
1. How does the uniform distribution differ from other distributions like normal or binomial?
2. How would you calculate the median for this distribution?
3. What is the interpretation of standard deviation in a uniform distribution?
4. How does the conditional probability formula change when dealing with other distributions?
5. How would the results change if the range were different, say 10 to 100?

**Tip:** When dealing with uniform distributions, remember that the mean and median are always located at the center of the range.

A random number generator picks a number from 5 to 69 in a uniform manner. Round answers to 4 decimal places when possible. a. The mean of this distribution is b. The standard deviation is c. The probability that the number will be exactly 19 is P(x = 19) = d. The probability that the number will be between 6 and 21 is P(6 < x < 21) = e. The probability that the number will be larger than 33 is P(x > 33) = f. P(x > 14 | x < 26) = g. Find the 85th percentile. h. Find the maximum for the lower quartile.

Solve a uniform distribution problem with numbers ranging from 5 to 69. Learn to find the mean, standard deviation, probabilities of certain ranges, the 85th percentile, and the maximum for the lower quartile. A step-by-step guide with key statistical concepts, formulas, and theorems for college-level students.

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