The Uniform Distribution (The Uniform Model)

**Problem 1:**
- A uniform distribution is defined over the interval from 2 to 6.
- What are the values for \( c \) and \( d \)?
- What is the mean of this uniform distribution?
- What is the standard deviation?
- Show that the total area is 1.00.
- Find the probability of a value more than 2.6.
- Find the probability of a value between 2.9 and 4.7.

**Problem 2:**
- Customers experiencing technical difficulty with their Internet cable hookup may call an 800 number for technical support. It takes the technician between 30 seconds to 11 minutes to resolve the problem. The distribution of this support time follows the uniform distribution.
- What are the values for \( c \) and \( d \) in minutes?
- What is the mean time to resolve the problem?
- What is the standard deviation of the time?
- What percent of the problems take more than 6 minutes to resolve?
- Suppose we wish to find the middle 50% of the problem-solving times. What are the end points of these two times?

**Problem 3:**
- The accounting department at Weston Materials Inc., a national manufacturer of unattached garages, reports that it takes two construction workers a mean of 33 hours and a standard deviation of 2 hours to erect the Red Barn model. Assume the assembly times follow the normal distribution.
- Determine the \( z \) values for 29 and 34 hours.
- What percent of the garages take between 31 hours and 35 hours to erect?
- What percent of the garages take between 29 hours and 34 hours to erect?
- What percent of the garages take 28.7 hours or less to erect?
- Of the garages, 5% take how many hours or more to erect?

**Problem 4:**
- A study of long-distance phone calls made from General Electric's corporate headquarters in Fairfield, Connecticut, revealed the length of the calls, in minutes, follows the normal probability distribution. The mean length of time per call was 4.2 minutes and the standard deviation was 0.50 minutes.
- What fraction of the calls last between 4.2 and 5 minutes?
- What fraction of the calls last more than 5 minutes?
- What fraction of the calls last between 4 and 6 minutes?
- As part of her report to the president, the director of communications would like to report the length of the longest (in duration) 4% of the calls. What is this time?

**Problem 5:**
- Explain (using your own words) the characteristics of a Uniform and Normal Distribution. They are both continuous distributions but how do they differ?

**Problem 6:**
- Let \( x \) be a normally distributed random variable having mean \( \mu = 30 \) and standard deviation \( \sigma = 5 \).
- Find the \( z \) value for each of the following observed values of \( x \):
- \( x = 25 \)
- \( x = 15 \)
- \( x = 30 \)
- \( x = 50 \)

**Problem 7:**
- Stanford–Binet IQ Test scores are normally distributed with a mean score of 100 and a standard deviation of 16.
- Write the equation that gives the \( z \) score corresponding to a Stanford–Binet IQ test score.
- Find the probability that a randomly selected person has an IQ test score:
- Over 140.
- Under 88.
- Between 72 and 128.