Topic:Review for Business Statistics 1 - Continuous Random Variables
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Contents |
Objective
This page is written to help students prepare for the midterm and final exams for Business Statistics 1.
It does not replace the need to review each topic or do the practice problems in each topic. Rather, it supplements those learning activities with a summary of steps you should follow as you study, and with some additional practice problems.
Preparing for Continuous Random Variable Problems
This section is intended to help you prepare for exam questions on continuous random variables.
Basic Preparation Steps
1. Review the following topics, being sure you can work the examples in each:
- Continuous and Normal Random Variables
- Probabilities and the Standard Normal Curve
- In particular recall the four practice problems we did found at the end of the topic and here
- More about the Standard Normal Curve
- In particular recall the Additional Normal Curve Problems we did here
2. Review Quiz 5 in detail
3. Try your hand at the additional practice problems given below
Additional Practice Problems
Practice Problem 1
Take out Quiz 5 and redo it. The solution is posted on Blackboard.
Practice Problem 2
Redo Practice Problems 2 and 3 on the Empirical and Chebychev rules found at the end of the topic Continuous and Normal Random Variables.
Practice Problem 3
What is the area under the standard normal distribution for the following cases?
a) P( 0 < z < 1.53 )
b) P( 0 < z < 0.70 )
c) P( z < 0.70 )
d) P( z > -0.55 )
e) P( -0.41 < z < 1.14 )
| Solution |
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Use the Table A.4 for these: a) 0.4370 b) 0.2580 c) 0.2580 + 0.5000 = 0.7580 d) 0.1554 + 0.3729 = 0.5283 |
Practice Problem 4
Use the table to find the following probabilities for the standard normal curve:
a) P( -1 < z < 1 )
b) P( -2 < z < 2 )
Do these remind you of anything? If not, see the discussion about the Empirical Rule in the topic Continuous Normal Random Variables.
| Solution |
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a) P( -1 < z < 1 ) = 0.3413 + 0.3413 = 0.6826 b) P( -2 < z < 2 ) = 0.4772 + 0.4772 = 0.9544 These should remind you of the empirical rule. z = +/- 1.0 is the same as the range +/- one standard deviation from the mean. As you know, this is ~68%. In fact, you've just proven the empirical rule. |
Practice Problem 5
Sketch the standard normal curves and the areas corresponding to each of the following.
a) 15% of students will spend at least how much time preparing?
b) 65% of students will spend at least how much time preparing?
c) 85% of students will spend at most how much time preparing?
d) 20% of students will spend at most how much time preparing?
| Solution |
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a) b) c) d) |
Practice Problem 6
What area would you look up in the table in each case in Practice Problem 5? What is the corresponding z-value?
| Solution |
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a) and the corresponding z-score is: z ~= 1.04 b) and the corresponding z-score is: z ~= - 0.385 (which is actually found by finding z = 0.385 and knowing from the sketch that z needs to be a negative number) c) and the corresponding z-score is: z ~= 1.04, which is the same as in a) d) and the corresponding z-score is: z ~= - 0.84 (which is actually found by finding z = 0.84 and knowing from the sketch that z needs to be a negative number) |
Summary Thoughts
- In previous topics we looked at discrete random variables - in Continuous and Normal Random Variables, we turned our attention to continuous random variables
- We saw that a continuous random variable has a continuous probability distribution for which probabilities are found by looking at areas under the curve for ranges of x (not by taking the sum of values like we did for the discrete case)
- The most common continuous distribution (and the only one we looked at) is the normal distribution, which we studied in Probabilities and the Standard Normal Curve and More about the Standard Normal Curve
- For the normal curve, we introduced three general approaches to solving problems:
- Solving for z
- Solving for x
- Solving for the mean
- In doing so, we introduced the following key equations:
(1)
(2)
(3)
where x is the value of the random variable in question, z is the corresponding z-score, 
is the mean and s is the standard deviation
