Statistics
Chapter 4
1. Book 4.1.1 page 169: 7 – 26, 33 – 36
2. Book 4.1.2 page 169: 1 – 6, 27 – 32, 37 – 44
3. Handout 4110
4. Handout 4120
5. Book 4.2.1 page 183: 5 – 14
6. Book 4.2.2 page 183: 15 – 20
7. Handout 4210
8. Handout 4220
9. Handout 4310
10. Book 4.3 page 192: 1 – 14
11. Book Uses and Abuses page 197
12. Book Chapter 4 Review page 198
13. Book Chapter 4 Quiz page 203
TEST
A random variable has a probability distribution whether it is discrete or continuous. The probability distribution is simply an assignment of probabilities to the specific values of the random variable or to a range of values of the random variable.
A discrete probability distribution lists every possible value the random variable can assume, together with its probability.
1. The probability distribution of a discrete random variable has a probability assigned to each value of the random variable, a value between 0 and 1, inclusive.
2. The sum of these probabilities must be 1.
Guidelines for constructing a discrete probability distribution
1. Make a frequency distribution for the possible outcomes, x.
2. Find the sum of the frequencies.
3. Find the probability of each possible outcomes, P(x), by dividing its frequency by the sum of the frequencies.
4. Check that each probability is between 0 and 1 and that the sum is 1.
Let’s look at a discrete probability distribution and its graph.
Example: Discrete probability distribution
Dr. Smith and Dr. Johnson developed a test to measure boredom tolerance. They administered it to a group of 200 statistics students between the ages of 25 and 35. The possible scores were 0, 1, 2, 3, 4, 5, and 6, with 6 indicating the highest tolerance for boredom. The test results for this group are shown in the table.
Test Scores for 200 Subjects
a. If a subject is chosen at random from this group, the probability that they have a score of 1 is 26/200, or 0.13. In a similar way, we can use the relative frequency to compute the probabilities for the other scores. These probability assignments make up the probability distribution. Notice that the scores are mutually exclusive. No one subject has two scores and the sum of the probabilities of all the scores is 1.
Probability Distribution of Scores on Boredom Tolerance Test
b. The graph of this distribution is simply a relative-frequency histogram in which the height of the bar over a score represents the probability of that score. Since each bar is one unit wide, the area of the bar over a score equals the height and thus represents the probability of that score. Since the sum of the probabilities is 1, the area under the graph is also 1.
It is important for you to establish the correspondence between the area under regions of the probability histogram and the probability that the random variable takes on a particular value.
c. The Spike Factory needs to hire someone with a score on the boredom tolerance test of 5 or 6 to operate the metal press machine. Since the scores 5 and 6 are mutually exclusive, the probability that someone in the group who took the boredom tolerance test made either a 5 or a 6 is the sum
P(5 or 6) = P(5) + P(6) = 0.08 + 0.02 = 0.10
Notice that to find P(5 or 6), we could have simply added the areas of the bars over 5 and over 6. One out of 10 of the group who took the boredom tolerance test would qualify for the position at The Spike Factory.
A statistical experiment is any process by which measurements are obtained. Examples are
1. Counting the number of eggs in a robin’s nest
2. Counting the number of defective light bulbs in a case of bulbs
3. Measuring daily rainfall
4. Measuring the weight of a polar bear cub
Let x represent a quantitative variable that is measured or observed in an experiment. We are interested in the numerical values that x can take on. So x = number of eggs in a robin’s nest and x = weight in kilograms of a polar bear cub would be examples of such quantitative variables. Furthermore, we say that the quantitative variable x is a random variable because the value that x takes on in a given experiment is a chance or random outcome. We will study two types of random variables: discrete random variables and continuous random variables.
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Discrete random variable: When the observations of a quantitative random variable can take on only a finite number of values or a countable number of values, we say that the variable is a discrete random variable.
What is a countable number of values? As an example, let the random variable x be the number of wells that are drilled before the first productive well is found. Then x could be any of the values 1, 2, 3 …. In theory, we have an infinite number of possibilities for the values of x. The set of values of x corresponds to the set of counting numbers. Therefore, this type of infinity is called countable, and we say x has a countable number of values.
In most of the cases we will consider, a discrete random variable will be the result of a count. For instance, the number of students in a certain section of a statistics course this term is a discrete random variable. The value must be a counting number such as 14, or 20, or 34, and so forth. The values 25.34 or 25 1/2 are not possible.
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Continuous random variable: When the observations of a quantitative random variable can take on any of the countless number of values in a line interval, we say that the variable is a continuous random variable.
We will see most continuous random variables occurring as the result of a measurement. For example, the air pressure in an automobile tire represents a continuous random variable. The air pressure could in theory take on any value from 0 lb/in2 or psi to the bursting pressure of the tire. Values such as 28.10 psi, 28.12 psi, and so forth are possible. Another example is the height of students in your statistics class. The heights could take on any value from a low of 3 feet to a high of 7.5 feet.
In general, measurements of quantities such as length, weight, volume, temperature, or time yield continuous random variables. If the temperature changes from 72°F to 73°F, for example, it must take on all the temperature values between 72 and 73. Temperatures cannot just jump from one reading to the next. Discrete random variables often come from counts, such as the number of passing scores on an exam or the number of weeds in a garden.
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Examples: Which of the following random variables are discrete and which are continuous?
a. The time it takes a student selected at random to register for the fall term. Time can take on any value, so this is a continuous random variable.
b. The number of bad checks drawn on a bank on a day selected at random. The number of bad checks can be only a whole number such as 0, 1, 2, 3, etc. This is a discrete variable.
c. The amount of gasoline needed to drive your car 200 miles. We are measuring volume, which can assume any value, so this is a continuous random variable.
d. The number of voters pick in a random sample of 50 registered voters in a district in order to find the number who voted in the last county election. This is a count, so the variable is discrete.
