Statistics

Statistics

5.2

Quiz 6

A quick quiz consists of a true/false question followed by a multiple-choice question with four possible answers (a,b,c,d). An Unprepared student makes random guesses for both answers.a. What is the probability of that both answers are correctb. Is guessing a good strategy?

Random VariablesEveryone look at figure 5-1 on page 204 of our book.• A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure

Random VariablesEveryone look at figure 5-1 on page 204 of our book.• A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure• A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula.

Random VariablesEveryone look at figure 5-1 on page 204 of our book.• A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure• A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula.

Random Variables• A discrete random variable has either a finite number of values or a countable number of values.

Random Variables• A discrete random variable has either a finite number of values or a countable number of values.• A continuous random variable has infinitely many values, and those value can be associated with measurements on a continuous scale without gaps or interruptions.

Random Variables• A discrete random variable has either a finite number of values or a countable number of values.• A continuous random variable has infinitely many values, and those value can be associated with measurements on a continuous scale without gaps or interruptions.

Random VariablesDetermine whether the given random variable is discrete or continuous.a. The total amount in (ounces) of soft drinks that you consumed in the past year.b. The number of cans of soft drinks that you consumed in the past year.c. The number of movies currently playing in U.S. theaters.d. The running time of a randomly selected movie.e. The cost of making a randomly selected movie.

Random VariablesWe use probability histograms to graph a probability distributionx

(number of peas with green pods)

P(x)

0 0.0011 0.0152 0.0883 0.2644 0.3965 0.237

Random VariablesWe use probability histograms to graph a probability distributionx

(number of peas with green pods)

P(x)

0 0.0011 0.0152 0.0883 0.2644 0.3965 0.237

0 1 2 3 4 50

0.10.20.30.40.5

Series 1

Axis Title

Axis Title

Random VariablesRequirements for a Probability Distribution1. where x assumes all possible values.2. for every individual value of x

Random VariablesRequirements for a Probability Distribution1. where x assumes all possible values.2. for every individual value of x.

Random VariablesBased on a survey conducted by Frank N. Magid Associates, Table 5-2 lists the probabilities for the number of cell phones in use per household. Does the table below describe a probability Distribution?

x P(x)

0 0.19

1 0.26

2 0.33

3 0.13

Random VariablesDoes (where can be 0, 1, 2, 3, or 4) determine a probability distribution?

Random VariablesDoes (where can be 0, 1, 2, 3, or 4) determine a probability distribution?

X P(x)

0 0/10

1 1/10

2 2/10

3 3/10

4 4/10

Random VariablesDoes (where can be 0, 1, 2, 3, or 4) determine a probability distribution?

X P(x)

0 0/10

1 1/10

2 2/10

3 3/10

4 4/10

Total 10/10=1

Random VariablesMean, Variance, and Standard Deviation• Mean for a probability distribution

Random VariablesMean, Variance, and Standard Deviation• Mean for a probability distribution• Variance for a probability distribution

Random VariablesMean, Variance, and Standard Deviation• Mean for a probability distribution• Variance for a probability distribution• Variance for a probability distribution

Random VariablesMean, Variance, and Standard Deviation• Mean for a probability distribution• Variance for a probability distribution• Variance for a probability distribution• Standard Deviation for a probability distribution

Random VariablesMean, Variance, and Standard Deviation• Mean for a probability distribution• Variance for a probability distribution• Variance for a probability distribution• Standard Deviation for a probability distributionLets do example 5 in excel!, and then do problem 3 on the worksheet

Random VariablesDetermine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small)

x P(x)

0 0.528

1 0.360

2 0.098

3 0.013

4 0.001

5 0+

Random VariablesDetermine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small)

0 0.528 0

1 0.360 0.360

2 0.098 0.196

3 0.013 0.039

4 0.001 0.004

5 0+ 0

Random VariablesDetermine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small)

0 0.528 0

1 0.360 0.360

2 0.098 0.196

3 0.013 0.039

4 0.001 0.004

5 0+ 0

Total 1 .599

Random VariablesDetermine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small)

0 0.528 0

1 0.360 0.360

2 0.098 0.196

3 0.013 0.039

4 0.001 0.004

5 0+ 0

Total 1 .599=0.6 =0.7

Random VariablesRound off rule for Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round to one decimal place.

Random VariablesRound off rule for Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round to one decimal place.Recall the range rule of thumb

Random VariablesRound off rule for Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round to one decimal place.Recall the range rule of thumb

Random VariablesRound off rule for Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round to one decimal place.Recall the range rule of thumb

Random VariablesRound off rule for Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round to one decimal place.Recall the range rule of thumb

Random VariablesUse the range rule of thumb to identify a range of values containing the usual number of peas with green pods. Based on this is it unusual to get only one pea with a green pod? Explain.

x (# of peas with green pods) P(x)

0 0+1 0+

2 0.0043 0.023

4 0.087

5 0.2086 0.311

7 0.2768 0.100

Random VariablesRare Event Rule for Inferential StatisticsIf, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct.

Random VariablesRare Event Rule for Inferential StatisticsIf, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct.• Unusually high number of successes: x successes among n trials is an unusually high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less.

Random VariablesRare Event Rule for Inferential StatisticsIf, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct.• Unusually high number of successes: x successes among n trials is an unusually high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less. • Unusually low number of successes: x successes among n trials is an unusually low number of successes if the probability of x or fewer successes is unlikely with a probability of 0.05 or less.

Random VariablesRare Event Rule for Inferential StatisticsIf, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct.• Unusually high number of successes: x successes among n trials is an unusually high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less. • Unusually low number of successes: x successes among n trials is an unusually low number of successes if the probability of x or fewer successes is unlikely with a probability of 0.05 or less.

Random Variablesa) Find the probability of getting exactly 3 peas with green pods . b) Find the probability of getting 3 or fewer peas with green pods. c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). d) Is 3 and unusually low number of peas with green pods? Why or why not?

x (# of peas with green

pods)

P(x)

0 0+

1 0+

2 0.004

3 0.023

4 0.087

5 0.208

6 0.311

7 0.276

8 0.100

Random Variablesa) Find the probability of getting exactly 3 peas with green pods . 0.023b) Find the probability of getting 3 or fewer peas with green pods. c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). d) Is 3 and unusually low number of peas with green pods? Why or why not?

x (# of peas with green

pods)

P(x)

0 0+

1 0+

2 0.004

3 0.023

4 0.087

5 0.208

6 0.311

7 0.276

8 0.100

Random Variablesa) Find the probability of getting exactly 3 peas with green pods . 0.023b) Find the probability of getting 3 or fewer peas with green pods. 0.027c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). d) Is 3 and unusually low number of peas with green pods? Why or why not?

x (# of peas with green

pods)

P(x)

0 0+

1 0+

2 0.004

3 0.023

4 0.087

5 0.208

6 0.311

7 0.276

8 0.100

Random Variablesa) Find the probability of getting exactly 3 peas with green pods . 0.023b) Find the probability of getting 3 or fewer peas with green pods. 0.027c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). Part (b)d) Is 3 and unusually low number of peas with green pods? Why or why not?

x (# of peas with green

pods)

P(x)

0 0+

1 0+

2 0.004

3 0.023

4 0.087

5 0.208

6 0.311

7 0.276

8 0.100

Random Variablesa) Find the probability of getting exactly 3 peas with green pods . 0.023b) Find the probability of getting 3 or fewer peas with green pods. 0.027c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). Part (b)d) Is 3 and unusually low number of peas with green pods? Why or why not? Yes since

x (# of peas with green

pods)

P(x)

0 0+

1 0+

2 0.004

3 0.023

4 0.087

5 0.208

6 0.311

7 0.276

8 0.100

Random VariablesExpected ValueThe expected value of a discrete random variable is denoted by E, and it represents the mean value of its outcomes. It is obtained by finding the value of

Random VariablesExpected ValueThe expected value of a discrete random variable is denoted by E, and it represents the mean value of its outcomes. It is obtained by finding the value of

Random VariablesYou are considering placing a bet on the number 7 in roulette or red for roulette.

Random Variables• If you bet $5 on the number 7 in roulette, the

probability of losing $5 is 37/38 and the probability making a net gain of $175 is 1/38. Let’s find the expected value if you bet on 7.

Random Variables• If you bet $5 on the number 7 in roulette, the

probability of losing $5 is 37/38 and the probability making a net gain of $175 is 1/38. Let’s find the expected value if you bet on 7.Event

Lose

Win(net)

Total

Random Variables• If you bet $5 on red, the probability of losing $5 is

20/38 and the probability making a net gain of $5 is 18/38. Let’s find the expected value if you bet on red.

Random Variables• If you bet $5 on red, the probability of losing $5 is

20/38 and the probability making a net gain of $5 is 18/38. Let’s find the expected value if you bet on red.Event

Lose

Win(net)

Total

Homework!!!

• 5-2: 1-17 odd ,21, 25, 27