(a)A _____________variable can be used to represent anticipated outcome of a study.

(b)Formally, random variable assigns _______________values to the outcomes of a random phenomenon

(2)A coin is tossed 3 times.Let T = a tail is observed and H = a head is observed.List the eight possible three-letter outcomes of tossing the coin 3 times in the table below and the associate probability 1/8 for each.The first two and the associated probabilities are already listed, (The prob. Will be 1/8 for each outcome)

Outcomes (toss 3 coins)

Probability

TTT

1/8

HTT

1/8

a.Based on the above table, what is the number of outcomes that contain two heads and one tail? ______________

b.Consider a random variable (X) that represents the number of heads in 3 tosses of a coin. So, X=0, for outcome TTT, since there are no heads, X=1, for the outcome TTH, since there is one head, X=2 for outcome HHT, there are two heads etc.

c.The random variable X can take 4 values: _____, ______, ______, _______

d.X represents the number of heads, in three tosses of a coin, and P(X) represents the probability associated with X, complete the last column in the table below.

e.

Outcomes (toss 3 coins)

X(no. of heads)

Probability

TTT

0

1/8

HTT

1

1/8

f.List the values of X(without repeats, i. e. write each value only once) and the associated probabilities P(X) in the table below (feel free to use fractions or the decimals), (Hint: 1 shows up three times, add 1/8 +1/8+1/8 for prob.)

X (no. of heads)

P(X)

0

1/8=0.125

1

3/8=0.375

2

3

Total

(g)What is the probability of getting two heads?P(X = 2) =

(h)What is the probability of getting less than two heads?P(X < 2) =

(3)X is an example of _______________random variable since it takes on a finite number of values

(4)Discrete random variables are generally results of the counts such as number of surgeries, number of cars, etc., so these variables can take a _____________number of values

5.Continuous random variables are results of measurements, such as weight, time, temperature etc. these variables can take any value in an interval, for example, think about the number of values (decimals and fractions) that are possible between numbers o and 1, 0.01. 0.1, 0.234 or any decimal between 0 and 1, so, continuous random variables can take _____________number of values

Expected value:

3.1.2

#6

(a)The mean of a random variable is called its _____________________.

(b)The expected value, which is the same as the mean, can be written as:

(c)To find the expected value, or, the mean of the random variable, multiply each possible value by its corresponding probability and _______________ these products.

7. (a)Calculate the Expected Value of X, where X represents the number of heads in 3 tosses of a fair coin: (the first two rows are completed for you)

X (no. of heads)

P(X)

X*P(X)

0

1/8=0.125

0*0.125=0

1

3/8=0.375

1*0.375=0.375

2

3/8=0.375

3

1/8=0.125

Total

(b)The expected value is the total of X*P(X). For above example, E(X)= ________

(8) At a particular hospital, there are typically 15 outpatient surgeries per day scheduled 30% of the time, 20 outpatient surgeries per day, scheduled 50% of the time, and 25 outpatient surgeries per day scheduled 20% of the time. Find the average (mean) number of surgeries per day. (same as the expected value)

X (no. of surgeries)

P(X)

X*P(X)

Total

For #8, the Expected Value=

Binomial Distribution:

(9) When in a random experiment (trial) there are only two possible outcomes it is known as the _______________random variable.The two possible outcomes are referred to as a success or a failure.For example a patient is diabetic or non-diabetic, or patient tested positive or negative for a virus.

(10) The Binomial Distribution describes the probability of k successes in n independent _____________trials.

(11) Is the binomial distribution a discrete or a continuous distribution? ________________________

(The following information in the textbox is optional. We will be using Minitab for calculating Binomial Probabilities for our class)

(12)What are the formulas for the mean and standard deviation of a binomial distribution? (This will be done manually)

mean = = st. dev. = =

(13)What are the 4 conditions to check if a distribution is a binomial distribution?

(14)The actual rate of unintended pregnancy for women taking the pill is 5%.A doctor prescribed the pill to 20 new patients.X is the count of the number of unintended pregnancies within one year.

(a)Find the probability that X= 2 (use n=20 and p= 5%= 0.05)

Minitab: Calc>Probability Distributions> Binomial..>Probability>Number of trials, Enter 20> Event probability, Enter 0.05>Click Input constant, Enter 2 (because X=2)

(b) Use Minitab to find the probability that X= 4

(c)Find the probability that X 4

(Notice the sign, this is the Cumulative probability, to find this we add the probabilities for X=0, X=1, X=2, X=3 and X=4 using Minitab, use the Cumulative prob. option, see below)

Minitab: Calc>Probability Distributions> Binomial..> Cumulative probability>Number of trials, Enter 20> Event probability, Enter 0.05>Click Input constant, Enter 4

(15) Use formulas from #12, for the mean and Std. Deviation of a Binomial Variable, to find the mean and the standard deviation for X in #14.(n=20, p=0.05)

3.3

Normal Distribution

Normal Distribution is very important because many variables, such as our heights, life expectancy, blood pressure etc. are have a near normal distribution.

16.What is the shape of a normal distribution?

17.What is the mean and standard deviation of a standard normal distribution?

18.Is the normal distribution a discrete or continuous distribution?

3.3.2

(19)A z-score is used to standardize an observation in a normal distribution.Write the formula to calculate the z-score, if :

x represents the normal variable,

(read as "mu") represents the mean of x

and ? represents the standard deviation of x

Z=

Z- score for a particular value tells us how far that value is from the mean in units of the standard deviations. For example, if Z=1.25, we know that the value is 1.25 std. deviations above the mean, if Z= -1.25, the value is 1.25 std. dev. below the mean.

(20) Z-scores can be used to compare the relative standing of values from two different populations or two different scales. For example, how do we compare one student's SAT score with another student's ACT score to find our who performed better?(Two different tests)

Consider:

Mean SAT score= _SAT= 1500, Std. Deviation for SAT= ?_SAT= 300

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5

Student A scores 1800 on SAT and student B scores 24 on ACT. Compare the performance of Student A vs Student B

(a)Calculate the Z score for student A

(b)Calculate the Z-score for student B

(c)Use the Z-scores to decide who performed better, student A or Student B?

Optional:

Look at the Guided Practice 3.19

3.3.3

Empirical Rule:

(21)The Empirical Rule is also known as the ____________________________rule

According to the empirical rule, what percentage of observations are within

i.one standard deviation of the mean?

ii.two standard deviations of the mean?

iii.three standard deviations of the mean?

(22)Consider the ACT scores with a normal distribution:

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5.

Based on the Empirical Rule:

(a)68% of the SAT scores fall between _____________and ____________________

(b)95% of the SAT scores fall between _____________and ____________________

(c)99.73% of the SAT scores fall between _____________and ____________________

3.3.4

Calculating Normal Probabilities:

(23)Normal Distribution is a _____________________probability distribution

(24)What is the total area under the normal curve?

(25)When the bell curve is drawn for a normal distribution, the value at the center = ___________

Normal Probability Calculations:

(26)Consider the SAT scores with a normal distribution, Mean SAT score= _SAT= 1500, Std. Deviation for SAT= ?_SAT= 300,

Find the % of students who scored below 1800? (which is the same as saying what is the probability that the given score is below 1800.

(For our class, we will be using Minitab to calculate the normal probabilities)

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By >Select X Value> Click Left Tail> X value: Enter 1800 >OK

Copy the graph below. (Notice 0.8413, this is the percentage of individuals who scored below 1800 on the SAT, or it is the percentile associated with the score 1800)

(27)Find the percentage of scores SAT scores that are above 1800.

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select X Value> Click Right Tail> X value: Enter 1800 >OK

Copy the graph below. (Notice 0.1587, that is the percentage of individuals who scored above 1800 on the SAT)

(28)Find the percentage of scores SAT scores that are between 1300 and 1800.

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By >Select X Value> Click Middle> X value 1: Enter 1300 > X value 2: Enter 1800 > OK

Copy the graph below. (Notice 0.5889, this is the percentage of individuals who scored between 1300 and 1800 on the SAT)

(29)Calculating the value of a variable for a given percentile:

(a)75% of the SAT scores are below what number?

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select Probability > Click Left Tail> Probability: 0.75>OK (Copy the graph below)

(Notice 1702, this means that 75% of the SAT scores are below the score 1702)

(b) 75% of the SAT scores are above what number?

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select Probability > Click Right Tail> Probability: 0.75>OK

(Notice 1298, this means that 75% of the SAT scores are above the score 1298)

(30)Consider the ACT scores with a normal distribution (Copy graphs for a thru e below)

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5

a)Find the percentage of ACT scores below 24

b) Find the percentage of ACT scores above 27

c) Find the percentage of ACT scores between 18 and 28

d)10% of the ACT scores are above what?

e)10% of the ACT scores are below what?

(a)A _____________variable can be used to represent anticipated outcome of a study.

(b)Formally, random variable assigns _______________values to the outcomes of a random phenomenon

(2)A coin is tossed 3 times.Let T = a tail is observed and H = a head is observed.List the eight possible three-letter outcomes of tossing the coin 3 times in the table below and the associate probability 1/8 for each.The first two and the associated probabilities are already listed, (The prob. Will be 1/8 for each outcome)

Outcomes (toss 3 coins)

Probability

TTT

1/8

HTT

1/8

a.Based on the above table, what is the number of outcomes that contain two heads and one tail? ______________

b.Consider a random variable (X) that represents the number of heads in 3 tosses of a coin. So, X=0, for outcome TTT, since there are no heads, X=1, for the outcome TTH, since there is one head, X=2 for outcome HHT, there are two heads etc.

c.The random variable X can take 4 values: _____, ______, ______, _______

d.X represents the number of heads, in three tosses of a coin, and P(X) represents the probability associated with X, complete the last column in the table below.

e.

Outcomes (toss 3 coins)

X(no. of heads)

Probability

TTT

0

1/8

HTT

1

1/8

f.List the values of X(without repeats, i. e. write each value only once) and the associated probabilities P(X) in the table below (feel free to use fractions or the decimals), (Hint: 1 shows up three times, add 1/8 +1/8+1/8 for prob.)

X (no. of heads)

P(X)

0

1/8=0.125

1

3/8=0.375

2

3

Total

(g)What is the probability of getting two heads?P(X = 2) =

(h)What is the probability of getting less than two heads?P(X < 2) =

(3)X is an example of _______________random variable since it takes on a finite number of values

(4)Discrete random variables are generally results of the counts such as number of surgeries, number of cars, etc., so these variables can take a _____________number of values

5.Continuous random variables are results of measurements, such as weight, time, temperature etc. these variables can take any value in an interval, for example, think about the number of values (decimals and fractions) that are possible between numbers o and 1, 0.01. 0.1, 0.234 or any decimal between 0 and 1, so, continuous random variables can take _____________number of values

Expected value:

#6

(a)The mean of a random variable is called its _____________________.

(b)The expected value, which is the same as the mean, can be written as:

(c)To find the expected value, or, the mean of the random variable, multiply each possible value by its corresponding probability and _______________ these products.

7. (a)Calculate the Expected Value of X, where X represents the number of heads in 3 tosses of a fair coin: (the first two rows are completed for you)

X (no. of heads)

P(X)

X*P(X)

0

1/8=0.125

0*0.125=0

1

3/8=0.375

1*0.375=0.375

2

3/8=0.375

3

1/8=0.125

Total

(b)The expected value is the total of X*P(X). For above example, E(X)= ________

(8) At a particular hospital, there are typically 15 outpatient surgeries per day scheduled 30% of the time, 20 outpatient surgeries per day, scheduled 50% of the time, and 25 outpatient surgeries per day scheduled 20% of the time. Find the average (mean) number of surgeries per day. (same as the expected value)

X (no. of surgeries)

P(X)

X*P(X)

Total

For #8, the Expected Value=

Binomial Distribution:

(9) When in a random experiment (trial) there are only two possible outcomes it is known as the _______________random variable.The two possible outcomes are referred to as a success or a failure.For example a patient is diabetic or non-diabetic, or patient tested positive or negative for a virus.

(10) The Binomial Distribution describes the probability of k successes in n independent _____________trials.

(11) Is the binomial distribution a discrete or a continuous distribution? ________________________

(The following information in the textbox is optional. We will be using Minitab for calculating Binomial Probabilities for our class)

(12)What are the formulas for the mean and standard deviation of a binomial distribution? (This will be done manually)

mean = = st. dev. = =

(13)What are the 4 conditions to check if a distribution is a binomial distribution?

(14)The actual rate of unintended pregnancy for women taking the pill is 5%.A doctor prescribed the pill to 20 new patients.X is the count of the number of unintended pregnancies within one year.

(a)Find the probability that X= 2 (use n=20 and p= 5%= 0.05)

Minitab: Calc>Probability Distributions> Binomial..>Probability>Number of trials, Enter 20> Event probability, Enter 0.05>Click Input constant, Enter 2 (because X=2)

(b) Use Minitab to find the probability that X= 4

(c)Find the probability that X 4

(Notice the sign, this is the Cumulative probability, to find this we add the probabilities for X=0, X=1, X=2, X=3 and X=4 using Minitab, use the Cumulative prob. option, see below)

Minitab: Calc>Probability Distributions> Binomial..> Cumulative probability>Number of trials, Enter 20> Event probability, Enter 0.05>Click Input constant, Enter 4

(15) Use formulas from #12, for the mean and Std. Deviation of a Binomial Variable, to find the mean and the standard deviation for X in #14.(n=20, p=0.05)

Normal Distribution

Normal Distribution is very important because many variables, such as our heights, life expectancy, blood pressure etc. are have a near normal distribution.

16.What is the shape of a normal distribution?

17.What is the mean and standard deviation of a standard normal distribution?

18.Is the normal distribution a discrete or continuous distribution?

(19)A z-score is used to standardize an observation in a normal distribution.Write the formula to calculate the z-score, if :

x represents the normal variable,

(read as "mu") represents the mean of x

and ? represents the standard deviation of x

Z=

Z- score for a particular value tells us how far that value is from the mean in units of the standard deviations. For example, if Z=1.25, we know that the value is 1.25 std. deviations above the mean, if Z= -1.25, the value is 1.25 std. dev. below the mean.

(20) Z-scores can be used to compare the relative standing of values from two different populations or two different scales. For example, how do we compare one student's SAT score with another student's ACT score to find our who performed better?(Two different tests)

Consider:

Mean SAT score= _SAT= 1500, Std. Deviation for SAT= ?_SAT= 300

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5

Student A scores 1800 on SAT and student B scores 24 on ACT. Compare the performance of Student A vs Student B

(a)Calculate the Z score for student A

(b)Calculate the Z-score for student B

(c)Use the Z-scores to decide who performed better, student A or Student B?

Empirical Rule:

(21)The Empirical Rule is also known as the ____________________________rule

According to the empirical rule, what percentage of observations are within

i.one standard deviation of the mean?

ii.two standard deviations of the mean?

iii.three standard deviations of the mean?

(22)Consider the ACT scores with a normal distribution:

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5.

Based on the Empirical Rule:

(a)68% of the SAT scores fall between _____________and ____________________

(b)95% of the SAT scores fall between _____________and ____________________

(c)99.73% of the SAT scores fall between _____________and ____________________

3.3.4

Calculating Normal Probabilities:

(23)Normal Distribution is a _____________________probability distribution

(24)What is the total area under the normal curve?

(25)When the bell curve is drawn for a normal distribution, the value at the center = ___________

Normal Probability Calculations:

(26)Consider the SAT scores with a normal distribution, Mean SAT score= _SAT= 1500, Std. Deviation for SAT= ?_SAT= 300,

Find the % of students who scored below 1800? (which is the same as saying what is the probability that the given score is below 1800.

(For our class, we will be using Minitab to calculate the normal probabilities)

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By >Select X Value> Click Left Tail> X value: Enter 1800 >OK

Copy the graph below. (Notice 0.8413, this is the percentage of individuals who scored below 1800 on the SAT, or it is the percentile associated with the score 1800)

(27)Find the percentage of scores SAT scores that are above 1800.

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select X Value> Click Right Tail> X value: Enter 1800 >OK

Copy the graph below. (Notice 0.1587, that is the percentage of individuals who scored above 1800 on the SAT)

(28)Find the percentage of scores SAT scores that are between 1300 and 1800.

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By >Select X Value> Click Middle> X value 1: Enter 1300 > X value 2: Enter 1800 > OK

Copy the graph below. (Notice 0.5889, this is the percentage of individuals who scored between 1300 and 1800 on the SAT)

(29)Calculating the value of a variable for a given percentile:

(a)75% of the SAT scores are below what number?

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select Probability > Click Left Tail> Probability: 0.75>OK (Copy the graph below)

(Notice 1702, this means that 75% of the SAT scores are below the score 1702)

(b) 75% of the SAT scores are above what number?

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select Probability > Click Right Tail> Probability: 0.75>OK

(Notice 1298, this means that 75% of the SAT scores are above the score 1298)

(30)Consider the ACT scores with a normal distribution (Copy graphs for a thru e below)

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5

a)Find the percentage of ACT scores below 24

b) Find the percentage of ACT scores above 27

c) Find the percentage of ACT scores between 18 and 28

d)10% of the ACT scores are above what?

e)10% of the ACT scores are below what?

(a)A _____________variable can be used to represent anticipated outcome of a study.

(b)Formally, random variable assigns _______________values to the outcomes of a random phenomenon

(2)A coin is tossed 3 times.Let T = a tail is observed and H = a head is observed.List the eight possible three-letter outcomes of tossing the coin 3 times in the table below and the associate probability 1/8 for each.The first two and the associated probabilities are already listed, (The prob. Will be 1/8 for each outcome)

Outcomes (toss 3 coins)

Probability

TTT

1/8

HTT

1/8

a.Based on the above table, what is the number of outcomes that contain two heads and one tail? ______________

b.Consider a random variable (X) that represents the number of heads in 3 tosses of a coin. So, X=0, for outcome TTT, since there are no heads, X=1, for the outcome TTH, since there is one head, X=2 for outcome HHT, there are two heads etc.

c.The random variable X can take 4 values: _____, ______, ______, _______

d.X represents the number of heads, in three tosses of a coin, and P(X) represents the probability associated with X, complete the last column in the table below.

e.

Outcomes (toss 3 coins)

X(no. of heads)

Probability

TTT

0

1/8

HTT

1

1/8

f.List the values of X(without repeats, i. e. write each value only once) and the associated probabilities P(X) in the table below (feel free to use fractions or the decimals), (Hint: 1 shows up three times, add 1/8 +1/8+1/8 for prob.)

X (no. of heads)

P(X)

0

1/8=0.125

1

3/8=0.375

2

3

Total

(g)What is the probability of getting two heads?P(X = 2) =

(h)What is the probability of getting less than two heads?P(X < 2) =

(3)X is an example of _______________random variable since it takes on a finite number of values

(4)Discrete random variables are generally results of the counts such as number of surgeries, number of cars, etc., so these variables can take a _____________number of values

5.Continuous random variables are results of measurements, such as weight, time, temperature etc. these variables can take any value in an interval, for example, think about the number of values (decimals and fractions) that are possible between numbers o and 1, 0.01. 0.1, 0.234 or any decimal between 0 and 1, so, continuous random variables can take _____________number of values

Expected value:

3.1.2

#6

(a)The mean of a random variable is called its _____________________.

(b)The expected value, which is the same as the mean, can be written as:

(c)To find the expected value, or, the mean of the random variable, multiply each possible value by its corresponding probability and _______________ these products.

7. (a)Calculate the Expected Value of X, where X represents the number of heads in 3 tosses of a fair coin: (the first two rows are completed for you)

X (no. of heads)

P(X)

X*P(X)

0

1/8=0.125

0*0.125=0

1

3/8=0.375

1*0.375=0.375

2

3/8=0.375

3

1/8=0.125

Total

(b)The expected value is the total of X*P(X). For above example, E(X)= ________

(8) At a particular hospital, there are typically 15 outpatient surgeries per day scheduled 30% of the time, 20 outpatient surgeries per day, scheduled 50% of the time, and 25 outpatient surgeries per day scheduled 20% of the time. Find the average (mean) number of surgeries per day. (same as the expected value)

X (no. of surgeries)

P(X)

X*P(X)

Total

For #8, the Expected Value=

Binomial Distribution:

(9) When in a random experiment (trial) there are only two possible outcomes it is known as the _______________random variable.The two possible outcomes are referred to as a success or a failure.For example a patient is diabetic or non-diabetic, or patient tested positive or negative for a virus.

(10) The Binomial Distribution describes the probability of k successes in n independent _____________trials.

(11) Is the binomial distribution a discrete or a continuous distribution? ________________________

(The following information in the textbox is optional. We will be using Minitab for calculating Binomial Probabilities for our class)

(12)What are the formulas for the mean and standard deviation of a binomial distribution? (This will be done manually)

mean = = st. dev. = =

(13)What are the 4 conditions to check if a distribution is a binomial distribution?

(14)The actual rate of unintended pregnancy for women taking the pill is 5%.A doctor prescribed the pill to 20 new patients.X is the count of the number of unintended pregnancies within one year.

(a)Find the probability that X= 2 (use n=20 and p= 5%= 0.05)

Minitab: Calc>Probability Distributions> Binomial..>Probability>Number of trials, Enter 20> Event probability, Enter 0.05>Click Input constant, Enter 2 (because X=2)

(b) Use Minitab to find the probability that X= 4

(c)Find the probability that X 4

(Notice the sign, this is the Cumulative probability, to find this we add the probabilities for X=0, X=1, X=2, X=3 and X=4 using Minitab, use the Cumulative prob. option, see below)

Minitab: Calc>Probability Distributions> Binomial..> Cumulative probability>Number of trials, Enter 20> Event probability, Enter 0.05>Click Input constant, Enter 4

(15) Use formulas from #12, for the mean and Std. Deviation of a Binomial Variable, to find the mean and the standard deviation for X in #14.(n=20, p=0.05)

Normal Distribution

Normal Distribution is very important because many variables, such as our heights, life expectancy, blood pressure etc. are have a near normal distribution.

16.What is the shape of a normal distribution?

17.What is the mean and standard deviation of a standard normal distribution?

18.Is the normal distribution a discrete or continuous distribution?

(19)A z-score is used to standardize an observation in a normal distribution.Write the formula to calculate the z-score, if :

x represents the normal variable,

(read as "mu") represents the mean of x

and ? represents the standard deviation of x

Z=

Z- score for a particular value tells us how far that value is from the mean in units of the standard deviations. For example, if Z=1.25, we know that the value is 1.25 std. deviations above the mean, if Z= -1.25, the value is 1.25 std. dev. below the mean.

(20) Z-scores can be used to compare the relative standing of values from two different populations or two different scales. For example, how do we compare one student's SAT score with another student's ACT score to find our who performed better?(Two different tests)

Consider:

Mean SAT score= _SAT= 1500, Std. Deviation for SAT= ?_SAT= 300

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5

Student A scores 1800 on SAT and student B scores 24 on ACT. Compare the performance of Student A vs Student B

(a)Calculate the Z score for student A

(b)Calculate the Z-score for student B

(c)Use the Z-scores to decide who performed better, student A or Student B?

Empirical Rule:

(21)The Empirical Rule is also known as the ____________________________rule

According to the empirical rule, what percentage of observations are within

i.one standard deviation of the mean?

ii.two standard deviations of the mean?

iii.three standard deviations of the mean?

(22)Consider the ACT scores with a normal distribution:

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5.

Based on the Empirical Rule:

(a)68% of the SAT scores fall between _____________and ____________________

(b)95% of the SAT scores fall between _____________and ____________________

(c)99.73% of the SAT scores fall between _____________and ____________________

Calculating Normal Probabilities:

(23)Normal Distribution is a _____________________probability distribution

(24)What is the total area under the normal curve?

(25)When the bell curve is drawn for a normal distribution, the value at the center = ___________

Normal Probability Calculations:

(26)Consider the SAT scores with a normal distribution, Mean SAT score= _SAT= 1500, Std. Deviation for SAT= ?_SAT= 300,

Find the % of students who scored below 1800? (which is the same as saying what is the probability that the given score is below 1800.

(For our class, we will be using Minitab to calculate the normal probabilities)

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By >Select X Value> Click Left Tail> X value: Enter 1800 >OK

Copy the graph below. (Notice 0.8413, this is the percentage of individuals who scored below 1800 on the SAT, or it is the percentile associated with the score 1800)

(27)Find the percentage of scores SAT scores that are above 1800.

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select X Value> Click Right Tail> X value: Enter 1800 >OK

Copy the graph below. (Notice 0.1587, that is the percentage of individuals who scored above 1800 on the SAT)

(28)Find the percentage of scores SAT scores that are between 1300 and 1800.

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By >Select X Value> Click Middle> X value 1: Enter 1300 > X value 2: Enter 1800 > OK

Copy the graph below. (Notice 0.5889, this is the percentage of individuals who scored between 1300 and 1800 on the SAT)

(29)Calculating the value of a variable for a given percentile:

(a)75% of the SAT scores are below what number?

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select Probability > Click Left Tail> Probability: 0.75>OK (Copy the graph below)

(Notice 1702, this means that 75% of the SAT scores are below the score 1702)

(b) 75% of the SAT scores are above what number?

Minitab> Graph>Probability Distribution Plot>View Probability>OK>Distribution: Normal> Mean: Enter 1500>Standard Deviation: Enter 300> Click Shaded Area (tab at the top)> Define Shaded Area By> Select Probability > Click Right Tail> Probability: 0.75>OK

(Notice 1298, this means that 75% of the SAT scores are above the score 1298)

(30)Consider the ACT scores with a normal distribution (Copy graphs for a thru e below)

Mean ACT score= _ACT= 21, Std. Deviation for ACT= ?_ACT= 5

a)Find the percentage of ACT scores below 24

b) Find the percentage of ACT scores above 27

c) Find the percentage of ACT scores between 18 and 28

d)10% of the ACT scores are above what?

e)10% of the ACT scores are below what?