## (solution) Question 1: Binomial Probabilities 1. Let y be a binomial random

Question 1:

Binomial Probabilities

1. Let y be a binomial random variable. Compute P(y) for each of the following situations:

a. , ?=.2,

b. , ?=.4,

c. , ?=.7,

2. Let y be a binomial random variable with and p=.4. Find the following values:

a. P (y?4)

b. P (y>4)

c. P (y<7)

d. P (y?6)

Question 2:

Over a long period of time in a large multinational corporation, 10% of all sales trainees are rated outstanding, 75% are rated as excellent, 10% are rated as satisfactory, and 5% are considered unsatisfactory. Fin the following probabilities for a sample of 10 trainees selected at random:

a. Two are rated as outstanding.

b. Two or more are rated as outstanding.

c. Eight of the ten are rated either outstanding or excellent.

d. None of the trainees is rated as unsatisfactory.

NOTE:

Below is an example on computing Binomial Probability using SAS

Question 3:

Let Y be a random variable having a binomial distribution with parameters

and p, i.e.,

Y ? Binomial (,p)

Using

i. the exact binomial distribution,

ii. the normal approximation to the binomial without the continuity

correction,

iii. the normal approximation to the binomial with the continuity correction,

Find,

P[Y ?1]

for

(1) p .1

(2) p .3

Does the continuity correction always yield a better approximation?

NOTE:

Compute normal probability and critical values using SAS.

Question 4:

Let y be a normal random variable with mean equal to 100 and standard deviation equal to 8. Find the following probabilities:

1. P(y ? 100)
2. P(y ? 105)
3. P(y < 110)
4. P(88 < y < 120)
5. P(100 < y < 108)

Question 5:

Using the standard normal curve area table, find the area under the normal curve between the following values:

1. z = 0.7 and z = 1.7
2. z = -1.2 and z = 0

. , ?=.2,

b. , ?=.4,

c. , ?=.7,

2. Let y be a binomial random variable with and p=.4. Find the following values:

a. P (y?4)

b. P (y>4)

c. P (y<7)

d. P (y?6)

Question 2:

Over a long period of time in a large multinational corporation, 10% of all sales trainees are rated outstanding, 75% are rated as excellent, 10% are rated as satisfactory, and 5% are considered unsatisfactory. Fin the following probabilities for a sample of 10 trainees selected at random:

a. Two are rated as outstanding.

b. Two or more are rated as outstanding.

c. Eight of the ten are rated either outstanding or excellent.

d. None of the trainees is rated as unsatisfactory.

NOTE:

Below is an example on computing Binomial Probability using SAS

/* SAS code for computation of binomial probabilities */

Options NoDate;

Title 'Binomial Probabilities';

Data A;

; * Define the number of trials n;

pi=.4; * Define the probability of success pi;

(pi,n,9); * P[X<];

(pi,n,8); * P[X<];

(pi,n,7); * P[X<];

(pi,n,6); * P[X<];

-prob7; * P[X= 8] = P[X<] - P[X<];

-prob6; * P[7<<] = P[X<] - P[X<];

-prob7; * P[X>7] = P[X>] = 1 - P[X<];

Label ; * Provide a descriptive label to the variable prob8;

Label ; * Provide a descriptive label to the variable prob_a;

Label ; * Provide a descriptive label to the variable prob_b;

Label ; * Provide a descriptive label to the variable prob_b;

Format prob9--prob_c 12.5; * Format all prob variables to allow enough room so ;

* That the label fits on one line;

Run;

Proc Print NoObs Label;

Var pi n prob8 prob_a prob_b prob_c;

Run;

Screen shot an example on computing Binomial Probability using SAS

Question 3:

Let Y be a random variable having a binomial distribution with parameters

and p, i.e.,

Y ? Binomial (,p)

Using

i. the exact binomial distribution,

ii. the normal approximation to the binomial without the continuity

correction,

iii. the normal approximation to the binomial with the continuity correction,

Find,

P[Y ?1]

for

(1) p .1

(2) p .3

Does the continuity correction always yield a better approximation?

NOTE:

Compute normal probability and critical values using SAS.

` `

Question 4:

Let y be a normal random variable with mean equal to 100 and standard deviation equal to 8. Find the following probabilities:

1. P(y ? 100)
2. P(y ? 105)
3. P(y < 110)
4. P(88 < y < 120)
5. P(100 < y < 108)

Question 5:

Using the standard normal curve area table, find the area under the normal curve between the following values:

1. z = 0.7 and z = 1.7
2. z = -1.2 and z = 0

Question 1:

Binomial Probabilities

1. Let y be a binomial random variable. Compute P(y) for each of the following situations:

a. n=10, ?=.2, y=3

b. n=4, ?=.4, y=2

c. n=16, ?=.7, y=12

2. Let y be a binomial random variable with n=8 and p=.4. Find the following values:

a. P (y?4)

b. P (y&gt;4)

c. P (y&lt;7)

d. P (y?6)

Question 2:

Over a long period of time in a large multinational corporation, 10% of all sales trainees are

rated outstanding, 75% are rated as excellent, 10% are rated as satisfactory, and 5% are

considered unsatisfactory. Fin the following probabilities for a sample of 10 trainees

selected at random:

a. Two are rated as outstanding.

b. Two or more are rated as outstanding.

c. Eight of the ten are rated either outstanding or excellent.

d. None of the trainees is rated as unsatisfactory.

NOTE:

Below is an example on computing Binomial Probability using SAS

/* SAS code for computation of binomial probabilities */

Options PageNo=1 NoDate;

Title 'Binomial Probabilities';

Data A;

n=10;

pi=.4;

prob9=probbnml(pi,n,9);

prob8=probbnml(pi,n,8);

prob7=probbnml(pi,n,7);

prob6=probbnml(pi,n,6);

prob_a=prob8-prob7;

prob_b=prob9-prob6;

prob_c=1-prob7;

Label prob8='P[X&lt;=8]';

prob8;

Label prob_a='P[X=8]';

prob_a;

Label prob_b='P[7&lt;=X&lt;=9]';

prob_b;

Label prob_c='P[X&gt;7]';

prob_b;

Format prob9--prob_c 12.5;

so ; *

*

*

*

*

*

*

*

*

* Define the number of trials n;

Define the probability of success pi;

P[X&lt;=9];

P[X&lt;=8];

P[X&lt;=7];

P[X&lt;=6];

P[X= 8] = P[X&lt;=8] - P[X&lt;=7];

P[7&lt;=X&lt;=9] = P[X&lt;=9] - P[X&lt;=6];

P[X&gt;7] = P[X&gt;=8] = 1 - P[X&lt;=7];

Provide a descriptive label to the variable * Provide a descriptive label to the variable

* Provide a descriptive label to the variable

* Provide a descriptive label to the variable

* Format all prob variables to allow enough room Run; * That the label fits on one line; Proc Print NoObs Label;

Var pi n prob8 prob_a prob_b prob_c;

Run; Screen shot an example on computing Binomial Probability using SAS Question 3:

Let Y be a random variable having a binomial distribution with parameters n=20

and p, i.e.,

Y ? Binomial (n=20,p)

Using

i. the exact binomial distribution,

ii. the normal approximation to the binomial without the continuity

correction,

iii. the normal approximation to the binomial with the continuity correction,

Find,

P[Y ?1]

for

(1) p =0.1

(2) p =0.3

Does the continuity correction always yield a better approximation?

NOTE:

Compute normal probability and critical values using SAS. Question 4:

Let y be a normal random variable with mean equal to 100 and standard deviation equal to

8. Find the following probabilities:

a.

b.

c.

d.

e. P(y ? 100)

P(y ? 105)

P(y &lt; 110)

P(88 &lt; y &lt; 120)

P(100 &lt; y &lt; 108) Question 5:

Using the standard normal curve area table, find the area under the normal curve between

the following values:

a. z = 0.7 and z = 1.7

b. z = -1.2 and z = 0

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Sep 13, 2020

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