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Ch4 How to Do it: Calculate Relative Frequency Probabilities ..., Lecture notes of Statistics

Simulate 50 births, where each birth results in a boy or girl. a) Sort the results, count the number of girls. b) Based on the result in (a), calculate the ...

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Math 227 Elementary Statistics-Minitab Handout Ch4, Ch5, and Ch6
1
Ch4
Ex) #1. Consider the following data. The random variable X represent the number of days patients stayed in
the hospital. Calculate Relative Frequency Probabilities.
Number of days stayed(X) Frequency
3 15
4 32
5 56
6 19
7 5
#2. Simulate 50 births, where each birth results in a boy or girl.
a) Sort the results, count the number of girls.
b) Based on the result in (a), calculate the probability of getting a girl when a baby is born.
c) The probability obtained in (b) is likely to be different from 0.5. Does this suggest that the
computer’s random number generator is defective? Why or why not?
#3. a) Simulate rolling a single die by generating 5 integers (5 trials) between 1 and 6. Count the number
of 3s that occurred and divide that number by 5 to get the empirical probability. Based on 5 trials,
what is the probability of getting 3s?
b) Repeat (a) for 25 trials. What is the probability of getting 3s?
c) Repeat (a) for 50 trials. What is the probability of getting 3s?
d) Repeat (a) for 100 trials. What is the probability of getting 3s?
e) Repeat (a) for 500 trials. What is the probability of getting 3s?
f) In your own words, generalize these results in a restatement of the Law of Large Numbers.
How to Do it:
Calculate Relative Frequency Probabilities from the table
1. In C1 enter the values of Xand name the column X.
2. In C2 enter the frequencies and name the column f.
3. Select Calc>Calculator.
4. Type Px in the box for Store result in variable:.
5. Click in the Expression box, then double-click C2 f and click the division operator.
6. Find Sum in the function list and click Selecton.
7. Double-click C2 f . (You should see ‘f’/Sum(‘f’) in the Expression box.)
8. Click OK.
Calculate Relative Frequency Probabilities from the original data in a worksheet.
1. Click Stat>Tables>Tally Individual Variables
2. Select the column needed (i.e. C1) in the Variables:
3. For Display click Counts and Percents.
Random Data
1. Click on Calc>Random Data>Integer
2. You will generate_____ rows of data (enter the amount of rows you want)
3. Store in a column (C1) or columns (C1-C10).
4. Minimum Value: enter minimum, for instance, the minimum for of a die is 1
5. Maximum Value: for instance, the maximum number for a die is 6
pf3
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Ch

Ex) #1. Consider the following data. The random variable X represent the number of days patients stayed in

the hospital. Calculate Relative Frequency Probabilities.

Number of days stayed(X) Frequency

#2. Simulate 50 births, where each birth results in a boy or girl.

a) Sort the results, count the number of girls.

b) Based on the result in (a), calculate the probability of getting a girl when a baby is born.

c) The probability obtained in (b) is likely to be different from 0.5. Does this suggest that the

computer’s random number generator is defective? Why or why not?

#3. a) Simulate rolling a single die by generating 5 integers (5 trials) between 1 and 6. Count the number

of 3s that occurred and divide that number by 5 to get the empirical probability. Based on 5 trials,

what is the probability of getting 3s?

b) Repeat (a) for 25 trials. What is the probability of getting 3s?

c) Repeat (a) for 50 trials. What is the probability of getting 3s?

d) Repeat (a) for 100 trials. What is the probability of getting 3s?

e) Repeat (a) for 500 trials. What is the probability of getting 3s?

f) In your own words, generalize these results in a restatement of the Law of Large Numbers.

How to Do it:

Calculate Relative Frequency Probabilities from the table

1. In C1 enter the values of X and name the column X.

2. In C2 enter the frequencies and name the column f.

3. Select Calc>Calculator.

4. Type Px in the box for Store result in variable:.

5. Click in the Expression box, then double-click C2 f and click the division operator.

6. Find Sum in the function list and click Selecton.

7. Double-click C2 f. (You should see ‘f’/Sum(‘f’) in the Expression box.)

8. Click OK.

Calculate Relative Frequency Probabilities from the original data in a worksheet.

1. Click Stat>Tables>Tally Individual Variables

2. Select the column needed (i.e. C1) in the Variables:

3. For Display click Counts and Percents.

Random Data

1. Click on Calc>Random Data>Integer

2. You will generate_____ rows of data (enter the amount of rows you want)

3. Store in a column (C1) or columns (C1-C10).

4. Minimum Value: enter minimum, for instance, the minimum for of a die is 1

5. Maximum Value: for instance, the maximum number for a die is 6

Ch

Ex) #1. It is known that 5% of the population is afraid of being alone at night. If a random sample of 20

Americans is selected, what is the probability that exactly 5 of them are afraid?

#2. If 63% of all women are employed outside the home,

(a) create a binomial table for a sample of 20 women;

(b) use the binomial table to find the probability that at least 10 are

employed outside the home.

#3. For the following probability distribution table,

(a) find the mean and standard deviation;

(b) draw a graph for the distribution.

x 0 1 2 3 4

P(x) 0.12 0.20 0.31 0.25 0.

Calculating Binomial Probability

1. Select Calc>Probability Distribution>Binomial.

2. Click Probability.

3. Enter Number of trials and probability of success.

4. Click the Input constant and type in the value.

5. Click OK

Constructing a Binomial Distribution

1. Select Calc>Make Patterned Data>Simple Set of Numbers.

2. Store patterned data in C1.

3. Enter the value of 0 for the first value and enter the value of n for the last value.

4. Click OK

5. Select Calc>Probability Distributions>Binomial.

6. Choose Probability for the option and enter the value for the number of trials.

7. Enter the value for the Probability of success.

8. Check the button for Input columns then enter the column (i.e. C1).

9. Enter the desired column for Optional Storage.

10. Click OK.

Finding Exact Values of  and  for a probability Distribution

1. Enter the values of X in C1 and enter the corresponding probabilities in C2.

2. Select Calc>Calculator

3. For  , enter the expression sum (C1  C2) and store result in C3.

4. Select Calc>Calculator

5. For  , enter the expression sqrt ( sum (C12C2)-C3*2) and store result in C4.

Graph a Binomial Distribution

1. Select Graph>Scatterplot , then Simple.

2. Double-click on C2 Ps or the Y variable and C1 X for the X variable.

3. Click [Data view], then Project lines, then [OK]. Deselect any other type of display that may be selected in this list.

4. Type an appropriate title, such as Binomial Distribution n=20, p=0.

5. Press Tab to the Subtitle 1, (you may type in Your Name)

6. Optional: Click [Scales] then [Gridlines] then check the box for Y major ticks.

7. Click [OK] twice.

Answer Key Ch4-Ch6 Handout

Ch

#1. Relative Frequency Probabilities.

Number of days stayed(X) Frequency(f) Px

#2. Simulate 50 births, where each birth results in a boy or girl.

Tally for Discrete Variables: C

C1 Count Percent 0 29 58. 1 21 42. N= 50

a) 21Girls(Answers vary.) 1s for baby girls and 0s for baby boys

b) 21/50=

c) No, 0.5 is only the theoretical probability. As the number of trials increases, the

empirical probability of an event will approach the theoretical probability.

#3. a)

C2 Count Percent 2 1 20. 3 1 20. 4 1 20. 6 2 40. N= 5

Tally for Discrete Variables: C

b)

C3 Count Percent 1 6 24. 2 4 16. 3 5 20. 4 1 4. 5 5 20. 6 4 16. N= 25

c)

Tally for Discrete Variables: C

C4 Count Percent 1 10 20. 2 11 22. 3 6 12. 4 8 16. 5 6 12. 6 9 18. N= 50 d)

Tally for Discrete Variables: C

C5 Count Percent 1 22 22.

e)

f) Law of large numbers: when a probability experiment is repeated a large number of times, the relative frequency probability of

  • 2 15 15.
  • 3 11 11.
  • 4 13 13.
  • 5 23 23.
  • 6 16 16.
  • N=
  • Tally for Discrete Variables: C
    • 1 81 16. C6 Count Percent
    • 2 74 14.
    • 3 74 14.
    • 4 84 16.
    • 5 105 21.
    • 6 82 16.
  • N=
  • Tally for Discrete Variables: C (Extra for N=1000)
    • 1 160 16. C7 Count Percent
    • 2 154 15.
    • 3 167 16.
    • 4 168 16.
    • 5 177 17.
    • 6 174 17.
  • N=

Ch

1. Central Limit Theorem

a. C

b.

SM

Frequency

50 52 54 56 58 60 62

40

30

20

10

0

Mean 56. StDev 2. N 200

Histogram (with Normal Curve) of SM

Yes, it appears to be normally distributed.

c.

Descriptive Statistics: SM

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q SM 200 0 56.012 0.160 2.262 50.125 54.564 56.147 57.

Variable Maximum SM 62.

d. 56.

x

Yes, it is close to   56. yes, according to the Central Limit Theorem,

x

e. 2.

x

  No, it is not close to   12. However, it is close to

n

. According to the Central Limit

Theorem, x

n

2. Cumulative Distribution Function

Normal with mean = 0 and standard deviation = 1

x P( X <= x ) 1.39 0.

3. Inverse Cumulative Distribution Function

Normal with mean = 0 and standard deviation = 1

P( X <= x ) x 0.025 -1.