# COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman - PowerPoint PPT Presentation

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COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick. Chapter 6 : Probability. Key Terms and Formulas: Don’t Forget Notecards. Probability (p. 165) Random Sample (p. 167) Independent Random Sample (p. 167)

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COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman

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COURSE: JUST 3900

TIPS FOR APLIA

Developed By:

John Lohman

Michael Mattocks

Aubrey Urwick

Chapter 6:

Probability

### Key Terms and Formulas: Don’t Forget Notecards

• Probability (p. 165)

• Random Sample (p. 167)

• Independent Random Sample (p. 167)

• Binomial Distribution (p. 185)

• Binomial Formulas:

• Mean:

• Standard Deviation:

• z-Score:

### Random Sampling

• Question 1: A survey of students in a criminal justice class revealed that there are 17 males and 8 females. Of the 17 males, only 5 had no brothers or sisters, and 4 of the females were also the only child in the household. If a student is randomly selected from this class,

• What is the probability of obtaining a male?

• What is the probability of selecting a student who has at least one brother or sister?

• What is the probability of selecting a female who has no siblings?

### Random Sampling

• p = 17/25 = 0.68

• p = 16/25 = 0.64

• p = 4/25 = 0.16

### Random Sampling With and Without Replacement

• Question 2:A jar contains 25 red marbles and 15 blue marbles.

• If you randomly select 1 marble from the jar, what is the probability of obtaining a red marble?

• If you take a random sample of n = 3 marbles from the jar and the first two marbles are both blue, what is the probability that the third marble will be red?

• If you take a sample (without replacement) of n = 3 marbles from the jar and the first two marbles are both red, what is the probability that the third marble will be blue?

### Random Sampling With and Without Replacement

• p= 25/40 = 0.625

• p = 25/40 = 0.625

• p = 15/38 = 0.395

• Remember that random sampling requires sampling with replacement.

Here, we did not replace the first two red

marbles that were drawn.

### Probability and Frequency Distributions

• Question 3: Consider the following frequency distribution histogram for a population that consists of N = 8 scores. Suppose you take a random sample of one score from this set.

• The probability that this score is equal to 4 is p(X = 4) = ____

• The probability that this score is less than 4 is p(X < 4) = ____

• The probability that this score is greater than 4 is p(X > 4) = __

### Probability and Frequency Distributions

• p(X = 4) = 4/8 = 0.500

• p(X < 4) = 3/8 = 0.375

• p(X > 4) = 1/8 = 0.125

### Properties of the Normal Curve

• Question 4: The scores for students on Dr. Anderson’s research methods test had a mean of µ = 80 and a standard deviation of σ = 5. Use the figure on the next slide to answer the following questions.

• A score of 65 is ___ standard deviations below the mean, while a score of 95 is ___ standard deviations above the mean. This means that the percentage of students with scores between 65 and 95 is ___.

• A score of 90 is ___ standard deviations above the mean. As a result, the percentage of students with scores below 90 is ___.

• You can infer that 84.13% of students have scores above ___.

### Properties of the Normal Curve

• A score of 65 is _3_standard deviations below the mean, while a score of 95 is _3_standard deviations above the mean. This means that the percentage of students with scores between 65 and 95 is _99.74%.

between -3σ and +3σ.

2.15 + 13.59 + 34.13 +

34.13 + 13.59 + 2.15 =

99.74%

75

80

85

95

65

70

90

### Properties of the Normal Curve

• A score of 90 is _2_ standard deviations above the mean. As a result, the percentage of students with scores below 90 is 97.72%.

13.59 + 34.13 + 34.13 +

13.59 + 2.15 + 0.13 =

97.72%

Score of 90.

or

100 – 2.15 – 0.13 = 97.72%

65

75

80

85

70

90

95

### Properties of the Normal Curve

• You can infer that 84.13% of students have scores above _75_.

Start from 100 and subtract

until you reach 84.13%.

84.13 % of students

scored above a 75.

100 – 0.13 – 2.15 – 13.59 –

34.13 - 34.13 = 84.13%

65

70

80

85

75

90

95

### The Unit Normal Table

• Question 5: Use the unit normal table (p. 699) to find the proportion of a normal distribution that corresponds to each of the following sections: (Hint: Make a sketch)

• z < 0.28

• z > 0.84

• z > -1.25

• z < -1.85

• p = 0.6103

• p = 0.2005

• p = 0.8944

• p = 0.0322

z < 0.28

z > 0.84

z > -1.25

z < -1.85

### Binomial Data

• Question 6: In the game Rock-Paper-Scissors, the probability that both players will select the same response and tie is p = 1/3, and the probability that they will pick different responses is q = 2/3. If two people play 72 rounds of the game and choose there responses randomly, what is the probability that they will choose the same response (tie) more than 28 times?

### Binomial Data

• Find µ and σ.

• Find z.

• Use unit normal table.

• p(X > 28.5) = p(z > 1.13) = 0.1292.

Don’t forget real limits.

We’re looking for the probability

Of MORE than 28. Hence, we

Use the upper real limit of 28.5.

### Binomial Data

• Question 7:If you toss a balanced coin 36 times, you would expect, on the average, to get 18 heads and 18 tails. What is the probability of obtaining exactly 18 heads in 36 tosses?

### Binomial Data

• Find µ and σ.

• Find z.

• Use the unit normal table to find the proportion between z and the mean for each z-value.

• p(X = 18) = p(z = ±0.17) = 0.0675 + 0.0675 = 0.1350

Don’t forget to use real limits.

X = 18 spans the interval from

17.5 to 18.5. Therefore, we have to find the z-score for both the upper and lower real limits.

• How does one know if a question is asking for random sampling with replacement or random sampling without replacement?

• Unless the question specifically states that the sample was taken without replacement, always assume that the sample took place with replacement.

• Remember the requirements for random samples:

• Every individual in the population must have an equal chance of being selected.

• The probability of being selected must stay constant from one selection to the next if more than one individual is being selected.