statistics - chapter 8: probability distributions

What is a probability distribution?

A list of all possible outcomes together with their probabilities

What is the probability distribution of a fair coin?

Heads = 1/2, Tails = 1/2

What is a binomial distribution?

A probability distribution with only two outcomes (success/failure)

Give examples of binomial situations

Coin flip (heads/tails); Dice roll (success = 6, failure = not 6)

What is the notation for a binomial distribution?

B(n, p)

What does n represent in B(n, p)?

Number of trials

What does p represent in B(n, p)?

Probability of success

State the 4 conditions for a binomial distribution

Fixed number of trials; Two outcomes; Independent trials; Constant probability

When can a binomial model be used?

When all 4 conditions are satisfied

What expansion is used in binomial probability?

(p + q)^n

What is the first step in finding binomial probabilities?

Identify p and q

What is the second step in finding binomial probabilities?

Expand (p + q)^n

What is the third step in finding binomial probabilities?

Find the term with p^x

What is the fourth step in finding binomial probabilities?

Substitute values and calculate

What term is used for 5 trials and 3 successes?

10p^3q^2

What is Pascal’s Triangle used for?

Finding binomial coefficients

How is Pascal’s Triangle formed?

Add the two numbers above

What calculator function gives binomial coefficients?

nCr

What does nCr represent?

Number of combinations

How do you find a range of probabilities?

Add individual probabilities

How do you find "at least one success"?

1 − P(0 successes)

What is the mean of a binomial distribution?

np

What does a normal distribution look like?

A smooth bell-shaped curve

Where is most data in a normal distribution?

Around the mean

What happens when standard deviation increases?

The curve becomes flatter

What is the notation for normal distribution?

N(μ, σ^2)

State the conditions for a normal distribution

Continuous data; Symmetrical; Mean ≈ median ≈ mode

When is a normal distribution not suitable?

When data is skewed

What percentage of data lies within 1 SD?

68%

What percentage lies within 2 SD?

95%

What percentage lies within 3 SD?

99.7% (or 99.8%)

How is area distributed in a normal distribution?

Half on each side of the mean

What percentage lies between mean and 1 SD?

34%

What percentage lies between mean and 2 SD?

47.5%

What percentage lies between mean and 3 SD?

49.9%

What is the 1 SD range for mean=30, SD=3?

27 to 33

What is the 2 SD range for mean=30, SD=3?

24 to 36

What is the 3 SD range for mean=30, SD=3?

21 to 39

What are the steps to sketch a normal distribution?

Mark 3 SDs; Draw axis; Sketch bell curve; Larger SD = flatter curve

What is the formula for number of SDs from the mean?

(value − mean) / standard deviation

What is a standardised score?

Number of SDs a value is from the mean

What is the formula for a standardised score?

(score − mean) / standard deviation

What does a positive z-score mean?

Above the mean

What does a negative z-score mean?

Below the mean

What does a z-score of 0 mean?

Value equals the mean

Why are standardised scores useful?

To compare different datasets

What is quality assurance?

Checking samples to ensure consistent quality

What is the aim of quality assurance?

Keep values close to a target

What are the steps in quality assurance?

Sampling; Calculate mean/median/range; Plot on control charts

What is a control chart?

A time series chart used for quality control

What is the target value?

The middle line

What are warning lines?

2 SD above and below the target

What happens if warning lines are crossed?

Check process and take another sample

What are action limits?

3 SD above and below the target

What happens if action limits are crossed?

Stop production and fix the issue

Where should most values lie on a control chart?

Within action limits