Lecture 3: Probability and Distribution of Sample Means

Probability (p)

• main def

  • Equation

• give example of flipping coins and example of 6-sided die

• Purpose of using probability

• def: the relative likelihood of an outcome

  • Equation: p (of A) = # of outcomes classified as A/total # of possible outcomes

• e.g.

  • Flipping coin, p (heads) = ½

  • Rolling 6 -sided die, p(any #) = 1/6

  • getting a king out of 52 cards, p (king) = 4/52 = 1/13

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• Purpose: to determine the type of sample likely to be obtained from the population

  • Inferential statistics: Often times however, the sample results are used to infer and draw conclusions about the population

Random sampling def

• what is the sample called

equal opportunity of each person being selected (descriptive statistics)

• sample called simple random samples

Independent random sampling

• main def

• what does it require for the sample to be called independent random sampling

• the selection of one person does not influence the selection of others

• requires sampling with replacement

  • e.g. ball gets put back into the pool (cannot remove the individual after sampling)

What type of data is probability (in what type of data representation)

• what does probability represent in it

• type of data that goes in a frequency distribution

• Probabilities are proportions of the distribution

  • e.g. in the image (probability of a score being 5 or higher is 2/10 = 20%)

  • If a probability is unlikely or unusual is important (results in finding treatments etc)

The normal distribution

• is the image showing a sample or population results?

• scores closer to the mean more common (scores near the mean is more probable), extreme values less common (improbable)

• image shows population

• tail: focus on the smaller parts of the distributions (often extreme values)

Z-transform

• for the z-score

  • mean = 0, SD scale = 1

• Symmetrical, mean = median = mode (normal distribution

• Link between score and SD units

• e.g. the difference between -1 and +1 = 34.13 + 34.13 = 68.26

The Unit Normal Table

• the body

• the tail

• relationship of body and tail

• the in between

• gives a proportion of cases in a part of a distribution in relationship to a particular Z score

  • To find on the table (don’t need to solve)

  • Body = all cases below z value

  • Tail - all cases after z

    • Tail + body = 1

  • Z score in relation to mean

Using the Unit Normal table

What proportion of cases corresponds to a Z value of greater than 1?

• What proportion of cases are between z = 0.34 and z = 0?

• What proportion of cases are between the mean and z = 0.21?

• What Z score corresponds to a point equal to, or greater, than 60% of all cases?

• What proportion of cases are between z = 0.34 and z = 0?

  • Look at tail

• What proportion of cases are between the mean and z = 0.21?

  • Look at D (between mean and z-score)

• What Z score corresponds to a point equal to, or greater, than 60% of all cases?

  • find body with 60% or 0.6

• Sometimes not exactly on the table: estimate to a reasonable amount

• There is a relationship between body and percentile rank (will be the same)

Binomial distributions

• number of outcomes

• equation

• What happens as n increases

• only two outcomes

• therefore p(A) + p(B) = 1 or p + q = 1, respectively

• As n increase, distribution approaches to normality

  • mean = pq, SD = sqrt(npq)

• Separate bar for discrete values or nominal type of data

Treatment and comparisons (L02)

• how to determine the treatment has an effect through z-scores

• is treatment of group different noticeably compared to the population: then effect is possible

• do z-score distributions

  • if the score is found in the tails (extremes) (e.g. Z=1.96), it is unlikely to be found through random sampling and are more likely to be a result of a treatment effect and outliers

How can outliers be defined through statistics- what do you deal with outliers

• How is it defined by z scores

• what value of SD is considered an outlier

• Three ways to deal with outliers

• Outliers are defined by very high or very low Z-score (or SD standard deviation)

• Criteria is 2 or 3 SD units to be considered as an outlier

  • Never less than 2

• What to do with outliers

  • Remove outliers (some may become outliers - where you draw the line of removing outliers)

    • Must still explain why you did it, and the rational

  • check errors

  • Outliers may be due to how you deal with the data

Issue about using population for comparing z-scores and SD

• what do we do instead, what do we use to prevent confounding variables

• impractical, confounding variables in the population

• Therefore, use control groups instead to control the confounding variables

  • Use of random selection

Sampling distribution

• multiple samples

• tracking the mean of all samples to create a frequency distribution of all those different sample

• Sampling distribution: the distribution of all possible sample means of a particular size (n) that can be drawn from the population

  • e.g. flip number is 10 for each sample (see the number of heads/tails that come up for each sample)

Central Limit Theorem

• hint: central = middle to the graph, limit = size

• as n approaches infinity (all samples are considered)

  • mean approaches mu synbol

  • SD approaches = signma/sqrt(n)

  • Approaching normality

• N doesn’t have to approach infinity, normal if

  • Data based are normal

  • Samples are very large (or large enough)

Standard Error of the Mean

• Law of large numbers

• SEM = sigma/(sqrt(n))

• law of large numbers: as n increases, and the sample mean represents the population mean

  • Or basically SEM decreases

• cannot change n size or anything in the experiment halfway through the experiment (questionable)

• The population of scores on the SAT forms a normal distribution with μ = 500 and σ = 100

• If you take a random sample of n = 16 from the population, what is the probability that the sample mean ≥ 525?

• Rephrase with proportion: Of all possible sample means (i.e. in the sampling distribution), what proportion has values greater than 525?