Stats midterm- Units 5-6

Random trial

A process that has multiple outcomes where the result on any particular trial is unknown

Sample space

Set of all possible outcomes. Shown with {a,b,v,c}

Event

The outcome we’re interested in. Show by E = {tail} or E = {2,3,4}

Discrete vs continuous variables

In the context of random trials can be for discrete or continuous variables

Random trials and sampling

Random trial is the act of selecting a sampling unit and taking a measurement. sample set is {more than 5 mins, less than 5 mins}… E = {more than 5 mins}

Probability of an event

proportion of times that an event would occur if a random trial was repeated a large amount of times

Probability and law of large numbers

With a smaller number of trials, there is more variation of the probabilty, but over with more trials, the probability converges on a single proportion (variation between trials becomes less significant)

Probability distributions

functions that describe probability over a range of events. Show the probability of observing an outcome within a range ov events as the area under the curve

• describe probability for entire sample space

• the area under the curve always equals one

• can be used to describe continuous and discrete random variables

Probability distributions for discrete variables

Shown as series of vertical bars, no space between them. Each event gets a separate bar, area of the bar = probability of that event. Vertical axis = probability mass

Probability distribution for continuous distribution

a single curve, the area under a part of the curve is the probability for events within a specific range. Vertical axis = probability density. If the range is 0, the probability is also 0. “Whats the probability of someone having waited exactly 5 minutes?” zero.

Calculating probability from a distribution

Area under the distribution curve for a given range

Range from a distribution

What range contains p=____? (eg. what daily minimum depth of snow do we expect 50% of the time during the winter")

Standard Normal Distribution

a normal distribution with a mean of 0, standard deviation of 1, and the x axis is the z score (how many standard deviations the value is from the mean)

• to convert your probabilities to standard normal distribution:

  • (value-mean)/standard deviation = z score

  • look at z-score distribution to find probability of eent range

If you’re trying to find a range based on a probability

1. find the z-score of the probabilitiy (x-0)/1

2. Set that equal to zscore of your sample to find the threshold number

Descriptive statistics

describe the characteristics of a specific sample for each measurement variable = you can make statements that apply to just your sample.

• each measurement variable has its own set of descriptive statistics

• descriptive statistics are any quantifyable characteristic of a sample

• they’re labeled using latin alphabet (p,m,s)

Population parameters

describe sampling population… quantifiable characteristic of a statistical population (eg. average of entire statistical population)

• population parameter = fixed value

• each measurement variable has its own set of population parameters

• descriptive statistics = not fixed (2 different samples have different mean values)

Estimation

descriptive statistics provide an estimate of the population parameter.

estimating =

Sampling distribution

probability distribution of a descriptive statistic (like mean) that would emerge if a statistical population was sampled repeatedly a large number of times

• can do it for any descriptive sample

• Shape of sample distribution is independant of statistical population as long as the sample is big enough (will be a bell even if the distribution is spiky or whatever)

• As sample size increases, variance between groups decreases (more likely to be concentrated around the mean of the statistical population)

Central limit theorum

• sampling distribution tends towards normal distribution as sample size gets larger

• the mean of the sampling distribution is the same as the mean of the statistical population

• Has standard error (standard deviation) of standard deviation of stat.pop/root sample size

• BUT assumes we know statistical population perfectly… but we have to estimate standard distribution

Standard error

Standard deviation of a sampling distribution (SE = standard deviation of statistical population/root(samplesize))

Chain of inference

• statistical population and sampling distribution are not directly observed, only the sample.

• observation of sample > infer about statistical population parameters > estimate sampling distribution

• BUT assumes we know statistical population perfectly… but we have to estimate standard distribution

Student’s T distribution

looks like a normal distribution, but with fatter tails to account for larger uncertainty

• as sample size increases, looks more like normal distribution

• basically to accomodate chain of inference because we don’t know the standard deviation of the statistical population - use our sample to estimate it but could be wrong, meaning our standard error could be wrong too.

Confidence intervals

describe range over x-axis of a sampling distribution that brackets a certain probability of where new samples may be found with ___ amount of certainty.

• estimate of how much uncertainty there is due to variation from sampling error

• can show with dots, and the lines through them = confidence interval

• start at the middle, move the lines apart until yo get the probability/% of certainty you’re looking for

• find interval (left and right t scores) that brackets that probability

• convert t scale back to raw scale (t = x-mean/standard deviation or error)