NSE 212 week 9: Probability (normal curve and z scores)

Descriptive statistics

used to summarize and describe data with regards to three characteristics

Types of descriptive statistics

• distribution of values

• central tendency

• variability

Inferential statistics

Based on the laws of probability and sampling distributions

• estimate population parameters from sample statistics

• provide objective measures to compare study findings to determine probability that their findings occurred by chance alone

Use of inferential statistics

• to test research hypothesis

• provide a means for drawing inferences about a population

• used to determine the probability that findings reflect actual population and is not chance alone

Probability

• basis for normal curve

• foundation for inferential statistics

• allows inferences about population from data of specific sample

• established connection between sample and populations

Range of probability

between 0% and 100%

Characteristics of normal distribution

• symmetrical

• same mean, median, mode

• asymptotic tail (events in tail are less likely to occur)

68% of scores( distribution of cases)

are between -1 & +1

• majority of scores

95% of scores (distribution of cases)

are between -2 & +2

99% of scores (distribution of cases)

are between -3 & +3

• least amount in tail

Z-score

used when you want to compare scores from two different distributions

tells you the SD away from mean the individual score is

Standard deviation → z score

Formula for z score

Z= (raw score-mean)/ standard deviation

How to calculate where a z score falls on the normal curve

draw a line of where the score -3to + 3 falls on the curve

add up distribution percentages between the mean and the z score

Sampling distributions

• sample means approximate population means

• samples such match as much of the characteristics of the mean as possible

Standard error of the mean

• a way to measure sampling error

• separate samples can be different even if drawn from same population

• measures approximate variability difference between sample and population mean

• the smaller the SEM the more accurate sample mean estimate of population mean

• increase sample size = increase accuracy of estimate

One tailed test (inferential statistics)

a test of directional hypotheses where the direction of the difference/relationship is predicted; uses only values at one end (tail) of a distribution to determine statistical significance

Two tailed test (inferential statistics)

a test of a nondirectional hypothesis where the direction of the difference/relationship is not stated; uses both ends (tails) of the sampling distribution to determine statistical significance

significance definition

study findings are likely due to some systematic influence i.e treatment or intervention and not due to chance alone

• error is always possible and it needs to be decided the amount of chance or risk they are willing to take that an error be made

Level of significance (alpha)

a number that expresses the probability that the result could have occurred purely by chance; risk of making a Type 1 error

most common used are 0.01(1%) and 0.05 (5%)

when the studies are limited use 0.05

you conduct a study and there is no statistically significant difference in observed groups what could be the reason?

• significance was sent to low (i.e 0.01) making type 2 error

• the same size was 30 ( too small)

• the power analysis was not done

What does it mean to have alpha set at 0.01?

that the chance of rejecting a true null is 1/100

that the chance of making a type 1 error is 1/100

Type 1 error

the probability of rejecting the null hypothesis when it is true(should have accepted it)

Type 2 error

the probability of accepting the null hypothesis when its false (should have rejected it)

When p is low

the null hypothesis must go

• you can also reject the null when it is equal to alpha

H_O

symbol for Null hypothesis

Degree of freedom (df)

• depends on the sample size

• different formula for each statistical test

• determines statistical significance of tests

Critical regions

• areas in the tail(s) that show extreme and very unlikely outcomes

• an area in sampling distribution representing values that are ‘ improbable’ if null is true

High probability samples are located in what part of the curve?

located in the centre of the distribution and have sample means close to the value specified in the null hypothesis

• likely to be obtained by chance

Low probability samples are located in what part of the curve?

located in the critical region (extreme tail of distribution)

• very unlikely to be obtained by chance

• 2.5% in each tail for two tailed

• 5% in the one tailed

what happens if value falls into the tail section?

we reject the null hypothesis and conclude that the treatment/intervention likely had an effect on the dependent variable and it was unlikely that the results happened by chance alone

Power analysis

used to reduce the risk of a type II error (accepting false null)

calculated out of 100% (0.80 means risk is 20%)

done prior to the research to determine sample size needed to support a significant result

• if differences between groups (effect size) will be small a large sample will be needed to detect

• if differences between groups (effect size) will be large a small sample size is needed to detect

Parametric statistical test

tests that involve assumptions, estimations of parameters, used with interval or ratio data

strong and prefered

• level of measurement for dependent ratio is interval or ratio

parametric data

the dependent variable is approximately normally distributed in the population​

• involves the estimation of at least one parameter (population characteristic) from the sample statistics​

Examples of parametric tests

t-tests, ANOVA, and Pearson’s r

Non parametric statistical tests

tests that do not involve rigourous assumptions; usually used with nominal or ordinal data

• not as powerful (sensitive) so more likely to fail in detecting a real difference​

• have less restrictive assumptions about the shape of the distribution than parametric tests

Examples of non parametric tests

The Chi-square test

Confidence interval (CI)

used and reported to further interpret study findings

reported as 95-99%

estimated range of values that provides a measure of certainty around the sample findings