Z-Scores and Probability

• Swinburne University of Technology acknowledges the historical and cultural significance of Australia's Indigenous history and its importance in contemporary education
• It also acknowledges the traditional custodians of the land, the Wurundjeri people, and pays respect to ancestors and elders, recognizing the continuing sovereignty of Aboriginal and Torres Strait Islander nations.

Revision of Week 4

• Z-scores locate individual scores within a distribution
• They indicate how many standard deviations a score is above or below the mean
• The mean of standardized distributions is always 0
• The standard deviation is always 1

Probability

• The probability is the chance or likelihood of an event occurring
• Probability proportions are decimal values between 0 and 1
• Probability percentages range between 0% and 100%
• The 68-95-99.7% Rule of Thumb relates to the percentage of data within 1, 2, and 3 standard deviations of the mean in a normal distribution

Learning Outcomes

• Calculate standard error for sample means and proportions
• Understand the use of sampling distributions for means and proportions
• Understand the steps of hypothesis testing
• Use a sampling distribution for hypothesis testing
• Carry out a one-sample Z-test
• Understand Type I and Type II errors

Sampling Distribution

• Z-scores can describe where a score is located in a distribution
• Z-scores and probabilities comprise single score information
• Z-scores determine where sample statistics fall within a distribution
• Samples provide an incomplete picture of the population

Sampling Variability and Distribution

• An important question is whether the sample truly represents the population
• Multiple samples may differ slightly
• The differences between samples is known as sampling variability
• With lots of random samples, a picture of the samples can be built, this is a sampling distribution

Distribution of Sample Means

• The distribution of sample means is all possible random samples of size n obtained from a population
• The entire population is covered with the samples
• All possible random sample values are needed to calculate probability
• Each sample will differ, creating sampling error
• Sampling error is the natural discrepancy between sample statistics and population parameters
• A method is needed to decide which sample truly represents the population

Characteristics of Sample Means Distribution

• Sample means cluster around the population mean
• Random sampling lets the samples truly reflect the population
• The distribution should be normal
• A larger sample size narrows the distribution, bringing the sample mean closer to the population mean
• Calculations become cumbersome with larger populations/samples
• To avoid multiple samples, the Central Limit Theorem is used

Central Limit Theorem (CLT)

• CLT shape results in normal distribution
• Samples from a normally distributed population should be normally distributed
• Regardless of the original distribution's shape, if n ≥ 30, the sampling distribution is almost perfectly normal

CLT Central Tendency

• The mean of all sample equals the population mean: µM = µ

CLT Variability

• The standard error depends on the size of the samples
• Standard error measures the average distance between the sample mean and the population mean
• The calculation is based on the population standard deviation and the sample size
• The magnitude of the sample is governed by the 'law of large numbers'

Standard Error Calculation Example

• IQ scores for a population of university students are normally distributed, with μ = 100 and σ = 15
• Standard error for sample size of 9 is 5
• Standard error for sample size of 25 is 3
• If n=9 the sample mean of 108 is unusual
• If n=25 the sample mean of 108 is unusual

Covered Distributions

• Original population of scores for individuals
• Any single sample drawn from a population
• Distribution of sample means are all possible random samples of n obtained from a population

Probability and Z-Scores

• Standardizing the distribution of sample means allows calculation of the probability associated with any specific sample
• The z-score provides the location of a sample mean in relation to all other possible sample means
• Negative z-scores are below the mean
• Central Limit Theorem, the mean of all sample means equals the mean of the population
• μM = μ

Probabilities and Populations

• The z-score for a sample mean IQ of 104, sample of 25 from a population of university students is 1.33
• μM = μ = 100
• σM = 3

Sample Distribution

• The primary use of the distribution of sample means is to calculate the probability associated with any specific sample
• Probability is equivalent to proportion
• Proportions determine probabilities since the distribution of sample means represents the entire set of all possible sample means
• If population of women heights are normally distributed with μ = 160cm and σ = 8cm and a random sample of n = 16 women, the probability of their mean height being less than 158 cm can be determined

Sampling Distribution for Proportions

• The same underlying theory applies to categorical data
• Sampling distribution proportion equals population proportion
• The standard error depends on sample size

Standard Error

• The standard error is the standard deviation of the sampling distribution of the proportion

Central Limit Theorem

• The mean of the sample proportions equals the population proportion
• Samples if taken from a normally distributed population should be normally distributed
• If np ≥ 10, and n(1 – p) ≥ 10 the sampling distribution is approximately normal

Proportions Example

• If a drug company claims 30% of people aged 18-30 vaccinate against measles
• Other drug companies suspect this claim is false. They believe a lower percentage get vaccinated
• When a random sample is taken, 6/40, this equates to 15%

Assessing a Normal

• np ≥ 10
• n(1 – p) ≥ 10

Distribution Variable

• The standard error of the samples depends on the distribution
• σp̂ = p 𝗑 ( 1− p ) √ n.30
• n=40

Normal Sampling

• When the distribution of sample proportions is normally distributed, 95% of sample proportions lie within 1.96 standard errors of the population proportion
• Any sample proportions that lie more than 1.96 standard errors away from the centre of the sampling distribution are considered unusual or unlikely, this is the critical region
• Distribution of sample proportions for samples of size 40 is taken from a population where the population proportion is equal to 0.30

Sampling Conclusion

• If there has been NO bias in a sample, then sample wasn't drawn from those aged between 18 and 30,
• population proportion of people aged between 18 and 30 who are vaccinated is different than 0.30, or
• The company's claim is not true as less than 30% of people aged between 18 and 30 years get a measles vaccination

Hand Example

• If 11.0% of the general population are left-handed
• Need to check if np ≥ 10, and n(1 – p) ≥ 10
• The mean of the sampling distribution of the proportion equals the population proportion= 0
• The standard deviation of the sampling distribution [standard error] depends on the size of the sample [n = 235] need to calculate it
• For a sample with 8.5%, insufficient evidence to suggest the percentage of STA10003 students who are left handed is less than 11%.

Summarizing Sampling Distributions

• Sampling error and standard error states a sample will not provide a 'perfectly accurate’ picture of the population
• The standard error provides a measure of the discrepancy and measures the average distance between the sample mean and population mean
• When the sampling distribution is approximately normal:
• 95% of sample means / sample proportions will fall within 1.96 standard errors;
• 5% will be beyond 1.96 standard errors this is the critical region

Overview Shape

• If the samples are taken from a normally distributed population, they should also be normally distributed
• For sample means regardless of the original distribution's shape, with a sample size is n ≥ 30, the sampling distribution is almost 'perfectly normal'
• For sample proportions the normality check is calculation np ≥ 10, and n(1 - p) ≥ 10

Overview Central Tendency

• For sample means the mean of the sample means equals the mean of the population µM = µ
• Sample proportions the mean of the sample proportions equals the population proportion

Hypothesis Testing

• Hypothesis tests use sample findings to make an inference [conclusion] about a population
• The aim is to rule out chance [sampling error] as an explanation of results
• Despite covering any hypothesis test [we'll be covering a few ...], there are some useful steps that can be used as a guide

Steps in the Process

• State the hypothesis by asking what is happening in the population
• Use this hypothesis to predict the characteristics the sample should have, including the critical region
• Obtain a sample from the population and calculate the sample statistic
• Compare the sample data findings to the hypothesis to reach a conclusion

Stating a Hypothesis

• State the null and alternative hypotheses (research hypothesis) , considering either directional or non-directional changes
• Choose an alpha level [∝ ] to decide critical region and will occur if Null Hypothesis is true

Obtaining Samples

• Collect some data and use introduction to z-statistic [one-sample z-test]

Comparing data

• Determine if there is sufficient evidence to convince

Null hypothesis, H0

• No change, difference, or relationship exists

Alternative Hypothesis, H1

• A change, difference, or relationship exists

Alpha Value

• If the null hypothesis is correct characteristics should [eg direction of change] including the ‘critical region'
• Use alpha value of 5% [∝ =.05] and will be used to make a decision about the Null Hypothesis [H0]
• A critical region for ∝ =.05 is z = -1.96 and z = 1.96 boundaries and the critical regions can change when using different boundary tests
• a z-score shows value [in critical region] we reject H0, proving evidence to suggest that the hypothesis is correct
• a z-score NOT within critical region there needs to be no rejecting of H0
• A value [p-value] is the Probability or chance of obtaining this result

Decision Making

• If p < 0.050 results show a significant reject of Ho
• If p ≥ 0.050 then results are NOT significant and should not reject Ho

Distribution shapes

• Distributions use two-tailed tests and one-tailed tests each with its individual areas either side

Three Step Process

• Collect some data to compute the test statistic
• Using Introduction to z-statistic [one-sample z-test]
• Which indicates the z-test statistic [z-score] is a standardized value which us to compare the actual difference between the sample mean and the hypothesized population mean
• = hypothesized population mean based on the Null Hypothesis

Results from Tests

• If the z-score NOT within critical region there is no evidence to suggest that the alternate hypothesis is correct and we should not reject Ho
• Z-score shows an extreme value inside within the critical region which indicates evidence to suggest that the alternate hypothesis is correct
• We reject Ho

Real World Example

• If the average IQ of university students is different (or not different) to that of the general adult population
• There must first be rejection of H0
• A test distribution indicates our the sample mean must fall inside with standard errors above the mean when sample data findings
• This is Unlikely if the true average student IQ was 100
• This test would prove the average IQ of university students is different to the general population

Hypothesis Tests w/ P-Values

• Alternative to using critical z-values to make a decision is to use a p-value
• Hypothesis Tests use sample findings to make an inference with reference to the Null Hypothesis H0
• Where Null Hypothesis [H0] Indicates change, difference, or relationship indicating change, difference, or relationship

Errors

• Hypothesis testing is an inferential process, making errors
• Error is based using information to reach a conclusion
• Sampling can lead to misleading to about the population

Type 1

• Type I error concludes as though a does have effect or there . is a difference, it is a False Positive
• The aim is to minimize risks to obtain a non-representative sample
• To adjust the level probability must used a lower and thus demands more evidence

Type II Errors

• Where Type II is known a False Negative

A Type II error occurs when the sample mean is not in the critical region even though there is an effect / difference / relationship Usually occurs when the has a small The consequences of a Type are usually not as serious as a A Type II error suggests that the do not show the results or

Errors and Decisions

• Type I error rejecting a true Ho creates a scenario where we should not reject
• Where Type II not rejecting a false Ho which suggest where we know an error should not

Type I/II Examples

• Using the legal system as analogy: Where is a NOT which can lead to the the risk of of making a a and demands more

Alpha Levels

• The minimize the Type error. is to be as the in of a is very

P-Value Conclusions

• High testing, higher than usual is more