26134 Business Statistics [email protected] Tutorial 12: REVISION THRESHOLD CONCEPT 5 (TH5): Theoretical foundation of statistical inference:

26134 Business Statistics [email protected]

Tutorial 12: REVISION

THRESHOLD CONCEPT 5 (TH5): Theoretical foundation of statistical inference: Collecting samples and drawing inference

WEEK 10

THRESHOLD CONCEPT 6 (TH6): Theoretical foundation of statistical inference: Building interval estimates and constructing hypothesis for statistical inference

WEEKS 11-12

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In statistics we usually want to statistically analyse a population but collecting data for the whole population is usually impractical, expensive and unavailable. That is why we collect samples from the population (sampling) and make inferences about the population parameters using the statistics of the sample (inferencing) with some level of accuracy (confidence level).

A population is a collection of all possible individuals, objects, or measurements of interest. A sample is a subset of the population of interest.

Sample Size N n

Statistical inference is the process of drawing conclusions about the entire population based on information in a sample by: • constructing confidence

intervals on population parameters

• or by setting up a hypothesis test on a population parameter

THRESHOLD 5: Normal Distribution• A random variable X is defined as a unique numerical value associated with

every outcome of an experiment.• If X follows a normal distribution, then it is denoted as X~N(μ,σ)• To find probabilities under the normal distribution, random variable X must be

converted to random variable Z that follows a standard normal distribution denoted as Z~N(μ=0,σ=1). We need to do this to standardise the distribution so we can find the probabilities using the tables.

• To convert random variable X to random variable Z, we calculate the z-score =(x- μ)/ σ

• Sampling distribution of the sample mean, X also follows a normal distribution by the CLT and it is denoted as X~N(μx=μ, σx=σ/√n)

• To convert random variable X to random variable Z, we calculate the z-score =(x- μ)/ (σ/ √n)

• If n/N>0.05, finite correction factor needs to be applied for the formula of the standard error, therefore

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Calculating Probabilities using normal distribution applying the complement rule and/or symmetry rule and/or interval rule

• Complement Rule P(Z>z)=1-P(Z<z)

• Symmetry Rule P(Z<-z)=P(Z>z)

• Interval Rule P(-z<Z<z)=P(Z<z)-P(Z<-z)

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THRESHOLD 5: Sampling Distribution of the Sample Mean

• Under the Central Limit Theorem (CLT), we can conclude that the sampling distribution of the sample mean is approximately normally distributed where:

• The original (population) distribution, from which the sample was selected, is normally distributed (regardless of sample size);

OR • If a sufficiently large sample size is taken, that is the sample size is

greater than or equal to 30. (regardless of the population distribution).

• Note that only ONE of these conditions need to be satisfied for this conclusion to be reached. 5

Finding Probabilities of the Mean

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THRESHOLD 6: Confidence Intervals

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Mean Mean

THRESHOLD 6: Hypothesis Testing

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We use hypothesis testing to infer conclusions about the population parameters based on analysing the statistics of the sample. Because in reality, we usually only have information about the sample.In statistics, a hypothesis is a statement about a population parameter.1. Formulate the hypothesis:

H0: population parameter = null parameterHa: population parameter ≠ null parameter (2-tailed)orHa: population parameter < null parameter (1-tailed/left tailed)orHa: population parameter > null parameter (1-tailed/right tailed)

2. Determine the level of significance α: Assumptions are if sample size is less than 30, we need to assume the distribution approaches normal. If sample size is more than 30, we need to assume the distribution approaches normal.

3. Determine the Test Statistic:

4. Determine the Critical Value: Compare test statistic with critical value.It is really helpful to draw the distribution up and shade the rejection region.

5. Make a decision rule and draw a conclusion in context of the problem.

or

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Critical Values from the z Distribution

STEP 5: Hypothesis Testing- write the decision rule and draw a

conclusion• For a left tail test (HA: μ < μ0), decision rule is: Reject H0 if ztest<-zα

• For a right tail test (HA: μ > μ0), decision rule is: Reject H0 if ztest>zα

• For a two tailed test (HA: μ ≠ μ0), decision rule is: Reject H0 if |ztest|>zα/2

• For a left tail test (HA: μ < μ0), decision rule is: Reject H0 if ttest<-tα,df=n-1

• For a right tail test (HA: μ > μ0), decision rule is: Reject H0 if ttest>tα,df=n-1

• For a two tailed test (HA: μ ≠ μ0), decision rule is: Reject H0 if |ttest|>|tα/2,df=n-1|

CONCLUSION: If the test statistic falls in the rejection region, we reject H0 and say that at 5% level of significance, there is sufficient evidence to conclude….If the test statistic fall in non-rejection region, we do not reject H0 and say that at 5% level of significance, there is not enough evidence to conclude….Make your conclusion in context of the problem.

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For a z-test statistic:

For a t-test statistic:

QUESTIONS:

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USE INTERVAL RULE

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The question is asking to choose a significance level that will have the lowest likelihood of making Type I error. Type I error is the likelihood of falsely rejecting the null hypothesis. Recall the null hypothesis from Activity 2 was, H0: μ=4000.The smaller the significance level, the lower the likelihood of making a type I error. So at the significance level of 0.1, there is a 10% chance of making a type I error, whereas at the significance level of 0.01, there is only 1% chance of making a type I error. So we select our significance level to be the lowest option given, which is 0.01.

Confidence Level