26134 Business Statistics Tutorial 11: Hypothesis Testing Introduction: Key concepts in this tutorial are listed below 1. Difference between one tailed and two tailed test 2. Steps in Hypothesis testing namely: a. STEP 1: formulate the hypothesis b. STEP 2: determine alpha(α), the level of significance c. STEP 3: determine the critical value d. STEP 4: determine the standardized test statistic e. STEP 5: write the decision rule and draw a conclusion (Source: Lecture Slide 18) 3. Choosing between z test and t-test. 1

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

General: Hypothesis Testing 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: The null hypothesis, denoted H 0 is a statement or claim about a population parameter that is initially assumed to be true. No “effect” or no “difference”. Is always an equality. The null hypothesis must specify that the population parameter is equal to a single value (definition from textbook page 459). (Eg. H 0 : population parameter=hypothesised null parameter) The alternative hypothesis, denoted by H a is the competing claim. What we are trying to prove. Claim we seek evidence for. (Eg. H a : population parameter ≠ or hypothesised null parameter) 2.Determine the level of significance α: related to the level of accuracy you want to be. 3.Determine the Test Statistic: a measure of compatibility between the statement in the null hypothesis and the data obtained. 4.Determine the Critical Value: the critical value helps you identify the rejection and non-rejection region. 5.Make a decision rule and draw a conclusion: Compare the value of the test statistic with the critical value and make your decision on whether you reject or do not reject the H 0. If the test statistic falls in the rejection region we reject H 0 and conclude that we have enough evidence to prove the alternative hypothesis is true at the α% level of significance. If the test statistic fall in non-rejection region, we do not reject H 0 and conclude that we do not have enough evidence to prove the alternative hypothesis is true at the α % level of significance. Make your conclusion in context of the problem.

STEP 1: Hypothesis Testing- formulate the null and alternative hypothesis H 0 : population parameter = null parameter H a : population parameter ≠ null parameter (2-tailed) or H a : population parameter < null parameter (1-tailed) (left-tailed) or H a : population parameter > null parameter (1-tailed) (right-tailed) 4

STEP 2: Hypothesis Testing- determine alpha α (level of significance) 5 Confidence Level

STEP 3: Hypothesis Testing: Determine the standardized test statistic 6 The test statistic is a measure of compatibility between the statement in the null hypothesis and the data obtained. If population standard deviation (sigma) is given then we find the z-test statistic. If population standard deviation is not given, we use the sample standard deviation and find the t-test statistic where degrees of freedom is n-1.

STEP 4:Hypothesis Testing: Determine the critical value The critical value helps you identify the rejection and non-rejection region. Note: If we use the t-test statistic, we can find the critical value on the t distribution table. 7 Confidence Level

STEP 5: Hypothesis Testing- write the decision rule and draw a conclusion For a left tail test (H A : μ < μ 0 ), decision rule is: Reject H 0 if z test <-z α For a right tail test (H A : μ > μ 0 ), decision rule is: Reject H 0 if z test >z α For a two tailed test (H A : μ ≠ μ 0 ), decision rule is: Reject H 0 if |z test |>z α/2 For a left tail test (H A : μ < μ 0 ), decision rule is: Reject H 0 if t test <-t α,df=n-1 For a right tail test (H A : μ > μ 0 ), decision rule is: Reject H 0 if t test >t α,df=n-1 For a two tailed test (H A : μ ≠ μ 0 ), decision rule is: Reject H 0 if |t test |>|t α/2,df=n-1 | CONCLUSION: If the test statistic falls in the rejection region, we reject H 0 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 H 0 and say that at 5% level of significance, there is not enough evidence to conclude…. Make your conclusion in context of the problem. 8 For a z-test statistic: For a t-test statistic:

9 μ=average life of a LED lamp H 0 : μ=3000 H a : μ>3000 (right-tailed test) 3) Because the sample size is 20, which is lower than 30, we need to assume the distribution is approaching to a normal distribution. 4) Critical Value Z α =1.645

10 μ=average life of a battery H 0 : μ=4000 H a : μ>4000 (right-tailed test) Because the sample size equals to 12, which is less than 30, we need to assume the distribution approaches normal. t critical = t α=0.05,df=n-1=10 = Because the t-stat = < = t-crit, we do not reject H 0. Therefore we conclude that we do not have enough evidence to prove that the average life of the battery exceeds 4000 hours at the 5% level of significance.

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12 H 0 : μ=4000 H a : μ>4000 (right-tailed test) Because the sample size equals to 500, which is more than 30, CLT (Central Limit Theorem) applies. Because the t-stat = < = t-crit, we can reject H 0. Therefore we conclude that we have enough evidence to prove that the average life of the battery exceeds 4000 hours at the 5% level of significance. μ=average life of a battery

13 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 Confidence Level

REVISION 14

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(μ x =μ, σ x =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 15

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) Interval Rule P(-z<Z<z)=P(Z<z)-P(Z<-z) 16

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. 17

Finding Probabilities of the Mean 18

THRESHOLD 6: Confidence Intervals 19 Mean

THRESHOLD 6: Hypothesis Testing 20 1.Formulate the hypothesis: H 0 : population parameter = null parameter H a : population parameter ≠ null parameter (2-tailed) or H a : 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