Business Statistics Topic 7 Hypothesis Testing

Business Statistics:Topic 7 Learning Objectives By the end of this topic you will be able to: define the principle of statistical inference formulate & distinguish the types of hypotheses recognise different types of errors in hypothesis testing recognise decision making using ‘Z’ test, ‘t’ test and ‘p’ value. Business Statistics:Topic 7

Business Statistics:Topic 7 Decision Making Business Statistics:Topic 7

Hypothesis testing Process Population Evidence to accept our claim We believe the average mileage is 10km (hypothesis) Sample mean, is 11 , test Draw Sample Business Statistics:Topic 7

What is a Hypothesis? A Statement(s) or claim about a population parameter, () developed for the purpose of testing. Hypothesis statement(s) are made before analysis. We claim that the average mileage is 10km/l Business Statistics:Topic 7

Business Statistics:Topic 7 Null Hypothesis A Null Hypothesis is a statement that nothing unusual occurs or will occur. We begin with the assumption it is true Designated H0 Always has equality sign: = Business Statistics:Topic 7

Alternative Hypothesis Opposite of null hypothesis, H0 Alternative hypothesis is a statement that something unusual occurs or will occur: Designated H1 or HA Always has inequality sign: ¹, <, or > Business Statistics:Topic 7

Business Statistics:Topic 7 Example 1 Test that the population mean is not 5 State the question statistically: m ¹ 5 State the opposite statistically: m = 5 Must be mutually exclusive & exhaustive Select the null hypothesis: m = 5 Always has an “=“ sign Hence the alternate hypothesis is: m ¹ 5 (two tailed test) Never has an “=“ sign. Business Statistics:Topic 7

Business Statistics:Topic 7 Example 2 Test that the population mean is less than 5 State the question statistically: m < 5 State the opposite statistically: m  5 Must be mutually exclusive & exhaustive Select the null hypothesis: m  5 Always has an “=“ sign Hence the alternate hypothesis is: m < 5 (one tailed test), Never has an “=“ sign. Business Statistics:Topic 7

Business Statistics:Topic 7 One or Two- tailed Test? Two-tailed test The alternative hypothesis, H1 has a “¹ “ sign One-tailed test The alternative hypothesis, H1 has a “>” sign (right tail test) or The alternative hypothesis, H1 has a “<“ sign (left tail test) Business Statistics:Topic 7

Business Statistics:Topic 7 Type of Test : z or t ? Business Statistics:Topic 7

Business Statistics:Topic 7 Level of Significance Probability of rejecting a true null hypothesis Represented by ‘a’ (alpha) Selected by researcher at start Typical values are .01, .05, .10 Same as confidence level of (1-a) 100% For example, a = .01 means 99% confidence level Used for getting the critical value stating the decision rule Business Statistics:Topic 7

Critical Value & Rejection Region The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. based on selected  value and the type of test (z or t) and whether a one or two-tailed test is used. Business Statistics:Topic 7

Critical Value & Rejection Region H0: m = 5 H1: m ¹ 5 For a two-tailed test (H1:   5) both the upper and lower tails marked by the critical values are the rejection region for H0 Business Statistics:Topic 7

Critical Value & Rejection Region H0: m = 5 H1: m > 5 For a one-tailed (right) test (H1:  >5) the rejection region is the right-hand tail marked by a positive critical value. Business Statistics:Topic 7

Critical Value & Rejection Region H0: m = 5 H1: m < 5 For a one-tailed (left) test (H1:  < 5) the rejection region is the left-hand tail marked by a negative critical value.   Business Statistics:Topic 7

Business Statistics:Topic 7 Decision Rule If the calculated test statistic lies in the rejection region, the null hypothesis is rejected Business Statistics:Topic 7

Business Statistics:Topic 7 Assumptions The population follows a normal distribution or The sample size is sufficiently large. Business Statistics:Topic 7

Steps in Hypothesis Testing Set up the null and alternative hypothesis H0, H1 based on the research question Decide the test to perform (z or t) depending on whether  is know or unknown State the decision rule Compute the test statistic (z or t) depending on the type of test used Make decision using the rule (reject / accept H0) Business Statistics:Topic 7

Z Test of Hypothesis for  Business Statistics:Topic 7

Business Statistics:Topic 7 Test Statistic – Z test Business Statistics:Topic 7

Critical Value & Decision Rule Two-tailed test When a = .05, 95% confidence level Critical values are 1.96 (from ‘z’ table) Note that for a two-tailed test the rejection region is on both tails Business Statistics:Topic 7

Critical Value & Decision Rule One-tailed (left) test When a = .05, 95% confidence level Critical value is -1.645 (from ‘z’ table) Note that for a one-tailed test the rejection region is on only one side: left tail Business Statistics:Topic 7

Critical Value & Decision Rule One-tailed (right) test When a = .01, 99% confidence level Critical value is 2.33 (from ‘z’ table) Note that for a one-tailed test the rejection region is on only one side: right tail Business Statistics:Topic 7

Business Statistics:Topic 7 Example The quality control manager believes that the lifetime of bulbs follows a normal distribution with a mean of 200 hours and a standard deviation of 20 hours. A sample of 100 bulbs showed a sample mean of 195 hours. The manager wants to test whether the mean lifetime of all bulbs is 200 at a 1% level of significance. Business Statistics:Topic 7

Business Statistics:Topic 7 Solution Is the mean life time of bulbs 200 hrs? Step 1 H0.:  = 200 H1:   200 (two-tailed test) Step 2 Z or t test?  known : Z test Business Statistics:Topic 7

Business Statistics:Topic 7 Step 3 a is 1% = .01 each tail area is a /2 = .005 critical value is 2.58 (Use Z table) Business Statistics:Topic 7

Business Statistics:Topic 7 Step 4 Step 5 Since the test statistic value, z = -2.5 does not lie in the rejection area, we do not reject H0. We conclude that the average lifetime of batteries is 200 hours. Business Statistics:Topic 7

t-Test of Hypothesis for  Business Statistics:Topic 7

Business Statistics:Topic 7 t-Test Statistic Business Statistics:Topic 7

Business Statistics:Topic 7 Example A particular branch of a well-known bank stores enough money during the weekends to satisfy its customer’s needs. The expected average withdrawal during the weekend is $500. When they looked at a sample of the last 16 weekend transactions, they found the average withdrawal to be $540 with a standard deviation of $70. At  = .05, is there evidence that the mean withdrawal has increased during weekends? It is assumed that the population follows a normal distribution. Business Statistics:Topic 7

Business Statistics:Topic 7 Solution Is the mean withdrawal more than $500? Step 1 H0.:   500 H1:  > 500 (one-tailed (right) test) Step 2 Z or t test?  unknown : t test Business Statistics:Topic 7

Business Statistics:Topic 7 Step 3 a is 5% = .05 upper tail is .05 Degrees of freedom v = n-1 = 16-1=15 Critical value, tn-1 is 1.7531 (Use t table) Business Statistics:Topic 7

Business Statistics:Topic 7 Step 4 Step 5 Since the test statistic value, t = 2.2857 lies in the rejection area, we reject H0. We conclude that the average money spent by the customers during weekend has increased. Business Statistics:Topic 7

Business Statistics:Topic 7 p-value test for  The p value is the smallest value of the significance level at which H0 can be rejected Assuming that the null hypothesis is true, the p value is the probability of obtaining a test statistic equal to or more extreme than the result obtained from the sample If the p value is less than , the null hypothesis is rejected If the p value is greater than or equal to , the null hypothesis is not rejected Business Statistics:Topic 7

p-value for a two-tailed test Business Statistics:Topic 7

p-value for a one-tailed(right) test Business Statistics:Topic 7

p-value for a one-tailed(left) test Business Statistics:Topic 7

Business Statistics:Topic 7 Example Is the mean life time of bulbs 200 hrs? Step 1 H0.:  = 200 H1:   200 (two-tailed test) Step 2  known : Business Statistics:Topic 7

Business Statistics:Topic 7 Step 3 p-value = 2  Probability {Z  2.5} = 2  0.0062 = 0.0124 Step4 Since the p-value is less than the  = .05, we reject the null hypothesis. Note that the decision is the same as before. Business Statistics:Topic 7

Business Statistics:Topic 7 Types of Error In reality H0 Is true H0 Is false Decision Type 1 error Probability  Reject H0 No error Type 2 error Probability  Accept H0 No error Business Statistics:Topic 7

Business Statistics:Topic 7 Types of Error Type 1 error has serious consequences As  increases  decreases Business Statistics:Topic 7

Business Statistics:Topic 7 Summary In this topic you have discussed: Decision Making processes Formulating and testing hypotheses Hypothesis testing of  when  is known Hypothesis testing of  when  is unknown p-value approach in hypothesis testing Types of errors involved in testing a hypothesis Business Statistics:Topic 7