Hypothesis testing

Inferential statistics Estimation Hypothesis testing

What is Hypothesis Testing? Hypothesis testing is a procedure, based on sample evidence and probability theory, used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected.

What is a Hypothesis? A Hypothesis is a statement about the value of a population parameter developed for the purpose of testing.  Examples of hypotheses made about a population parameter are:  The mean monthly income for systems analysts is $3,625.  Twenty percent of all customers at Bovine’s Chop House return for another meal within a month.

State Hypothesis Nullhypothesis H 0 – Negation: alternate hypothesis H 1 H 0 : No significance relationship H 0 : No significance difference H 0 : Always contains „=„ symbol

Example Compare the life expectancy of men and Women State H0 and H1

Steps of Hypothesis Testing

Errors when we make decision H 0 true H 0 not true Retain H 0 Right decision (1-  ) Error Type II (  ) Reject H 0 Error Type I (  ) Right decision (1-  ) H0: my leg is healthy

Definitions Null Hypothesis H 0 : A statement about the value of a population parameter.  Alternative Hypothesis H 1 : A statement that is accepted if the sample data provide evidence that the null hypothesis is false.  Level of Significance: The probability of rejecting the null hypothesis when it is actually true.  Type I Error: Rejecting the null hypothesis when it is actually true.

Definitions Type II Error: Accepting the null hypothesis when it is actually false.  Test statistic: A value, determined from sample information, used to determine whether or not to reject the null hypothesis.  Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

A classification of tests According the number of samples One sample Two samples More than 2 samples

Usage of spss Search p-vale, sig.. If sig Reject H0 – Else Retain H0

ONE SAMPLE TESTS

We will examine Distribution of a variable Population mean Population standard. deviation A proportion in the Population

Population standard deviation ( σ ) is known Population standard deviation ( σ ) is not known Testing for the Population Mean When can we use z or t statistics? – If n>=100 – If 30<=n<100 but the distribution is not skewed to the right strongly (skewness<+1) – If n<30 and there is a normal distribution

EXAMPLE 1 The processors of Fries’ Catsup indicate on the label that the bottle contains 16 ounces of catsup. The standard deviation of the process is 0.5 ounces. A sample of 36 bottles from last hour’s production revealed a mean weight of ounces per bottle. At the 0.05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle is different from 16 ounces?

EXAMPLE 1 continued Step 1: State the null and the alternative hypotheses: H 0 :  = 16; H 1 :   16  Step 2: Identify the test statistic. Because we know the population standard deviation, the test statistic is z.  Step 3: Compute test statistics.

EXAMPLE 1 continued Step 4: State the decision rule: z =1.96 Retain H 0 if z is within (-1.96;1.96) Step 5: Make a decision Retain the null hypothesis. We cannot conclude the mean is different from 16 ounces.

EXAMPLE 2 Roder’s Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than $400. The level of significance is set at A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation to be $38. Should Lisa conclude that the population mean is greater than $400, or is it reasonable to assume that the difference of $7 ($407-$400) is due to chance?

EXAMPLE 2 continued Step 1: H 0 : µ $400 Step 2: Because standard deviation of the population is not known we can use the t distribution as the test statistic. Step 3: Compute test statisics

EXAMPLE 2 continued Step 4: Decision rule: t 0.95 (171)=1.65 Retain H 0 if t is within (-∞;1.65) Step 5: Make a decision. H 0 is rejected. Lisa can conclude that the mean unpaid balance is greater than $400.

Example 3 The current rate for producing 5 amp fuses at Neary Electric Co. is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. It is also known that the hourly production follow a normal distribution. At the 0.05 significance level can Neary conclude that the new machine is faster?

Example 3 continued Step 1: State the null and the alternate hypothesis. H 0 : µ 250 Step 2: Find a test statistic. It is the t distribution because the population standard deviation is not known and the sample size is less than 30, but normal distribution is assumed. Step 3: Compute a test statistic.

Example 3 continued Step 4: State the decision rule. t 0.95 (9)=1.833 Retain H 0 if t is within (-∞;1.833) Step 5: Make a decision. The null hypothesis is rejected. The mean number produced is more than 250 per hour.

Tests Concerning Proportion A Proportion is the fraction or percentage that indicates the part of the population or sample having a particular trait of interest. The sample proportion is denoted by p and is found by:

Test Statistic for Testing a Single Population Proportion The sample proportion is p and π is the population proportion.

EXAMPLE 4 In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution. A new solicitation letter that has been drafted is sent to a sample of 200 people and 45 responded with a contribution. At the 0.05 significance level can it be concluded that the new letter is more effective?

Example 4 continued Step 1: State the null and the alternate hypothesis. H 0 : π 0.15 Step 2: Find a test statistic. The z distribution is the test statistic. Step 3: Compute the test statistic.

Example 4 continued The null hypothesis is rejected. More than 15 percent are responding with a pledge. The new letter is more effective. Step 4: State the decision rule. z 0.95 =1.65 Retain H 0 if z is within (-∞;1.65) Step 5: Make a decision.