HYPOTHESIS TESTING

Statistical Methods Estimation Hypothesis Testing Inferential Statistics Descriptive Statistics Statistical Methods

Statistical estimation Population Random sample Parameters Statistics Every member of the population has the same chance of being selected in the sample estimation

Hypothesis Testing Population I believe the population mean age is 50 (hypothesis). Mean  X = 20 Random sample Reject hypothesis! Not close.

Reason and Intuition

What is a Hypothesis? A statistical hypotesis is simply a claim about a population that can be put to a test by drawing a random sample A belief about a population parameter Parameter is population mean, proportion, variance Must be stated before analysis I believe the mean GPA of this class is 3.5! © T/Maker Co.

Statistical inference, Role of chance. Formulate hypotheses Collect data to test hypotheses Reason and intuitionEmpirical observation Scientific knowledge

Formulate hypotheses Collect data to test hypotheses Accept hypothesisReject hypothesis C H A N C E Random error (chance) can be controlled by statistical significance or by confidence interval Systematic error Statistical Cont..

Testing of hypotheses Type I and Type II Errors  - level of significance 1-  - power of the test No study is perfect, there is always the chance for error

Errors in Making Decision 1.Type I Error Reject true null hypothesis Has serious consequences Probability of Type I Error is  (alpha)‏ Called level of significance 2.Type II Error Do not reject false null hypothesis Probability of Type II Error is  (beta)‏

 &  Have an Inverse Relationship   You can’t reduce both errors simultaneously!

Factors Affecting  1.True value of population parameter Increases when difference with hypothesized parameter decreases Significance level,  Increases when  decreases Population standard deviation,  Increases when  increases Sample size, n Increases when n decreases

p-Value Probability of obtaining a test statistic more extreme (  or  than actual sample value, given H0 is true Called observed level of significance Smallest value of  for which H 0 can be rejected 1.Used to make rejection decision If p-value  , do not reject H 0 If p-value < , reject H 0

Testing of hypotheses Definition of p-value. 95% 2.5% If our observed age value lies outside the black lines, the probability of getting a value as extreme as this if the null hypothesis is true is < 5%

The smaller the p-value, the more unlikely the null hypothesis seems an explanation for the data Interpretation for the example If results falls outside green lines, p<0.05, if it falls inside green lines, p>0.05

Rejection Region HoHo Value Critical Value  Sample Statistic Rejection Region Nonrejection Region Sampling Distribution 1 –  Level of Confidence Observed sample statistic Significant level

Rejection Cont.. HoHo Value Critical Value Critical Value 1/2   Sample Statistic Rejection Region Rejection Region Nonrejection Region Sampling Distribution 1 –  Level of Confidence Sampling Distribution Level of Confidence Observed sample statistic

Build hypothesis 0 - Null hypothesis H 0 - there is no difference a Alternative hypothesis H a - question explored by the investigator Statistical method are used to test hypotheses The null hypothesis is the basis for statistical test.

Null Hypothesis H 0 A statement about the value of a population parameter Null Hypothesis 1.What is tested 2.Has serious outcome if incorrect decision made Always has equality sign: =,  or  Designated H 0 (pronounced H-oh)‏ Specified as H 0 : µ = some numeric value – Specified with = sign even if  or  – Example, H 0 : µ = 3

Alternative Hypothesis Opposite of null hypothesis Always has inequality sign: ,  or  Designated H a Specified Ha:  , , or  some value Example, H a :  < 3 Alternative Hypothesis H 1 : A statement that is accepted if the sample data provide evidence that the null hypothesis is false

Test statistic z, t, F,  2 A value, determined from sample information, used to determine whether or not to reject the null hypothesis.

Identifying Hypotheses Steps

Example 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.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?

Step 1 State the null and the alternative hypotheses H 0 : µ = 16 H a : µ  16 Step 3 Identify the test statistic. Because we know the population standard deviation, the test statistic is z. Step 2 Select the significance level. The significance level is.05. Step 4 State the decision rule. Reject H 0 if z > 1.96 or z < or if p <.05. Step 5 Make a decision and interpret the results.

Computed z of 0.24 < Critical z of 1.96, Do not reject the null hypothesis. Step 5: Make a decision and interpret the results. We cannot conclude the mean is different from 16 ounces.

Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown Here  is unknown, so we estimate it with the sample standard deviation s. As long as the sample size n > 30, z can be approximated using

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.05. 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?

Step 1 H 0 : µ < $400 H 1 : µ > $400 Step 2 The significance level is.05. Step 3 Because the sample is large we can use the z distribution as the test statistic. Step 4 H 0 is rejected if z > 1.65 Step 5 Make a decision and interpret the results.

o Computed z of 2.42 > Critical z of 1.65, Step 5 Make a decision and interpret the results. Lisa can conclude that the mean unpaid balance is greater than $400.

The critical value of t is determined by its degrees of freedom equal to n-1. Testing for a Population Mean: Small Sample, Population Standard Deviation Unknown The test statistic is the t distribution.

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. The production hours are normally distributed. A sample of 10 randomly selected hours from last month revealed that the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. At the.05 significance level can Neary conclude that the new machine is faster?

Step 4 State the decision rule. There are 10 – 1 = 9 degrees of freedom. Step 1 State the null and alternate hypotheses. H 0 : µ < 250 H 1 : µ > 250 Step 2 Select the level of significance. It is.05. Step 3 Find a test statistic. Use the t distribution since  is not known and n < 30. The null hypothesis is rejected if t > 1.833

o Computed t of >Critical t of Step 5 Make a decision and interpret the results. The mean number of amps produced is more than 250 per hour.

The sample proportion is p and  is the population proportion. The fraction or percentage that indicates the part of the population or sample having a particular trait of interest. Proportion Test Statistic for Testing a Single Population Proportion

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.05 significance level can it be concluded that the new letter is more effective?

Step 1 State the null and the alternate hypothesis. H0: p <.15 H1: p >.15 Step 2 Select the level of significance. It is.05. Step 3 Find a test statistic. The z distribution is the test statistic. Step 4 State the decision rule. The null hypothesis is rejected if z is greater than Step 5 Make a decision and interpret the results.

Because the computed z of 2.97 > critical z of 1.65, the null hypothesis is rejected. More than 15 percent responding with a pledge. The new letter is more effective. Step 5: Make a decision and interpret the results.

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