Deliverable 4 - Hypothesis Tests

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Module 04 – Hypothesis Testing

Class Objectives:

· Define Hypothesis Testing

· Identify the null and alternative hypotheses

· Find the test statistic with and without knowing

· Make a decision using the p-value method

· Make a decision using the Critical Value Method

Module 04 - Part 1

· A ________________________________________________ is a statistical test that is used to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population.

A hypothesis test examines two opposing hypotheses about a population: the null hypothesis and the alternative hypothesis.

· The _________________________________________________ is the statement being tested.

· Usually the null hypothesis is a statement of "no effect" or "no difference".

· Denoted by __________.

· The __________________________________________________ is the statement you want to be able to conclude is true.

· Denoted by ___________ or ____________.

· The alternative hypothesis could be _________, ___________, or __________.

Example. It’s an accepted fact that ethanol boils at 173.1°F; you have a theory that ethanol actually has a different boiling point, of over 174°F. Give the null and alternative hypothesis.

Example. A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage. Give the null and alternative hypothesis.

Hypothesis testing steps

· For our purposes, stick with a 95% confidence ().

7. Make a decision based on the method used.

Calculating the Test Statistic

If we know know we use:

If is unknown, we use :

Note: In most cases, we will be using the second formula to calculate the test statistic. Also, is found in the null hypothesis.

Example 1.

The author obtained times of sleep for randomly selected adult subjects included in the National Health and Nutrition Examination Study, and those times (hours) are listed below. Here are the unrounded statistics for this sample: n = 12, = 6.83333333 hours, s = 1.99240984 hours. A common recommendation is that adults should sleep between 7 hours and 9 hours each night. Test the claim that the mean amount of sleep for adults is less than 7 hours using a 0.05 significance level.

1-3. Give the Null and Alternative Hypothesis.

4. Give the significance level.

5. Calculate the test statistic. (Is it z or t?)

Hypothesis Testing Methods

To test a hypothesis, we have two methods we can use:

Note: Both methods will result in the same conclusion: whether or not to reject the null hypothesis .

The main difference is the following:

· The critical value finds the z or t value where the rejection region begins, and then compares the test statistic to that number.

· The P-value finds the probability or the test statistic.

Critical Value Method

The Critical Value Method finds the z or t value of where the rejection region begins.

· The ____________________________________________________ (or _______________________________________) is the area corresponding to all value of the test statistic that causes us to reject the null hypothesis.

With the critical value method of testing hypotheses, we make a decision by comparing the test statistic to the critical value(s).

Screen Clipping

How to Find the Critical Value

1. Identify if it is left-tailed, right-tailed, or two tailed using the alternative hypothesis.

2. Calculate the C.V. as follows

a. If it is left-tailed, use t.inv(, df)

b. If it is right-tailed, use t.inv(, df)

c. If it is two-tailed, use t.inv.2t(, df)

Making a Decision

1. Identify if it is left-tailed, right-tailed, or two tailed using the alternative hypothesis.

2. Make a decision as follows

a. Left-tailed

I. If test statistic < critical value reject H0 and evidence supports claim

II. If test statistic > critical value do not reject H0 and evidence does not support claim

b. Right-tailed

I. If test statistic < critical value do not reject H0 and evidence does not support claim

II. If test statistic > critical value reject H0 and evidence supports claim

c. Two-tailed

I. If test statistic < negative critical value or if test statistic > positive critical value reject H0 and evidence does not support claim

II. If test statistic is between the two critical values do not reject H0 and evidence supports claim

Example 1 Cont’d

1-3. Give the Null and Alternative Hypothesis.

4. Give the significance level.

5. Calculate the test statistic. (t)

6. Find Values (Critical Value Method)

7. Make a Decision(Critical Value Method)

8. Restate in nontechnical terms.

P-Value Method

· In a hypothesis test, the ___________________________ is the probability of getting a value of the test statistic that is at least as extreme as the test statistic obtained from the sample data, assuming that the null hypothesis is true.

How to find the P-Value

1. Identify if it is left-tailed, right-tailed, or two tailed using the alternative hypothesis.

2. Calculate the p-value as follows

a) If it is left-tailed, use t.dist(t, df, true)

b) If it is right-tailed, use t.dist(t, df, true)

c) If it is two-tailed,

I. If the test statistic is positive, use the right-tailed method.

II. If the test statistic is negative, use the left-tailed method.

Making a Decision Using the P-Value

For the p-value, making a decision does not matter if you are left or right tailed.

A. If p-value , reject H0 and evidence supports claim.

B. If p-value , do not reject H0 and evidence does not support claim

Example 1 Cont’d

1-3. Give the Null and Alternative Hypothesis.

4. Give the significance level.

5. Calculate the test statistic. (t)

6. Find Values (P-Value Method)

7. Make a Decision(P-Value Method)

8. Restate in nontechnical terms.

Module 04 – Part 2

In a hypothesis test, we are testing a claim be applied to an entire population.

You can test a claim about a _______________________________________, a ___________________, a ___________________________________________________, or a ______________________________.

Hypothesis Testing: Steps 1, 2, and 3

1. Identify the claim to be tested.

2. Give symbolic form.

3. Identify the Null and Alternative Hypothesis.

4. Select significane level.

· ____________ (“alpha”) represents the significance level.

· In practice, we either take 5% or 1% level of significance.

· This translates to 95% confidence and 99% confidence respectively.

· Example: __________________________________________________________________________________________________________________________________________________________________________________

· Caution: When thinking about the significance level, make sure you take into consideration if it’s two-tailed, right-tailed, or left-tailed.

5. Identify the test statistic.

· Recall:

· Use the _______________________ if you know (“sigma, population standard deviation). This is unusual.

· When we have a sample and we DON’T know , use the _____________________:

· Caution: Make sure you use parentheses in Excel!

· In Excel: _______________________________________

6. To find your answer to whether you should reject the null hypothesis or not, you have two avenues you can take.

Screen Clipping

Critical Value Method

Pre-work: Determine the type of test: two-tailed, right-tailed, or left-tailed.

1. Find the ________________________________ depending on the type of test:

· If it is one-tailed (left or right), use the ______________________________ function.

· If it is two-tailed , use the _________________________________ function.

2. Draw a graph with the critical value determining the rejection region.

3. Make a decision to either reject the null hypothesis or not.

· ___________________________ if the test statistic falls into the rejection region.

· _____________________________________ if the test statistic does NOT fall into the rejection region.

P-Value Method

1. Use the ______________________ function to find the probability of the test statistic (found in step 5).

2. Make a decision to either reject the null hypothesis or not.

· _____________________________ if p-value

· ______________________________________ if p-value

8. Restate Decision in nontechnical terms.

· If you reject the null hypothesis, you should conclude: _________________________________________________________________________________________________________________________________________________________________________________________________________.

· If you fail to reject the null hypothesis, you should conclude:

_________________________________________________________________________________________________________________________________________________________________________________________________________.

· Note: when you are failing to reject the null hypothesis, this does NOT mean that we are accepting it. We have NOT proven the null hypothesis so we cannot accept it!

Example. A biologist was interested in determining whether sunflower seedlings treated with an extract from Vinca minor roots resulted in a lower average height of sunflower seedlings than the standard height of 15.7 cm. The biologist treated a random sample of n = 33 seedlings with the extract and found the sample mean to be 13.664 and the sample standard deviation to be 2.544.

Example. The administrator at your local hospital states that on weekends the average wait time for emergency room visits is 10 minutes. Based on discussions you have had with friends who have complained on how long they waited to be seen in the ER over a weekend, you dispute the administrator's claim. You decide to test you hypothesis. Over the course of a few weekends you record the wait time for 40 randomly selected patients. The average wait time for these 40 patients is 11 minutes with a standard deviation of 3 minutes. Do you have enough evidence to support your hypothesis that the average ER wait time exceeds 10 minutes? You opt to conduct the test at a 5% level of significance.

Example. A clinical trial was conducted to test the effectiveness of the drug zopiclone for treating insomnia in older subjects. Before treatment with zopiclone, 16 subjects had a mean wake time of 102.8 min. After treatment with zopiclone, the 16 subjects had a mean wake time of 98.9 min and a standard deviation of 42.3 min. Assume that the 16 sample values appear to be from a normally distributed population, and test the claim that after treatment with zopiclone, subjects have a mean wake time of less than 102.8 min.

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