Inference for the Mean and Proportion

Inference for the Mean and ProportionSections 8.1 & 8.2

Cathy Poliak, [email protected] in Fleming 11c

Department of MathematicsUniversity of Houston

Lecture 23 - 2311

Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Sections 8.1 & 8.2 Lecture 23 - 2311 1 / 25

Outline

1 Hypothesis Tests for Mean

2 Matched Pairs Test

3 Hypothesis for Proportions


Popper Set Up

Fill in all of the proper bubbles.

Make sure your ID number is correct.

Make sure the filled in circles are very dark.

This is popper number 19.


Steps of a Significance Test

When performing a significance test, we follow these steps:1. Check assumptions.

2. State the null and alternative hypothesis.

3. Graph the rejection region, labeling the critical values.

4. Calculate the test statistic.

5. Find the p-value. If this answer is less than the significance level,α, we can reject the null hypothesis in favor of the alternativehypothesis.

6. Give your conclusion using the context of the problem. Whenstating the conclusion give results with a confidence of(1− α)(100)%.


Interpreting the Weight of Evidence against the NullHypothesis

If we are not given α we can interpret the results as follows:

If the p-value for testing H0 is less than:I 0.1, we have some evidence that H0 is false.

I 0.05, we have strong evidence that H0 is false.

I 0.01, we have very strong evidence that H0 is false.

I 0.001, we have extremely strong evidence that H0 is false.

If the p-value is greater than 0.1, we say that there is no evidencethat H0 is false.


Assumptions of the Tests

For a z-test:1. An SRS of size n from the population.2. Known population standard deviation, σ.3. Either a Normal population or a large sample (n ≥ 30).

For a t-test:1. An SRS of size n form the population.2. Unknown population standard deviation.3. Either a Normal population or a large sample (n ≥ 30).


Popper #19 Questions

A can of Pepsi is supposed to have a mean volume of 12 ounces. Bothoverfilling and under-filling are undesirable. If either occurs, the machine thatfills the cans has to be readjusted.

1. The bottling company wants to set up a hypothesis test so that themachine has to be readjusted if the null hypothesis is rejected. Set upthe null and alternative hypothesis for this test.

a) H0 : µ = 12 and Ha : µ 6= 12 c) H0 : x̄ = 12 and Ha : x̄ 6= 12b) H0 : µ = 12 and Ha : µ < 12 d) H0 : µ = 12 and Ha : µ > 12

2. Once completing the test, they readjusted the machine but they did nothave to. This is an example of:

a) Type I error c) Type III errorb) Type II error d) correct decision


Example

Mr. Murphy is an avid golfer. Suppose he has been using the samegolf clubs for quite some time. Based on this experience, he knowsthat his average distance when hitting a ball with his current driver (thelongest-hitting club) under ideal conditions is 200 yards with astandard deviation of 9. After some preliminary swings with a newdriver, he obtained the following sample of driving distances:

205 198 220 210 194 201 213 191 211 203

He feels that the new club does a better job. Do you agree?



Mean Textbook Costs

An association of college bookstores reported that the average amountof money spent by students on textbooks for the Fall 2010 semesterwas $325.16. A random sample of 75 students at the local campus ofthe state university indicated an average bill for textbooks for thesemester in question to be $312.34 with a standard deviation of$76.42. Do these data provide significant evidence that the actualaverage bill is different from the $325.16 reported? Test at the 1%significance level.



Mean Amount of Coffee Dispensed

A coffee machine dispenses coffee into paper cups. Here are theamounts measured in a random sample of 20 cups.

9.9, 9.7, 10.0, 10.1, 9.9, 9.6, 9.8, 9.8, 10.0, 9.5,9.7, 10.1, 9.9, 9.6, 10.2, 9.8, 10.0, 9.9, 9.5, 9.9

The machine is supposed to dispense a mean of 10 ounces. Is theresignificant evidence to conclude that the mean is 10 ounces?



Popper#19 Questions

3. The average monthly rent for one-bedroom apartments inChattanooga has been $700. Because of the downturn in the realestate market, it is believed that there has been a decrease in theaverage rental. The correct hypotheses to be tested are:

a) H0 : µ = 700, Ha : µ < 700 c) H0 : µ = 700, Ha : µ 6= 700b) H0 : x̄ = 700, Ha : x̄ < 700 d) H0 : x̄ = 700, Ha : x̄ 6= 700

4. In a two-tailed hypothesis test situation HA : µ 6= µ0, the teststatistic is determined to be t = -2.423. The sample size has beenn = 41. The p-value for this test is:

a) -0.01 b) +0.01 c) -0.02 d) +0.02


Popper#19 Questions

5. Which one of the following statements is false?a) If the p-value is 0.0004, there seems to be significance and the null

hypothesis does not seem to be highly likely. With the level ofsignificance,α= 0.01.

b) The alternative hypothesis proposes what we should conclude if wefind the null hypothesis to be unlikely.

c) A test statistic is used to measure the difference between theobserved and what is expected under the null hypothesis.

d) If the p-value is 0.0004, the null hypothesis seems highly likely. Withthe level of significance, α= 0.01.


Matched Pairs

Matched pairs is a special test when we are comparingcorresponding values in data.

This test is used only when our data samples are DEPENDENTupon one another (like before and after results).

Matched pairs t – test assumptions:1. Each sample is an SRS of size n from the same population.2. The test is conducted on paired data (the samples are NOT

independent).3. Unknown population standard deviation.4. Either a Normal population or large samples (n ≥ 30 ).

Hypotheses - H0 : µd = 0 and Ha : µd 6= 0 or µd < 0 or µd > 0.Where µd is the mean of the differences.


Conduct the test

The following is the data from the six neighborhood, construct ahypothesis test to answer the question: Does this indicate that thenumber of reported crimes have dropped?

Neighborhood 1 2 3 4 5 6Before 18 35 44 28 22 37After 21 23 30 19 24 29



Inference for a Population Proportion

For these inferences, p0 represents the given populationproportion and the hypothesis will be

I H0 : p = p0I Ha : p 6= p0 or p < p0 or p > p0

Conditions:1. The sample must be an SRS from the population of interest.2. The population must be at least 10 times the size of the sample.3. The number of successes and the number of failures must each be

at least 10 (both np̂ ≥ 10 and n(1− p̂) ≥ 10).

Recall, the statistic used for proportions is: p̂ = # of successes# of observations = x

n .

For tests involving proportions that meet the above conditions, wewill use the z-test statistic:

z =p̂ − p0√p0(1−p0)

n


Example

A new shampoo is being test-marketed. A large number of 16-ouncebottles were mailed out at random to potential customers in the hopethat the customers will return an enclosed questionnaire. Out of the1,000 returned questionnaires, 575 indicated that they like theshampoo and will consider buying it when it becomes available on themarket. Perform a hypothesis test to determine if the proportion ofpotential customers is more than 50%.



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Inference for the Mean and Proportion