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SECTION 1 TEST OF A SINGLE PROPORTION

SECTION 1 TEST OF A SINGLE PROPORTION. If the data we are analyzing are nominal data, the hypothesis might be a statement about: The value of the proportion,  , of population members that have a certain characteristic (one of the categories of the nominal variable).

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SECTION 1 TEST OF A SINGLE PROPORTION

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  1. SECTION 1TEST OF A SINGLE PROPORTION • If the data we are analyzing are nominal data, the hypothesis might be a statement about: • The value of the proportion, , of population members that have a certain characteristic (one of the categories of the nominal variable).

  2. SECTION 1TEST OF A SINGLE PROPORTION • For example, the hypothesis might be a statement about the proportion of • Students who are interested in graduate school • Vaccinated patients who remain cancer free • CEOs who use computers as a major tool • People who are unemployed

  3. SECTION 1TEST OF A SINGLE PROPORTION Two-Tail Tests of Proportions • We will use precisely the same five steps that we have been following for any hypothesis test: • Step 1:Set up the null and alternative hypotheses test • Step 2:Pick the value of  and find the rejection region. • Step 3:Calculate the test statistic. • Step 4:Decide whether or not to reject the null hypothesis. • Step 5:Interpret the statistical decision in terms of the stated problem.

  4. SECTION 1TEST OF A SINGLE PROPORTION • Five-step hypothesis testing procedure is identical to the one we have been using. • The test statistic is the same as that used for a two-tail test of proportions and the rejection regions are the same as those used for one-tail tests of the mean.

  5. SECTION 1TEST OF A SINGLE PROPORTION One-Tail Test of Proportions • Step 1: Set up the null and alternative hypotheses. • There are two possible ways to set up a one-tail test of proportions. • Upper-Tail Test • Ho:  [a specific number] • HA: > [a specific number]

  6. SECTION 1TEST OF A SINGLE PROPORTION • Lower-Tail Test • Ho: [a specific number] • HA:< [a specific number] • Step 2: Select the value of “” and find the rejection region. • Step 3: Calculate the test statistic.

  7. SECTION 1TEST OF A SINGLE PROPORTION • We have seen in the previous section that the appropriate test statistic is • Steps 4 and 5 remain the same.

  8. SECTION 1COLLECTING DATA FROM TWO POPULATIONS P-level • The value of the p-level represents a decreasing index of the reliability of a result. • The higher the p-level, the less we can believe that the observed relation between variables in the sample is a reliable indicator of the relation between the respective variables in the population.

  9. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • Specifically, the p-level represents the probability of error that is involved in accepting our observed result as valid, that is, as "representative of the population." • For example, the p-level of .05 (i.e.,1/20) indicates that there is a 5% probability that the relation between the variables found in our sample is a "fluke."

  10. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • In many areas of research, the p-level of .05 is customarily treated as a "border-line acceptable" error level. • For more info: http://www.fiu.edu/~howellip/P-value.pdf

  11. Problem 17 (Page 65 or 153) • Fowle Marketing Research Inc., bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less. If a longer mean survey time is necessary, a premium rate is charged. Suppose a sample of 35 surveys shows a sample mean of 17 minutes and a sample standard deviation of 4 minutes. Is the premium rate justified?

  12. Problem 17 (Page 65 or 153) • Formulate the null and alternative hypotheses for this application. • Compute the value of the test statistic.

  13. Problem 17 (Page 65 or 153) • What is the p -value? • Using =0.01, what is your conclusion? • Since P-value= 0.0015 < = 0.01, we reject H0, that means the premium rate is justified.

  14. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • When you are comparing characteristics of two different populations, you must have a sample from each of the populations. These samples are usually selected independently of each other. • In other words, the selection of one sample should not have any effect on the selection of the second sample.

  15. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • We will label all of the parameters of one population with a subscript 1 and all the parameters of the second population with a subscript 2. • It does not matter which population you label 1 or 2. The populations and samples are shown in Figure 13.1.

  16. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • Consider the question posed about whether men or women spend more money on frozen foods. • We could label the population of males as population 1 and the population of females as population 2.

  17. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • If we do this then the parameters and statistics corresponding to the male population will be identified with a subscript 1 and those describing the female population will carry a subscript 2. • For this example, we would select a sample of men shoppers and a separate sample of women shoppers.

  18. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • We would ask all members of the sample how much money they spent on frozen foods in the past week. It is not necessary that the sample sizes be equal, but if possible it is desirable to have both sample sizes (n1, n2) greater than or equal to 30. • The reason for this stems from the fact that the Central Limit Theorem generally applies when the sample size is 30 or greater.

  19. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • Remember that we developed the Z test statistic based on the knowledge that the sample mean, (X-bar), has a normal distribution. • However, often a single sample is selected and a qualitative variable is used to identify two populations for comparison. For the food shopper example, we might select one sample of shoppers and then record the gender of the respondent as partof the data.

  20. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • This means that the data can then be divided into the two comparison-populations after the data have been collected. • If at the same time you collect data on the age of the person as "under 40" or "40 and over" then you can also compare the average frozen food expenditure for younger buyers to that of older buyers.

  21. SECTION 1COLLECTING DATA FROM TWO POPULATIONS • Clearly, spending differences that are identified by gender or age could be of great assistance in developing a marketing strategy.

  22. SECTION 1HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION MEANS • Regardless of the particular case, each hypothesis test will follow the five-step procedure • Step 1: Set up the null and alternative hypotheses. • Step 2: Pick the value of and find the rejection region. • Step 3: Calculate the test statistic and the p value. • Step 4: Decide whether or not to reject the null hypothesis. • Step 5: Interpret the statistical decision in terms of the stated problem.

  23. SECTION 1LARGE-SAMPLE TESTS OF THE DIFFERENCE IN TWO POPULATION MEANS Large-Sample Tests of Two Means with Known Standard Deviations • The basic test concerning two population means occurs when we want to know whether the two samples come from populations with equal means and we assume that the population standard deviations are known.

  24. SECTION 1LARGE-SAMPLE TESTS OF THE DIFFERENCE IN TWO POPULATION MEANS • The first step of the procedure is to construct the null and alternative hypotheses. As with tests of a single mean, there are three different ways to set up the hypothesis test. • Notice that the equals sign is always part of the null hypothesis and hypotheses are statements about the relationship between the size of the mean of population 1 and the mean of population 2.

  25. SECTION 1LARGE-SAMPLE TESTS OF THE DIFFERENCE IN TWO POPULATION MEANS • The tests do not give you information about the value of the means, only about how the value of 1 compares to the value of 2. • You can see that there are two ways to state each of the hypotheses. • For each setup shown above the first way of writing the test makes a statement about the relative value of 1 to 2.

  26. SECTION 1LARGE-SAMPLE TESTS OF THE DIFFERENCE IN TWO POPULATION MEANS • The second way of writing the same test makes a statement about the value of the difference, 1-2. • They are equivalent to each other. Look at the two-sided test. • Clearly, if 1=2 then the difference between them must be zero. It is also possible to test for differences of values other than zero.

  27. SECTION 1LARGE-SAMPLE TESTS OF THE DIFFERENCE IN TWO POPULATION MEANS Large-Sample Tests of Two Means with Unknown Standard Deviations • The large-sample test for the difference between two population means requires that both of the sample sizes be greater than 30. • Since the sample sizes are large, each individual sample standard deviation is a good estimate of the corresponding unknown population standard deviation.

  28. SECTION 1LARGE-SAMPLE TESTS OF THE DIFFERENCE IN TWO POPULATION MEANS • So, we simply use each of the sample standard deviations in the formula instead of the corresponding values of σ. • The test statistic becomes • The rejection region depends on and whether the test is one-sided or two-sided.

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