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Overview

Overview. 10.1 Inference for Mean Difference—Dependent Samples 10.2 Inference for Two Independent Means 10.3 Inference for Two Independent Proportions. p-Value Method. Method for carrying out hypothesis test p-Value measures how well data fits null hypothesis. p-Value.

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Overview

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  1. Overview • 10.1 Inference for Mean Difference—Dependent Samples • 10.2 Inference for Two Independent Means • 10.3 Inference for Two Independent Proportions

  2. p-Value Method • Method for carrying out hypothesis test • p-Value measures how well data fits null hypothesis.

  3. p-Value • Probability of observing a sample statistic (such as x or Zdata) at least as extreme as observed statistic assuming null hypothesis is true. • Roughly speaking, represents probability of observing sample statistic if the null hypothesis is true. • Since term “p-value” means “probability value,” always lies between 0 and 1.

  4. Rejection Rule When performing a hypothesis test using the p-value method: Reject H0 when the p-value is less than a.

  5. Assessment of the Strength of Evidence Table 9.5 Strength of evidence against the null hypothesis for various levels of p-value

  6. 10.1 Inference for Mean Difference—Dependent Samples Objectives: By the end of this section, I will be able to… • Distinguish between independent samples and dependent samples. • Perform hypothesis tests for the population mean difference for dependent samples using the p-value method and the critical value method.

  7. Independent Samples and Dependent Samples • Two samples are independent when the subjects selected for the first sample do not determine the subjects in the second sample. • Two samples are dependent when the subjects in the first sample determine the subjects in the second sample. • The data from dependent samples are called matched-pair or paired samples.

  8. Example 10.1 - Dependent or independent sampling? Indicate whether each of the following experiments uses an independent or dependent sampling method. a. A study wished to compare the differences in price between name-brand merchandise and store-brand merchandise. Name-brand and store-brand items of the same size were purchased from each of the following six categories: paper towels, shampoo, cereal, ice cream, peanut butter, and milk. b. A study wished to compare traditional acupuncture with usual clinical care for a certain type of lower-back pain. The 241 subjects suffering from persistent non-specific lower-back pain were randomly assigned to receive either traditional acupuncture or the usual clinical care. The results were measured at 12 and 24 months.

  9. Example 10.1 continued Solution • a. For a given store, each name-brand item in the first sample is associated with exactly one store-brand item of that size in the second sample. • Items in the first sample determine the items in the second sample • Example of dependent sampling • b. Randomly assigned to receive either of the two treatments • Thus, the subjects that received acupuncture did not determine those who received clinical care, and vice versa. • Example of independent sampling.

  10. Paired Sample t Test for the Population Mean: The Critical Value Method Step 1 • State the hypotheses and the rejection rule. • Use one of the hypothesis test forms from Table 10.2 below. • State clearly the meaning of μd.

  11. Paired Sample t Test for the Population Mean: The p-value Method Step 2 • Enter the columns as lists in your calculator and find the difference between the two columns. • Use the option STATS>TESTS>2:T-TEST.. • Inpt: DATA • Choose the list with the differences. • Obtain the p-value.

  12. Paired Sample t Test for the Population Mean: The Critical Value Method • Step 3 • If p < α, reject the null hypothesis. Step 4 • State the conclusion and the interpretation.

  13. 10.2 Inference for Two Independent Means

  14. Sampling Distribution of x1-x2 • Random samples drawn independently from populations with population means μ1 andμ2 and either Case 1: The two populations are normally distributed, or Case 2: The two sample sizes are large (at least 30), then the quantity

  15. Sampling Distribution of x1-x2 continued • Approximately a t distribution • Degrees of freedom equal to the smaller of n1 - 1 and n2 – 1 • x1 and s1 represent the mean and standard deviation of the sample taken from population 1, • x2 and s2 represent the mean and standard deviation of the sample taken from population 2.

  16. Standard Error of x1-x2 • Standard error of the statistic is • It measures the size of the typical error in using to measure μ1- μ2.

  17. Hypothesis Test for the Difference in Two Population Means p-Value Method Step 1 • Select the option STATS>TESTS>4: 2-SampTTEST. • Enter the given information. Step 2 • Find tdata.

  18. Hypothesis Test for the Difference in Two Population Means continued Step 3 • Find the p-value using calculator. Step 4 • State the conclusion and interpretation. • Compare the p-value with a.

  19. 10.3 Inference for Two Independent Proportions

  20. Sampling Distribution of p1 - p2 • Independent random samples from two populations • The quantity

  21. Sampling Distribution of p1 - p2 continued • Has an approximately standard normal distribution when the following conditions are satisfied: x1 ≥ 5, (n1 - x1) ≥ 5, x2 ≥ 5, (n2 - x2) ≥ 5 • p1 and n1 represent the sample proportion and sample size of the sample taken from population 1 with population proportion p1;

  22. Sampling Distribution of p1 - p2 continued • p2 and n2 represent the sample proportion and sample size of the sample taken from population 2 with population proportion p2; • q1 = 1 - p1 and q2 = 1 - p2.

  23. Standard Error of p1 - p2 • Standard error of the statistic p1 - p2 • Where q1 = 1 - p1 and q2 = 1 - p2. • The standard error measures the size of the typical error in using p1 - p2 to estimate p1 - p2.

  24. Hypothesis Test for the Difference in Two Population Proportions: p-Value Method • Two independent random samples • Taken from two populations • Population proportions p1 and p2 • Required conditions: x1 ≥ 5, (n1 - x1) ≥ 5, x2 ≥ 5, and (n2 - x2) ≥ 5.

  25. Hypothesis Test for the Difference in Two Population Proportions: p-Value Method Step 1 • State the hypotheses and the rejection rule • Use one of the forms from Table 10.19 page 576 • State the meaning of p1 and p2 • Reject H0 if the p-value is less than a.

  26. Hypothesis Test for the Difference in Two Population Proportions: p-Value Method Step 2 • Find Zdata. • Zdata follows an approximately standard normal distribution if the required conditions are satisfied.

  27. Hypothesis Test for the Difference in Two Population Proportions: p-Value Method Step 3 • Find the p-value using calculator. Step 4 • State the conclusion and interpretation.

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