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Chapter 12: Testing Hypotheses

Chapter 12: Testing Hypotheses. Motivating Example: UNT Stadium. Research Question : Do more than 50% of all UNT students support the new stadium? Study Options Census : Ask all 35,000+ students Is this feasible? Sample : Ask a smaller random sample

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Chapter 12: Testing Hypotheses

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  1. Chapter 12: Testing Hypotheses

  2. Motivating Example: UNT Stadium • Research Question: Do more than 50% of all UNT students support the new stadium? • Study Options • Census: Ask all 35,000+ students • Is this feasible? • Sample: Ask a smaller random sample • A random sample is likely to be representative • There is still some uncertainty because we are not studying the whole population

  3. Motivating Example: UNT Stadium • Sample: Suppose we study a random sample of 400 students • Variable and Values: Support for new stadim (yes, no) • Results: Suppose 55% of these 400 students support the new stadium • Question: Based on the results in the sample, can we say that more than 50% of allUNT studens support the new stadium? • Answer: We will use hypothesis testing to help us answer this question

  4. Goal in Hypothesis Testing • Goal: Understand something about a population • Population: All UNT students • The “Something”: Percentage in the populationwho support the new stadium • Reaching the Goal: Use information from a sample • Sample: 400 UNT students • Information: Percentage in the sample who support the new stadium

  5. Hypothesis Testing: The evaluation of hypotheses about population parameters using sample statistics Parameters and Statistics Mean: Population mean of a variable (parameter = μ) evaluated using the sample mean (statistic = ) Proportion: Population proportion of a variable (paramter = π) using the sample proportion (statistic = p) Hypothesis Testing

  6. 4 Types of Hypotheses • Type 1: Population mean (μ) of a variable for one group • Example:The average number of children per family, among all families in Texas, is different from 2.2 (μ ≠ 2.2) • Type 2: Population proportion (π) of a variable for one group • Example: The proportion of all voters in the U.S. who support Ralph Nader is less than 0.05 (π < 0.05)

  7. 4 Types of Hypotheses • Type 3: Comparing population means (μ1 and μ2) of a variable for two groups • Example: The mean income is higher among all Asian Americans than among all non-Hispanic white Americans (μ1 > μ2) • Type 4: Comparing population proportions (π1 and π2) of a variable for two groups • Example: The proportion of all adults who support legalizing marijuana is lower for women than for men in California (π1 < π2)

  8. Hypothesis Testing Assumptions • Assumption 1: Sampling Methods • One Group: Random sample from a population • Two Groups: Independent random samples from two populations • Assumption 2: Level of Measurement • Mean(s): Interval-Ratio • Proportion(s): Nominal

  9. Hypothesis Testing Assumptions • Assumption 3: Sampling Distribution is Normal • One Group: Sample size of 50 or more • Two Groups: Sample size for each group is 50 or more • Justification: Central Limit Theorem • When the sample size is 50 or more, the sampling distribution of the mean and of the proportion will be approximately normal

  10. Hypothesis Testing Assumptions: Application to Motivating Example • Assumption 1: Sampling Methods • A random sample was selected • Assumption 2: Level of Measurement • We are working with a proportion • The variable is nominal (yes, no) • Assumption 3: Sampling Distribution is Normal • The sample size is 400 • The Central Limit Theorem suggests that the sampling distribution of the proportion will be approximately normal

  11. Research Hypothesis (H1) • A statement that is the hypothesis of interest • It represents what the researcher thinks is happening in the population • The wording of the problem will indicate how it’s written

  12. Null Hypothesis (H0) • A statement of “no difference” • When working with one group, it indicates that the mean or proportion is equal to a hypothesized value • When working with two groups, it indicates that the means or proportions for the two groups are equal to each other

  13. Writing H1 and H0 • Parameters: Written in terms of population parameters • This is because we are trying to understand something about a population parameter • The parameters are carefully defined when writing hypotheses • Direction • The research hypothesis (H1) is written in terms of a specific direction • Direction is determined from the research question • See examples on the next slide • The null hypothesis (H0) is always written as being “equal to”

  14. Wording and the Direction for H1 • Wording for “Not Equal” (≠) • Different from • Not equal to • Not the same • Wording for “Less Than” (<) • Less • Smaller • Lower • Wording for “Greater Than” (>) • More • Greater • Higher

  15. Writing H1 and H0: Possible Pairings

  16. Writing H1 and H0: Application to Motivating Example • Research Question: Do more than 50% of all UNT students support the new stadium? • Parameter: Letπ represent the proportion of all UNT students who support the new stadium • Research Hypothesis: H1: π > 0.50 • Null Hypothesis: H0: π = 0.50

  17. Writing H1 and H0 Together • Typically, the hypotheses are written together with the null hypothesis listed first • In the motivating example, we would write the hypotheses in this way: H0: π = 0.50 H1: π > 0.50

  18. Testing the Null Hypothesis H0 • Idea • We do not test H1 directly • Instead, we test H0 • Procedure • We initially assume that H0 is true • We see if there is evidence to reject H0 in favor of H1 • Notes • We cannot prove whether H0 or H1 are true or false • This is because we have information from a sample

  19. Evidence to Reject H0 • We will have evidence to reject H0 if… • There is a small probability of seeing the results in the sample assumingthe null hypothesis is true or • There is a small probability that the results in the sample are due to sampling error alone • We calculate this probability from our sample

  20. Testing the Null Hypothesis H0: Application to Motivating Example • Assumption: We assume initially that the null hypothesis is true • We assume that the proportion of all UNT students who support the new stadium is equal to 0.50 • Probability: There is a probability of 0.02 that we would see a sample proportion of 0.55 when the population proportion is 0.50

  21. Testing the Null Hypothesis H0: Application to Motivating Example • Evidence: This probability is small, so we have evidence to reject the null hypothesis in favor of the research hypothesis • P-Value: This probability is called the p-value

  22. P-Value • Definition 1: The probability of seeing the results in the sample if the null hypothesis is true • Definition 2: The probability that the results in the sample are due to sampling error alone • Note: The p-value is not the probability that H0 is true • Determining the P-Value: We need to compute a z-statistic (or a t-statistic to be discussed later)

  23. Z-Statistic and P-Value • Z-Statistic: Use when testing hypotheses about proportions • Formula for One Group: • p = the sample proportion • π0 = the hypothesized value of π

  24. Getting the P-Value • Step 1: Compute the z-statistic • Step 2: Look up the z-statistic in Column A of Appendix B • Step 3: Read across to the value in Column C of Appendix B • Step 4: Use the value in Column C to determine the p-value • This depends on the tailed-ness of the test (next slide)

  25. Tailed-ness and P-Value • One-Tailed Test: When H1has either < or > • The p-value is the value given in Column C • Two-Tailed Test: When H1has ≠ • Multiply the value in Column C by 2 to get the p-value

  26. Z-Statistic and P-Value:Application to Motivating Example • Z-Statistic: p = 0.55 and π0 = 0.50 • P-Value • Look up 2.00 in Column A of Appendix B • Read across to Column C (0.0228) • The p-value is 0.0228 (one-tailed test)

  27. Alpha (α) • Question: How do we know when the p-value is considered small? • Answer: We set our threshold in advance • Alpha (α) • This is the threshold value that we set in advance • For this class, you will be given α • If it isn’t explicitly stated, assume α = 0.05

  28. Alpha (α) • Typical Values for α • 0.05 (most common) • 0.01 • 0.001 • Interpretation of α • The probability of rejecting H0 when it is true • The risk of rejecting H0 when it is actually true

  29. Alpha (α), P-Value, and Decision • Comparison: Compare the p-value with α • Decision: If our p-value is less than α, we have evidence to reject H0 in favor of H1 • Statistically Significant: When the p-value is smaller than α, we say that the z-statistic (or t-statistic) is statistically significant

  30. Alpha (α), P-Value, and Decision:Application to Motivating Example • Recall that our p-value was 0.0228 • Suppose that we selected α = 0.05 • The p-value (0.0228) is less than α (0.05) • We have evidence to reject H0 in favor of H1 • Suppose instead that we selected α = 0.01 • Thep-value (0.0228) is not less than α (0.01) • We do not have evidence to reject H0 in favor of H1

  31. Interpretation of Results • Interpretation: The final step involves translating the decision back into the wording of the research hypothesis • If p < α: We have evidence to suggest that more than 50% of allUNT students support the new stadium • If p ≥α: We do not have evidence to suggest that more than 50% of allUNT students support the new stadium

  32. Steps in Hypothesis Testing • Step 0: See if the assumptions are met • This won’t need to be done for this class • Step 0.5: Select alpha (α) • You will be given alpha (or if it’s not given, assume α = 0.05) • Step 1: State the null and research hypotheses • Step 2: Compute a t- or z-statistic • Step 3: Determine the p-value • Step 4: Decide whether there is evidence to reject the null hypothesis by comparing the p-value to α • Step 5: Interpret the results using the wording of the research hypothesis

  33. One-Sample Proportion (Z-Statistic) • Research Question: Is the proportion of all adults in the U.S. who believe that marijuana should be legalized less than 0.25? (Use α = 0.05) • Sample: Random sample of 930 adults from among all adults in the U.S. • Variable: They were asked whether they thought marijuana should be legalized • Results: 22.69% of the 930 adults in the sample believed that marijuana should be legalized

  34. One-Sample Proportion (Z-Statistic) • Step 1: State H0 and H1 • Let π represent the proportion of all adults in the U.S. who support legalizing marijuana H0: π = .25 H1: π < .25 • Step 2: Compute the z-statistic • Formula: • p = the sample proportion • π0 = the hypothesized value of π

  35. Step 2(continued) Our Example: π0 = 0.25, p = 0.2269, N = 930 Step 3: Determine the p-value Look up 1.63 in Column A of Appendix B Read over to Column C (0.0516) Our p-value is 0.0516 (one-tailed test) One-Sample Proportion (Z-Statistic)

  36. One-Sample Proportion (Z-Statistic) • Step 4: Evidence to reject H0? • Our p-value (0.0516) is not less than α (0.05) • We do not have evidence to reject H0 • Step 5: Interpret the results • We do not have evidence to suggest that the proportion of all adults in the U.S. who believe that marijuana should be legalized is less than 0.25

  37. In-Class Exercise (ICE) #1 • Axl Rose wanted to conduct a hypothesis test to find out whether more than 35% of all adults in Texas own guns. He randomly selected 770 adults in Texas, and he found that 37% of the respondents own guns.  • State the research and null hypotheses and carefully define the population parameter of interest. • Compute the z-statistic for the hypothesis test. • Decide whether a one-tailed or two-tailed test is appropriate. • Determine the p-value for the hypothesis test. • Decide whether there is evidence to reject the null hypothesis (use α = 0.05). • Interpret the results of the hypothesis test in terms of the proportion of adults in Texas who own guns.

  38. Two-Sample Proportion (Z-Statistic) • Research Question: Is there a difference between the proportion of all adult men and women in the U.S. who experience a UFO abduction? (Use α = 0.01) • Sample: Random sample of 381 men and 549 women from among all adults in the U.S. • Variable: They were asked whether they ever experienced a UFO abduction • Results: 28% of men and 19% of women in the sample experienced a UFO abduction

  39. Two-Sample Proportion (Z-Statistic) • Step 1: State H0 and H1 • Let π1 represent the proportion of all adult men in the U.S. who experienced a UFO abduction • Let π2 represent the proportion of all adult women in the U.S. who experienced a UFO abduction H0: π1 = π2 H1: π1 ≠ π2

  40. Step 2: Compute the z-statistic p1and p2 are the sample proportions for groups 1 and 2 N1 and N2 are the sample sizes for groups 1 and 2 Sp1-p2 is the estimated standard error of the difference in sample proportions Two-Sample Proportion (Z-Statistic)

  41. Step 2(continued) p1 = 0.28 and N1 = 381 p2 = 0.19 and N2 = 549 Two-Sample Proportion (Z-Statistic)

  42. Step 2(continued) Step 3: Determine the p-value Look up 3.16 in Column A of Appendix B Read over to Column C (0.0008) Our p-value is 2*0.0008 = 0.0016 (two-tailed test) Two-Sample Proportion (Z-Statistic)

  43. Two-Sample Proportion (Z-Statistic) • Step 4: Evidence to reject H0? • Our p-value (0.0016) is less than α (0.01), so we have evidence to reject H0 in favor of H1 • Step 5: Interpret the results • We have evidence to suggest that there is a difference between the proportion of all adult men and women in the U.S. who experienced a UFO abduction

  44. In-Class Exercise (ICE) #2 • Dr. Pelé wanted to conduct a hypothesis test to find out whether the percentage of all men who ever attended a live soccer match differed between the United Arab Emirates (UAE) and Oman. Dr. Pelé studied random samples of 310 men from UAE and 387 men from Oman. He found that 35% of men from UAE and 25% of men from Oman had ever attended a live soccer match. • State the research and null hypotheses and carefully define the population parameters of interest. • Compute the z-statistic for the hypothesis test. • Decide whether a one-tailed or two-tailed test is appropriate. • Determine the p-value for the hypothesis test. • Decide whether there is evidence to reject the null hypothesis (use α = 0.01). • Interpret the results of the hypothesis test in terms of the percentage of men who ever attended a live soccer match in the UAE and Oman.

  45. One-Sample Mean (T-Statistic) • Research Question: Is the average number of cans of Monster Energy drank per week among all college students in Texas greater than 12? (Use α = 0.02) • Sample: Random sample of 1496 college students from among all college students in Texas • Variable: They were asked to report the number of cans of Monster Energy they drank during the past week • Results: The average number of cans drank in the sample was 13.04 with a standard deviation of 3.074

  46. One-Sample Mean (T-Statistic) • Step 1: State H0 and H1 • Let μ represent the average number of cans of Monster Energy drank per week among all college students in Texas H0: μ = 12 H1: μ > 12

  47. One-Sample Mean (T-Statistic) • Step 2: Compute the t-statistic • Formula: • is the sample mean • S is the sample standard deviation • N is the sample size • μ0 is the hypothesized value of the population mean

  48. T-Statistic and Degrees of Freedom • T-Distribution • The t-statistic is associated with the t-distribution • The t-distribution looks much like the normal distribution, but it is generally a little flatter • Degrees of Freedom (df) • The shape of the t-distribution is governed by its degrees of freedom • It is interpreted as the amount of information used to calculate the t-statistic (WTF?!?!?) • Calculating the Degrees of Freedom • One Group: N – 1 • Two Groups: N1 + N2 – 2

  49. Getting the P-Value for a T-Statistic • Compute the t-statistic and degrees of freedom (df) • Find the df in the first column of Appendix C • When df is greater than 120, use the last row (df = ∞) • Read across the row until you find the t-statistic • The t-statistic is often between 2 values; use both of those values to determine the approximate p-value • Read up to the top rows of the table to find the p-value • Top row for a one-tailed test • Second row for a two-tailed test

  50. Example 1: Suppose df = 120 and t = 1.98 The p-value for a one-tailed test is 0.025; 0.05 for a two-tailed test Getting the P-Value for a T-Statistic

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