Please answer the following questions in order.

CHAPTER 17 & 20 One-sample t test Note: For this set of multiple-choice questions, we assume the distributions of x and x are Normally distributed. In a real-world setting, a researcher sets the hypotheses and level of significance (a) BEFORE collecting data. So there is a time gap between setting hypotheses and obtaining sample data. However, in a homework or exam problem, typically you will be given all information right away - the researcher's belief, the sample mean and standard deviation. Just be aware that this is NOT the case in real life. Since we are using the f tables in our book, we can only obtain an estimate for a P-value (lower and upper bounds). If we use computer software, we can obtain an exact P-value. 1. Which of the following is an example of statistical inference? a. Drawing a boxplot b. Calculating a standard deviation c. Conducting a hypothesis test d. Drawing a histogram 2. In conducting a hypothesis test, a researcher chooses the level of significance (a) collecting data. Hint: You can think of a as a standard against which a researcher evaluates the data evidence. If the researcher looks at the data evidence BEFORE setting the standard, it would be considered cheating! a. before b. after3. In a statistical test of hypotheses, "the data are statistically significant" means that we have enough evidence for the researcher's belief (the alternative hypothesis). We say that the data are "statistically significant at level a "if Hint: See the box Statistical Significance on p399 (and the paragraph "Nonetheless..." before that). What does a mean? Recall in #31 in Ch 3 & 15 multiple choice, before collecting data, the researchers consider a probability smaller than 1% a "small probability". In that example, the 1% is the a. That is, a sample mean with a tail probability smaller than 1% is considered very unlikely to happen by pure chance alone if we only select one single sample - so unlikely to happen by pure chance alone that if it does happen, it will cast doubt on the validity of our assumption of H. The more "extreme" your sample mean is, the more evidence you have against the assumption. So you can think of a as a standard to determine if your evidence is extreme enough! a. a = 0.05. b. a = 5% c. the P-value is less than or equal to a. d. the P-value is larger than a. 4. Here is the interpretation of a small P-value: If our assumption of the population mean stated in the null hypothesis is indeed then it is very to observe a sample mean as extreme as or even more extreme than the one obtained in our statistical study. In other words, if we are to select one sample only, we expect such extreme sample mean to happen by pure chance alone. But when it does happen, our extreme sample mean cast doubt on the validity of our assumption about the population mean. Therefore, the smaller the P-value, the stronger the evidence a. true, unlikely, do not, does, against the null hypothesis b. true, unlikely, do, does not, for the null hypothesis c. false, likely, do not, does, against the null hypothesis d. false, likely, do, does not, for the alternative hypothesis - 2. 5. Suppose a researcher sets a to 5%. After collecting and analyzing the data, the researcher found that P-value s 0.05. Now, P-value S a means that the sample mean is very unlikely to happen just by pure chance alone if the assumption (the null hypothesis) is indeed true. How unlikely? In the long run (that is, if the researcher repeats the statistical studies many times), that the sample mean would happen by pure chance alone is at most once per samples! That is, the sample mean is considered very unlikely based on the a set by researcher BEFORE collecting the evidence! Hint: 5% =- a. 10 c. 100 b. 20 d. 2006. A P-value is the Hint: See the box "Test Statistic and P-value" on p396. The formula for the one-sample t test statistic is on p453. t = a. probability, when the null hypothesis is false, of obtaining a sample result that is at least as unlikely (or as extreme) as what is observed b. probability, when the null hypothesis is true, of obtaining a sample result that is at least as unlikely (or as extreme) as what is observed C. I test statistic (that is, P-value = / test statistic) d. Both b and c. 7. Which of the following would be strong evidence against the null hypothesis? a. a larger P-value b. a smaller P -value c. a more extreme / test statistic (more extreme means farther away from the center) d. Both b and c - 3. 8. The standard error of the sample mean is Hint: n = sample size, N = population size. See the box "Standard Error" on p452. a. b . C. d. 9. We use the standard error of the sample mean when is unknown. a. X b. 10. We use the / distribution tables instead of the standard Normal table when is unknown. a. X b. C. d 11. When engaging in weight-control (fitness/fat burning) types of exercise, a person is expected to attain about 60% of their maximum heart rate. For 20-year-olds, this rate is approximately 120 bpm. Does a 20-year-old have a lower rate on average? A researcher conducted a hypothesis test and set the level of significance a at 5%. (Assume that the maximum heart rates for 20-year-olds are Normally distributed.) What does u represent? a. Population mean heart rate of the entire US population when engaging in weight-control (fitness/fat burning) types of exercise. Population mean heart rate of college students when engaging in weight-control (fitness/fat burning) types of exercise. C. Population mean heart rate of 20-year-olds when engaging in weight-control (fitness/fat burning) types of exercise. d. Population mean heart rate of 10-year-olds when engaging in weight-control (fitness/fat burning) types of exercise.12. Refer to the weight-control problem above. What are the null and alternative hypotheses of the test? a. Ha: H = 120 H. : A 120 c. Ha : H = 120 H. : H # 120 d. Ho : X = 120 H. : X * 120 13. Refer to the weight-control problem above. The researcher collected a simple random sample of one hundred 20-year-olds, and the sample mean was found to be 107 bom with a standard deviation of 45 bpm. Based on the data, the value of the one-sample / statistic is a. 2.8888 C. -2.8888 b. 3.5891 d. -3.4012 14. Refer to the weight-control problem above. The P-value for the one-sample / test is Hint: Use the closest degrees of freedom. a. larger than 0.05. b. between 0.025 and 0.05. c. between 0.001 and 0.025. d. below 0.001. 15. Refer to the weight-control problem above. The P-value for the one-sample / test is than the level of significance (a). Therefore we the null hypothesis (Ho). Remark: Notice that the word "reject" is only used on the null hypothesis. We don't use that term on the alternative hypothesis. a. larger, DO NOT reject b. larger, reject C. smaller, DO NOT reject smaller, reject - 5 - 16. Refer to the weight-control problem above. What is your conclusion? Hint: Hypotheses are statements about the population. Consequently, the conclusion of a hypothesis test is also a statement about the population, not the sample. a. There is enough evidence to conclude that the population mean heart rate of 20-year-olds when engaging in weight-control (fitness/fat burning) types of exercise is less than 120 bpm. b. There is not enough evidence to conclude that the population mean heart rate of 20-year-olds when engaging in weight-control (fitness/fat burning) types of exercise is less than 120 bpm. c. There is enough evidence to conclude that population mean heart rate of 20-year-olds when engaging in weight-control (fitness/fat burning) types of exercise is less than 107 bpm. d. There is enough evidence to conclude that the sample mean heart rate of 20-year-olds when engaging in weight-control (fitness/fat burning) types of exercise is less than 120 bpm.17. The physician-recommended dosage of a new medication is 14 mg. Actual administered doses vary slightly from dose to dose and are Normally distributed. A representative of a medical review board wishes to see if there is any evidence that the mean administered dosage is more than recommended. The representative conducted a hypothesis test and set a at 1%. What does u represent? a. Population mean administered dosage of the new medication. b. Population mean administered dosage of all medications in US. c. Sample mean administered dosage of a new medication. d. Both a and c. 18. Refer to the new medication problem above. What are the null and alternative hypotheses of the test? a. Ho: H = 14 H. : p 14 C. Ho : H = 14 H. : p # 14 d. Ho : H = 14 H. : X > 14 - 6. 19. Refer to the new medication problem above. A sample of 16 doses at random was selected and the sample mean was found to be x = 14.12 mg with sample standard deviation s = 0.24 mg. Based on the data, the value of the one-sample r statistic is 1.39 c. 3.66 b. 2 d. 3.98 20. Refer to the new medication problem above. The P-value for the one-sample / test is Hint: Use the closest degrees of freedom. a. larger than 0.10. b. between 0.05 and 0.10. c. between 0.025 and 0.05. d. below 0.025. 21. Refer to the new medication problem above. The P-value for the one-sample / test is than the level of significance (a). Therefore we the null hypothesis (Ho). a. larger, DO NOT reject b. larger, reject smaller, DO NOT reject d. smaller, reject 22. Refer to the new medication problem above. What is the conclusion? a. There is enough evidence to conclude that the population mean administered dosage of the new medication is greater than 14 mg. b. There is not enough evidence to conclude that the population mean administered dosage of the new medication is greater than 14 mg- C. There is not enough evidence to conclude that the sample mean administered dosage of the new medication is greater than 14.12 mg. d. There is enough evidence to conclude that the sample mean administered dosage of a new medication is less than 14.12mg.In the next several questions, we investigate the role played by the level of significance (a) in hypothesis testing. In conducting a hypothesis test, a researcher needs to choose a specific a before collecting data and then compare the P-value with that specific a. For learning purpose, we will compare our evidence against different levels of significance (a) to see how the size of a matters. 28. We are conducting a left-tail hypothesis test. Assume that a = 0.05 (5%). The / statistic from a sample of 25 observations is / = -2.11. Based on this information, a. we would reject the null hypothesis b. we would NOT reject the null hypothesis 29. Refer to the question above. Assume that the / test statistic remains the same (i.e. / = -2.11). However, if a = 0.01 (1%), that is, a is smaller this time, then a. We would reject the null hypothesis. b. We would NOT reject the null hypothesis. 30. Refer to the two questions above. The smaller the a, the it is to reject the null hypothesis, and therefore the trustworthy your conclusion is. Hint: a is a probability number that researchers choose before collecting data. When we write our statistical report, we need to report clearly what a is. Smaller a implies the following: - smaller P-values are needed to reject the null hypothesis, which means - the less likely it is to reject the null hypothesis, which means - the less likely it is to conclude that the alternative hypothesis is true, which means - the higher standard for your data evidence to meet before you can conclude that the alternative hypothesis is true a. easier, less b. harder, more c. easier, more d. harder, less 31. You conduct a statistical test of hypotheses and find that your data are statistically significant at a =0.05. In other words, your P-value is equal to or smaller than 0.05. Does it imply that your data are also statistically significant at a = 0.01? Hint: For this question, we are NOT given the exact P-value. All we know is that the P-value is equal to or smaller than 0.05 (that is at most 0.05). So your P-value lies somewhere in this interval 0.05 0. 1 Now, based on this information, we ask: "Would the P-value also be smaller than 0.01?" In other words, if the P-value is smaller than 0.05, does it imply that it is also smaller than 0.01?31. You conduct a statistical test of hypotheses and find that your data are statistically significant at a =0.05. In other words, your P-value is equal to or smaller than 0.05. Does it imply that your data are also statistically significant at a = 0.01? Hint: For this question, we are NOT given the exact P-value. All we know is that the P-value is equal to or smaller than 0.05 (that is at most 0.05). So your P-value lies somewhere in this interval 0.05 0. ] Now, based on this information, we ask: "Would the P-value also be smaller than 0.01?" In other words, if the P-value is smaller than 0.05, does it imply that it is also smaller than 0.01? - 10 - To answer this question, pick a P-value between 0 and 0.05 e.g. P-value = 0.03. Then compare 0.03 with a = 0.01. Which one is smaller? This question illustrates the importance of stating what your a is in your statistical report! Practice: Let's play around with the numbers: If your data are significant at a = 4%, does it mean that the data are also significant at a = 7%? a. Yes b. No c. Cannot be determined. 32. Which of the following is true of the / test statistic of a hypothesis test? a. Its value must always be positive. b. Its value lies between 0 and 1 inclusively. c. It measures how far the sample mean in a statistical study is from the population mean in terms of standard error (=) i.e. it tells you how many standard errors the sample mean is from the population mean which is assumed to be true tentatively. d. All of the above. 33. In doing statistical inference, which of the following is NOT considered a random variable? Hint: A random variable is a variable that takes on different numerical values due to randomness. For example, x = height of a student x = average sample mean height of a group of 8 students Now, a population mean is considered a fact-its value doesn't change even though it is unknown to the researcher. a. X b. M C. d. All of the above are random variables