Chapter 06.01 Exercise Question 07 6.1.7 The following histogram is the distribution of the amount of time, in minutes, that some statistics students said they watched television the previous day. Using this graph to estimate the mean and median age for the amount of television watching time, we can conclude: You can't tell which is larger without the actual data. The mean is larger than the median. The mean and the median have the same value. The mean is smaller than the median. Chapter 06.01 Exercise Question 10 6.1.10 A sample of heights (in inches) of 20 female statistics students is as follows: 72, 65, 61, 71, 66, 62, 68, 70, 66, 62, 72, 66, 77, 63, 65, 67, 68, 59, 73, 70 Find the median height. Round your answer to 1 decimal place, e.g. 0.5. the absolute tolerance is +/-0 Find the first and third quartiles. Round your answers to 1 decimal place, e.g. 0.5. Q1= , Q3= Find the inter-quartile range. Round your answer to 1 decimal places, e.g. 0.5. IQR= the absolute tolerance is +/-0.1 Where on the box plot would you find the inter-quartile range? The distance between the points The length of the box The height of the box The point at the far left Chapter 06.01 Exercise Question 14 6.1.14 The dotplots show the average monthly temperatures (degrees F) for San Francisco, CA, and Raleigh, NC. Determine the median temperature for each city. What do these numbers tell us about the two data sets? San Francisco: 57, Raleigh: 59.5. With regards to median average monthly temperature, the two cities are quite similar. San Francisco: 56, Raleigh: 58.5. With regards to median average monthly temperature, the two cities are quite different. San Francisco: 57, Raleigh: 59.5. With regards to median average monthly temperature, the two cities are quite different. San Francisco: 56, Raleigh: 58.5. With regards to median average monthly temperature, the two cities are quite similar. Determine the inter-quartile range for each city. What do these numbers tell us about the two data sets? Raleigh: IQR = 26; San Francisco: IQR = 10. With regards to IQR, the variability in monthly average temperature in Raleigh is much larger than in San Francisco. Raleigh: IQR = 29; San Francisco: IQR = 12. With regards to IQR, the variability in monthly average temperature in Raleigh is much larger than in San Francisco. Raleigh: IQR = 26; San Francisco: IQR = 10. With regards to IQR, the variability in monthly average temperature in Raleigh is much smaller than in San Francisco. Raleigh: IQR = 29; San Francisco: IQR = 11. With regards to IQR, the variability in monthly average temperature in Raleigh is much smaller than in San Francisco. Chapter 06.01 Exercise Question 27 6.1.27 Statistics students found the sugar content of breakfast cereals, in grams per serving, for those placed on the high shelves of the store versus the low shelves. The data file CerealSugar.txt has these results. Copy and paste this data set into the Descriptive Statistics applet. If you want to compare the sugar content of cereals on high versus low shelves, what would be the explanatory variable and what would be the response? Explanatory: sugar content, response: shelf level Explanatory: shelf level, response: sugar content Based on just how the distributions look, do you think there is an association between where the cereal is located and its sugar content? If so, in what way? The distributions suggest more sugar content on high shelves (an association). The distributions suggest the same sugar content on high and low shelves (no association). The distributions suggest less sugar content on high shelves (an association). Find the mean grams of sugar per serving for the cereal on the low shelves and the mean for those on the high shelves. With this new information, do you think there is an association between where the cereal is located and its sugar content? Mean for low shelves: 13; mean for high shelves: 9. These values support an association, since they are different. Mean for low shelves: 13; mean for high shelves: 13. These values support no association, since they are the same. Mean for low shelves: 11.925; mean for high shelves: 9.625. These values support an association, since they are different. Mean for low shelves: 9.625; mean for high shelves: 9.625. These values support no association, since they are the same. Find the median grams of sugar per serving for the cereal on the low shelves and the median for those on the high shelves. With this new information, do you think there is an association between where the cereal is located and its sugar content? Median for low shelves: 11.925; median for high shelves: 9.625. These values support an association, since they are different. Median for low shelves: 13; median for high shelves: 9. These values support an association, since they are different. Median for low shelves: 9; median for high shelves: 9. These values support no association, since they are the same. Median for low shelves: 11.925; median for high shelves: 11.925. These values support no association, since they are the same. Are the median sugar contents between high and low shelved cereal farther apart or closer together than the means are? The medians are farther apart than the means are. The medians are closer together than the means are. Chapter 06.02 Exercise Question 14 6.2.14 In a survey of introductory statistics students, an instructor asked her students how many flip-flops they have. Suppose that the intent is to study whether there is an association between number of flip-flops and a person's sex. Use the 2SD method and the information available in the previous exercises to find a 95% confidence interval for the difference in average number of flip-flops that men have and that women have. Round to two decimal places. , Interpret the confidence interval reported in part (a) in the context of the study. We are 95% sure that females own between 1.25 and 6.11 fewer flip-flops on average than males in the population. We are 95% sure that males own between 1.25 and 6.11 fewer flip-flops on average than females in the population. We are 95% sure that females own between 1.25 and 6.11 more flip-flops on average than males in the population. We are 95% sure that males own between 1.25 and 6.11 more flip-flops on average than females in the population. The conclusions from this confidence interval agree with the conclusions from the reported p-value of 0.001. True False Chapter 06.02 Exercise Question 15 6.2.15 In a survey of introductory statistics students, an instructor asked her students how many flip-flops they have. Suppose that the intent is to study whether there is an association between number of flip-flops and a person's sex. Here are some summary statistics about the data: The following is output from carrying out the randomization test using the Multiple Means applet. Hint: Pay close attention to the label on the x-axis. The shaded region depicts the p-value. True False Notice that the mean of the randomization test is 0.014 and the standard deviation is 2.348. Use the information available to find and report the standardized statistic in this context. 1.57 8.06 2.13 2.93 Chapter 06.02 Exercise Question 24 6.2.24 Rising mercury levels due to industrial pollution is a recent concern. Many fish have high levels of mercury. Data were collected on random samples of tuna from 1991 to 2010 and are available in the file Tuna. Each row in the data set represents the mercury level of a different fish. Do fresh Yellowfin tuna have different levels of mercury on average compared to fresh Albacore tuna? Use the fresh tuna data to answer the following questions. Is this an experiment or an observational study? Explain how you know. Observational study: Researchers did not assign variable values to the fish. Experiment: Researchers assigned variable values to the fish. Identify the observational units. Each fish Levels of mercury Identify the explanatory and response variable. Also, identify whether each is quantitative or categorical. Explanatory: mercury= level (quantitative), response: type of tuna (categorical) Explanatory: type of= tuna (categorical), response: mercury level (quantitative) Explanatory: mercury= level (categorical), response: type of tuna (quantitative) Explanatory: type of = tuna (quantitative), response: mercury level (categorical) Define (in words) the parameters of interest of this study. Also, assign symbols to the parameters. The average mercury level in albacore tuna (Albacore); the average mercury level in yellowfin tuna (Yellowfin) The number of albacore tuna (Albacore); the number of yellowfin tuna (Yellowfin) State the null and alternative hypotheses in words as well as using symbols. Null: The average mercury level in albacore tuna is the same as in yellowfin tuna (Albacore = Yellowfin). Alt: The average mercury level in albacore tuna is different from yellowfin tuna (Albacore Yellowfin). Null: The average mercury level in albacore tuna is different from yellowfin tuna (Albacore Yellowfin). Alt: The average mercury level in albacore tuna is the same as in yellowfin tuna (Albacore = Yellowfin). Null: The average mercury level in albacore tuna is greater than yellowfin tuna (Albacore > Yellowfin). Alt: The average mercury level in albacore tuna is the same as in yellowfin tuna (Albacore = Yellowfin). Null: The average mercury level in albacore tuna is the same as in yellowfin tuna (Albacore = Yellowfin). Alt: The average mercury level in albacore tuna is greater than yellowfin tuna (Albacore > Yellowfin). What statistic will you use? Difference of sample medians Difference in sample means Chapter 06.02 Exercise Question 25 6.2.25 Rising mercury levels due to industrial pollution is a recent concern. Many fish have high levels of mercury. Data were collected on random samples of tuna from 1991 to 2010 and are available in the file Tuna. Each row in the data set represents the mercury level of a different fish. With randomization techniques we are able to analyze any test statistic we desire. What if we wanted to look at whether there was a difference in the median values of mercury in the Yellowfin tuna compared to the Albacore tuna? Use the Multiple Means applet with the Median option to answer the following questions. Is this an experiment or an observational study? Explain how you know. Observational study: Researchers did not assign variable values to the fish. Experiment: Researchers assigned variable values to the fish. Identify the observational units. Each fish The years 1991 to 2010 The two types of tuna Levels of mercury Identify the explanatory and response variable. Also, identify whether each is quantitative or categorical. Explanatory: mercury level (quantitative), response: type of tuna (categorical) Explanatory: type of tuna (categorical), response: mercury level (quantitative) mercury level (categorical), response: type of tuna (quantitative) Explanatory: type of tuna (quantitative), response: mercury level (categorical) Define (in words) the parameters of interest of this study. The median mercury level in albacore tuna; the median mercury level in yellowfin tuna The number of albacore tuna; the number of yellowfin tuna State the null and alternative hypotheses in words. Null: The median mercury level in albacore tuna is the same as in yellowfin tuna Alt: The median mercury level in albacore tuna is greater than in yellowfin tuna. Null: The median mercury level in albacore tuna is different from that in yellowfin tuna. Alt: The median mercury level in albacore tuna is the same as in yellowfin tuna. Null: The median mercury level in albacore tuna is greater than in yellowfin tuna. Alt: The median mercury level in albacore tuna is the same as in yellowfin tuna. Null: The median mercury level in albacore tuna is the same as in yellowfin tuna Alt: The median mercury level in albacore tuna is different from that in yellowfin tuna. What statistic will you use? Difference of sample means Difference in sample medians Use an appropriate applet to carry out a simulation analysis. Report the p-value. the absolute tolerance is +/-0.02 Use the 2SD rule to determine a 95% confidence interval. ( , ) Chapter 06.03 Exercise Question 14 6.3.14 In an article titled \"Unilateral Nostril Breathing Influences Lateralized Cognitive Performance\" that appeared in Brain and Cognition (1989), researchers Block et al. published results from an experiment involving assessments of spatial and verbal cognition when breathing through only the right versus left nostril. The subjects were 30 male and 30 female right-handed introductory psychology students who volunteered to participate in exchange for course credit. Initial testing on spatial and verbal tests revealed the following summary statistics. Note that the scores on the spatial task can range from 0 to 40, whereas those on the verbal task can go from 0 to 20. The distributions are not strongly skewed on either scale or for males or females. Consider comparing males to females with regard to performance on the spatial assessment task. State the appropriate null and alternative hypotheses in the context of the study. Null: The average spatial score for men is different than for women. Alt: The average spatial score for men is the same as for women. Null: The average spatial score for men is the same as for the women. Alt: The average spatial score for men is greater than for women. Null: The average spatial score for men is greater than for women. Alt: The average spatial score for men is the same as for women. Null: The average spatial score for men is the same as for women. Alt: The average spatial score for men is different than for women. Explain why it is valid to use the theory-based method for producing a p-value to test the hypotheses stated in part (a). The sample sizes are less than 20 in each group, and the data are not strongly skewed within either group. The sample sizes are less than 20 in each group, and the data are strongly skewed within each group. The sample sizes are at least 20 in both groups, and the data are strongly skewed within each group. The sample sizes are at least 20 in both groups, and the data are not strongly skewed within either group. Use an applet to carry out the appropriate test to produce a p-value to test the hypotheses stated in part (a) and interpret the p-value. (Round the test statistic to 2 decimal places, e.g. 1.58, and the p-value to 4 decimal places, e.g. 0.5863.) p-value = The probability of obtaining a t-statistic of smaller is or larger or or if the null hypothesis is true. Find a 95% confidence interval for the difference in mean scores of males and females with regard to performance on spatial assessments. (Round your answers to 3 decimal places, e.g. 0.586.) 95% CI: ( , ) We are 95% confident that men have an average spatial score between and higher than women. Based on your p-value, state a conclusion in the context of the study. We have strong evidence that the average spatial score is significantly higher for men than for women. This is a cause-and-effect conclusion, and the result be confidently generalized to a broader population since the sample was obtained randomly from a larger population. We have no evidence that the average spatial score is significantly higher for men than for women. This is not necessarily a cause-and-effect conclusion, nor can the result be confidently generalized to a broader population since the sample was not obtained randomly from a larger population. We have strong evidence that the average spatial score is significantly higher for men than for women. This is not necessarily a cause-and-effect conclusion, nor can the result be confidently generalized to a broader population since the sample was not obtained randomly from a larger population. We have no evidence that the average spatial score is significantly higher for men than for women. This is a cause-and-effect conclusion, and the result be confidently generalized to a broader population since the sample was obtained randomly from a larger population. Repeat the investigation comparing males and females, this time on verbal performance. State the appropriate null and alternative hypotheses in the context of the study. Null: The average verbal score for men is the different than for women. Alt: The average verbal score for men is same as for women. Null: The average verbal score for men is the same as for women. Alt: The average verbal score for men is different than for women. Null: The average verbal score for men is the same as for women. Alt: The average verbal score for men is less than for women. Null: The average verbal score for men is different than for women. Alt: The average verbal score for men is greater than for women. Explain why it is valid to use the theory-based method for producing a p-value to test the hypotheses stated in part (f). The sample sizes are less than 20 in each group and the data are strongly skewed within each group. The sample sizes are at least 20 in both groups and the data are not strongly skewed within either group. The sample sizes are at least 20 in both groups and the data are strongly skewed within each group. The sample sizes are less than 20 in each group and the data are not strongly skewed within either group. Use an applet to carry out the appropriate test to produce a p-value to test the hypotheses stated in part (f) and interpret the p-value. (Round the test statistic to 2 decimal places, e.g. 1.58.) p-value < 0.01 0.001 0.1 0.0001 0.05 The probability of obtaining a t-statistic of or larger or smaller is less than 0.05 0.01 0.1 0.0001 0.001 if the null hypothesis is true. Find a 95% confidence interval for the difference in mean scores of males and females with regard to performance on verbal assessments. Interpret the interval. (Round your answers to 2 decimal places, e.g. 0.58.) 95% CI: ( , ) We are 95% confident that men have average verbal scores between and lower than women. Based on your p-value, state a conclusion in the context of the study of verbal tests. We have no evidence that the average verbal score is significantly higher for women than for men. This is not necessarily a cause-and-effect conclusion, nor can the result be confidently generalized to a broader population since the sample was not obtained randomly from a larger population. or We have strong evidence that the average verbal score is significantly higher for women than for men. This is a cause-and-effect conclusion, and the result can be confidently generalized to a broader population since the sample was not obtained randomly from a larger population. We have strong evidence that the average verbal score is significantly higher for women than for men. This is not necessarily a cause-and-effect conclusion, nor can the result be confidently generalized to a broader population since the sample was not obtained randomly from a larger population. We have no evidence that the average verbal score is significantly higher for women than for men. This is a cause-and-effect conclusion, and the result can be confidently generalized to a broader population since the sample was not obtained randomly from a larger population. Chapter 06.03 Exercise Question 21 6.3.21 A psychology study (Rutchick, Slepian, and Ferris, 2010) investigated whether using a red pen causes people to assign lower scores than using a blue pen. A group of 128 students in an introductory undergraduate psychology class were asked to grade an eighth grader's essay on a scale of 0-100. Half of the student graders were randomly assigned to use a red pen while grading the essay, and the other half were randomly assigned to use a blue pen. The researchers reported that the group using a red pen gave an average score of 76.20, and the SD of the scores was 12.29. For the group using blue pens, the average score was 80.00, and the SD of the scores was 9.36. State the null and alternative hypotheses to be tested. (Highlight one in below) Null: The average score given when using a red pen is the same as with a blue pen. Alt: The average score given is lower when using the red pen than the blue pen. Null: The average score given is lower when using the red pen than the blue pen. Alt: The average score given when using a red pen is the same as with a blue pen. Without having access to the 128 scores that were assigned to the essays, can you feel comfortable in applying a two-sample t-test using the summary statistics? Not totally comfortable; would like to know that there are not large outliers or strong skewness in the scores in either group Totally comfortable What additional information is needed to apply a two-sample t-test using the summary statistics? The population mean The sample size in each group The sample mean of each group The skewness of each group Assume that neither group is skewed, and use an appropriate applet to conduct a two-sample t-test. Report the values of the test statistic and p-value. (Round the test statistic to 2 decimal places, e.g. 1.58, and the p-value to 3 decimal places, e.g. 0.586.) t= ,p= Summarize your conclusion from this test. We have moderate evidence of a difference in the average scores assigned to the essay between the two different colors of pens. We have very strong evidence of a difference in the average scores assigned to the essay between the two different colors of pens. We have no evidence of a difference in the average scores assigned to the essay between the two different colors of pens. We have strong evidence of a difference in the average scores assigned to the essay between the two different colors of pens. Is it legitimate to draw a cause-and-effect conclusion based on these data and your test result? No, because random assignment was not used to assign pen colors to students. No, because random sampling was not used to select the students. Yes, because random assignment was used to assign pen colors to students. Yes, because random sampling was used to select the students. What would you advise the researchers about how broadly they can generalize the conclusions of this study? For example, would you feel comfortable in generalizing the results of this study to eighth-grade teachers or to college professors? They can generalize to all teachers and college professors. They shouldn't generalize too far because these were introductory psych students grading the papers. They can generalize to all eighth-grade teachers, since the students were grading eighth-grade papers. Do you think the study's conclusions would be stronger if the subjects had been actual teachers or college professors? Why do you think the researchers used introductory students as their subjects? No, it would not be stronger with a different population, using undergraduate students is a good sample for teachers. Yes, it would be stronger with a different population, but using undergraduate students is easy/convenient. Yes, it would be stronger with a different population, but using undergraduate students is a good sample for teachers. No, it would not be stronger with a different population, but using undergraduate students is easy/convenient