Exam 4: Hypothesis Tests

Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.

Use the following to answer the questions below: A study conducted by the National Center of Health Statistics collects data on Vitamin D levels. In 2011-2014, in a sample of 3,929 non-Hispanic Blacks showed that 17.5% were Vitamin D deficient. A 95% confidence interval based on the sample is (0.152, 0.200). -In January 2015 Gallup reported the results from a survey of 167,000 U.S. adults from January - June 2014 for the Gallup-Healthways Well-Being Index. Based on self-reported height and weight data, they found that 62.8% of U.S. adults are overweight or obese. A 95% confidence interval for the proportion of U.S. adults that are overweight or obese is (0.626, 0.63). Does this interval support the claim that "two-thirds of Americans are overweight or obese"?

Use the following to answer the questions below: A Division III college men's basketball team is interested in identifying factors that impact the outcomes of their games. They plan to use "point spread" (their score minus their opponent's score) to quantify the outcome of each game this season; positive values indicate games that they won while negative values indicate games they lost. They want to determine if "steal differential" (the number of steals they have in the game minus the number of steals their opponent had) is related to point spread; positive values indicate games where they had more steals than their opponent. The data for the games they played this season displayed in the provided table. Assuming that this season was a typical season for the team, they want to know if steal differential is positively correlated with point spread. -Describe how you would generate a single randomization sample in this situation, and identify the statistic you would calculate for each sample.

Use the following to answer the questions below: A certain species of tree has an average life span of 130 years. A researcher has noticed a large number of trees of this species washing up along a beach as driftwood. She takes core samples from 27 of those trees to count the number of rings and measure the widths of the rings. Counting the rings allows the researcher to determine the age of each tree. The average age of the trees in the sample is approximately 120 years. One of her interests is determining if this sample provides evidence that the average age of the driftwood is less than the 130 year life span expected for this type of tree. If the average age is less than 130 years it might suggest that the trees have died from unusual causes, such as invasive beetles or logging. -Use the provided randomization distribution (based on 100 samples) to estimate the p-value for this sample.

A Type I error occurs by

Using a significance level of 5%, the appropriate conclusion for a test with a p-value of 0.0421 would be:

Use the following to answer the questions below: -Describe how you would generate a single randomization sample in this situation, and identify (using the appropriate notation) the sample statistic you would record for each sample.

Use the following to answer the questions below: In May 2012, President Obama made history by revealing his support of gay marriage. Around that time, the Gallup Organization polled 1,024 U.S. adults about their opinions on gay/lesbian relations and gay marriage. They found that 54% of those sampled viewed gay/lesbian relations as "morally acceptable" and that 50% felt that gay marriage should be legal. -A 90% confidence interval for the proportion of American adults who support gay marriage is (0.475, 0.524). Does this confidence interval provide evidence that the percentage of American adults who support gay marriage differs from 50%? State the significance level you are using.

Use the following to answer the questions below: A certain species of tree has an average life span of 130 years. A researcher has noticed a large number of trees of this species washing up along a beach as driftwood. She takes core samples from 27 of those trees to count the number of rings and measure the widths of the rings. Counting the rings allows the researcher to determine the age of each tree. Her data are displayed in the provided table. One of her interests is determining if this sample provides evidence that the average age of the driftwood is less than the 130 year life span expected for this type of tree. If the average age is less than 130 years it might suggest that the trees have died from unusual causes, such as invasive beetles or logging. -Describe how you would generate a single randomization sample in this situation, and identify the statistic you would calculate for each sample.

Use the following to answer the questions below A student in an introductory statistics course investigated if there is evidence that the proportion of milk chocolate M&M's that are green differs from the proportion of dark chocolate M&M's that are green. She purchased a bag of each variety, and her data are summarized in the following table. -Define the appropriate parameter(s) and state the hypotheses for testing if the proportion of green M&M's differs for milk chocolate and dark chocolate M&M's.

Use the following to answer the questions below: A certain species of tree has an average life span of 130 years. A researcher has noticed a large number of trees of this species washing up along a beach as driftwood. She takes core samples from 27 of those trees to count the number of rings and measure the widths of the rings. Counting the rings allows the researcher to determine the age of each tree. The average age of the trees in the sample is approximately 120 years. One of her interests is determining if this sample provides evidence that the average age of the driftwood is less than the 130 year life span expected for this type of tree. If the average age is less than 130 years it might suggest that the trees have died from unusual causes, such as invasive beetles or logging. -Define the appropriate parameter(s) and state the hypotheses for testing if this sample provides evidence that the average age of the driftwood along this beach is less than 130 years.

Use the following to answer the questions below A student in an introductory statistics course investigated if there is evidence that the proportion of milk chocolate M&M's that are green differs from the proportion of dark chocolate M&M's that are green. She purchased a bag of each variety, and her data are summarized in the following table. -Use the provided randomization distribution (based on 100 samples) to estimate the p-value for this sample. Use your p-value to make a decision about these hypotheses. Be sure to word your decision in the context of the problem. Include an assessment of the strength of your evidence.

Use the following to answer the questions below: A certain species of tree has an average life span of 130 years. A researcher has noticed a large number of trees of this species washing up along a beach as driftwood. She takes core samples from 27 of those trees to count the number of rings and measure the widths of the rings. Counting the rings allows the researcher to determine the age of each tree. Her data are displayed in the provided table. One of her interests is determining if this sample provides evidence that the average age of the driftwood is less than the 130 year life span expected for this type of tree. If the average age is less than 130 years it might suggest that the trees have died from unusual causes, such as invasive beetles or logging. -Use technology to create a randomization distribution with at least 1,000 values for testing these hypotheses and estimate the p-value. What conclusion would you make at the 10% significance level?

Use the following to answer the questions below: The makers of a popular brand of laundry detergent have discovered a new secret ingredient that they believe will boost the cleaning power of their detergent. The new ingredient is expensive, and if they use it, they would have to increase the retail price of the detergent (and they worry that the price increase will cause them to lose customers). However, they believe that if the improved detergent gets clothes drastically cleaner, customers will recognize that it is worth the extra cost. They conduct an experiment to compare the performance of the new and old formulas at removing grass stains, red wine, and chocolate from white t-shirts. Each cleaned shirt was rated on a scale from 1 (stain did not get removed) to 10 (no evidence of the stain) by trained experts. They compared the average rating for the new and old formulas. -Suppose they find that the average rating for the shirts cleaned with the new formula was 8.2 and the average rating for the shirts cleaned with the old formula is 8.0 Do you think these results are practically significant? Briefly explain.

Use the following to answer the questions below: A Division III college men's basketball team is interested in identifying factors that impact the outcomes of their games. They plan to use "point spread" (their score minus their opponent's score) to quantify the outcome of each game this season; positive values indicate games that they won while negative values indicate games they lost. They want to determine if "steal differential" (the number of steals they have in the game minus the number of steals their opponent had) is related to point spread; positive values indicate games where they had more steals than their opponent. The data for the first five games are in the provided table as an example. The correlation between point spread and steal differential for the games they played this season is about Assuming that this season was a typical season for the team, they want to test if this sample provides evidence that steal differential is positively correlated with point spread. -Use the provided randomization distribution (based on 100 samples) to estimate the p-value for this sample. Use the p-value to make a decision about these hypotheses. Be sure to word your decision in the context of the problem.

Use the following to answer the questions below: A Division III college men's basketball team is interested in identifying factors that impact the outcomes of their games. They plan to use "point spread" (their score minus their opponent's score) to quantify the outcome of each game this season; positive values indicate games that they won while negative values indicate games they lost. They want to determine if "steal differential" (the number of steals they have in the game minus the number of steals their opponent had) is related to point spread; positive values indicate games where they had more steals than their opponent. The data for the first five games are in the provided table as an example. The correlation between point spread and steal differential for the games they played this season is about Assuming that this season was a typical season for the team, they want to know if steal differential is positively correlated with point spread. -Use the provided randomization distribution (based on 100 samples) to determine if this sample provides evidence that point spread and steal differential are positively correlated. Use a 10% significance level to make your conclusion.

Use the following to answer the questions below: Match each p-value to the most appropriate conclusion. - "The result is significant at the 5% level but not at a 1% level."

Use the following to answer the questions below: A certain species of tree has an average life span of 130 years. A researcher has noticed a large number of trees of this species washing up along a beach as driftwood. She takes core samples from 27 of those trees to count the number of rings and measure the widths of the rings. Counting the rings allows the researcher to determine the age of each tree. Her data are displayed in the provided table. One of her interests is determining if this sample provides evidence that the average age of the driftwood is less than the 130 year life span expected for this type of tree. If the average age is less than 130 years it might suggest that the trees have died from unusual causes, such as invasive beetles or logging. -Use technology to create a randomization distribution with at least 1,000 values for testing these hypotheses and estimate the p-value. Use your p-value and a 5% significance level to make a decision about these hypotheses. Be sure to word your decision in the context of the problem.

Use the following to answer the questions below: A study described in Attention, Perception, and Psychophysics investigated the impacts of multi-tasking on people who play video games and those who don't. Participants in the study were asked to perform three visually demanding tasks with (dual-task) and without (single-task) answering unrelated questions over the phone. One of the tasks involved tracking multiple circles moving around on a computer monitor. At the 5% significance level, the authors of the study concluded "tracking accuracy was significantly worse in the dual-task condition" for both people who play video games and those who do not. -What conclusion would the authors have made at the 10% significance level?

The following figure shows a randomization distribution for the hypotheses versus The statistic used for each sample is Which of the two possible sample results provides the most evidence against ?