ST 311 Midterm 2

We decided to construct a 99% confidence interval for the true proportion of NCSU undergraduates who are in the College of Sciences from a random sample of 500 NCSU undergraduates. The resulting interval is (0.083, 0.157). Which of the following is a correct interpretation of 99% confidence? a) About 99% of all confidence intervals constructed in this way will contain the true proportion of NCSU undergraduates who are in the College of Sciences. b) There is a 99% probability that the true proportion of NCSU undergraduates who are in the College of Sciences is between 0.083 and 0.157. c) We are 99% confident that the sample proportion of these 500 NCSU undergraduates who are in the College of Sciences is between 8.3% and 15.7%. d) There is a 99% probability that our interval (0.083, 0.157) contains the true proportion of NCSU undergraduates who are in the College of Sciences.

a) About 99% of all confidence intervals constructed in this way will contain the true proportion of NCSU undergraduates who are in the College of Sciences

Vaidehi believes that how people nap has been changed by the pandemic and working from home. Prior to the pandemic, it is believed that people, on average, napped for 120 minutes each week. She randomly samples 150 people and obtains a sample mean of 160 minutes napped per week. Which set of hypotheses below can be used to investigate the question: Has the mean number of minutes napped per week increased since the pandemic started? a) 𝐻0: 𝜇 = 120 𝑣𝑠. 𝐻𝑎: 𝜇 > 120 b) 𝐻0: 𝜇 = 160 𝑣𝑠. 𝐻𝑎: 𝜇 > 160 c) 𝐻0: 𝜇 = 120 𝑣𝑠. 𝐻𝑎: 𝜇 < 120 d) 𝐻0: 𝜇 = 120 𝑣𝑠. 𝐻𝑎: 𝜇 = 160

a) H0: u = 120 vs. Ha: u > 120

Joon wants to measure the amount of time (mins.) college students spend on their phones. A previous study claimed that the average student spends 240 minutes on their phone each day. Joon randomly samples 200 students and obtains a sample mean of 200 mins. Which of the hypotheses would be the correct null and alternative hypothesis if Joon wanted to investigate if there is a difference? a) H0: 𝜇 = 240 vs Ha: 𝜇 ≠ 240 b) H0: 𝜇 = 240 vs Ha: 𝜇 > 240 c) H0: 𝜇 = 200 vs Ha: 𝜇 < 200 d) H0: 𝜇 = 200 vs Ha: 𝜇 ≠ 200

a) H0: u = 240 vs Ha: u =/ 240

A researcher wishes to study the relationship between social media use and anxiety. They ask participants to record their usage (in minutes per day) of social media, and their daily anxiety levels. What type of study is described above? a) Observation, because the researcher is not intervening in the participants social media use. b) Observation, because the participants are reporting their anxiety levels themselves. c) Experiment, because the researcher hopes to claim that social media causes more anxiety. d) Experiment, because the number of minutes on social media is assigned to participants.

a) Observation, because the researcher is not intervening in the participants social media use

One goal of statistical inference is to: a) Use the parameter to make inferences on the statistic. b) Use convenience sampling to get a good sample of the population. c) Select a representative sample from which we can find a statistic to make inferences on the population parameter. d) Include sampling bias in our sampling method to best represent the population of interest.

c) Select a representative sample from which we can find a statistic to make inferences on the population parameter

If a p-value is smaller than the significance level, we a) Reject the null hypothesis, because it is unlikely to observe the sample statistic assuming the null hypothesis is true. b) Accept the null hypothesis, because it is NOT unlikely to observe the sample statistic assuming the null hypothesis is true. c) Fail to reject the null hypothesis, because it is NOT unlikely to observe the sample statistic assuming the null hypothesis is true. d) Fail to reject the null hypothesis, because it is unlikely to observe the sample statistic assuming the null hypothesis is true

a) Reject the null hypothesis, because it is unlikely to observe the sample statistic assuming the null hypothesis is true

Which of the following is true about confidence intervals? a) The purpose of a confidence interval is to learn about the population parameter. b) The purpose of a confidence interval is to remove sampling variability. c) Both A and B. d) A confidence interval is guaranteed to contain the population proportion.

a) The purpose of a confidence interval is to learn about the population parameter

Suppose you have a random sample of 16 NCSU students and would like to construct a 90% confidence interval for the average height of the students. What is the critical value you should use in the confidence interval? a) 1.341 b) 1.753 c) 1.282 d) 1.645

b) 1.753

The teacher at Bob's school with the highest performing students gets a $20 gift card. Bob tells his students that a new study states that taking a nap before an exam will improve scores. Because of this, he encourages his students to take a nap before the exam to perform better. During the exam, he includes a question at the end, asking if the student took a nap or not. He finds that 70% of students indicated they took a nap and their average exam score was 5 points higher than those who didn't. Which of the following experimental design errors did Bob make? a) Bob used a block design instead of matched-pairs b) Bob failed to assign a randomized treatment and control group c) Bob failed to report a p-value d) All of the above

b) Bob failed to assign a randomized treatment and control group

Joe and Charlie would like to estimate the average weight of the NCSU students and construct a corresponding 95% confidence interval. Joe collects a random sample of 50 students, while Charlie only finds 20 students randomly and asks for their weight. What is the relationship between the critical values (i.e. confidence coefficients) Joe and Charlie use? a) Joe's critical value is larger. b) Charlie's critical value is larger. c) They are the same. d) We don't have enough information to tell.

b) Charlie's critical value is larger

In a manufacturing process, let p be the constant probability of an item being defective. The managers would like to know information about p. From a randomly selected sample of size 100, they obtained a 95% confidence interval (0.0186, 0.0394) for p. Which of the following is TRUE about the margin of error (MOE)? a) The MOE for this 95% confidence interval is 0.0208. b) If they increase the sample size, the MOE will decrease. c) If they increase the confidence level, the MOE will decrease. d) If they increase both the sample size and the confidence level, the MOE will be decrease even more.

b) If they increase the sample size, the MOE will decrease

In hypothesis test of single proportion, we always reject the null hypothesis when: a) P-value is less than 0.05, because it is the probability that the null hypothesis is true. b) P-value is less than the significance level 𝛼 which depends on the context. c) The population proportion 𝑝0 under the null hypothesis falls within the 95% confidence interval. d) None of the above.

b) P-value is less than the significance level 𝛼 which depends on the context

Which of the following is true about the margin of error? a) The margin of error does not account for random sampling variability. b) The margin of error does not account for sampling bias. c) The margin of error depends on the sample size but not on the confidence level. d) A margin of error of 5% means there is a 5% chance the interval does not capture the true value of the parameter.

b) The margin of error does not account for sampling bias

A biologist would like to analyze the nitrogen content of two specific varieties of red clover plants (Kenland and Marathon) exposed to three strains of Rhizobium: 3D0K1, 3DOK5, and 3DOK7. In preparation for this study, the biologist planted 36 samples of one variety of red clover (Kenland) and 42 samples of the other (Marathon). Assume the biologist has asked you, the researcher, to analyze the data. What are the treatments? a) The measured nitrogen content values for all 78 samples. b) The three strains of Rhizobium, 3D0K1, 3DOK5, and 3DOK7. c) The two red clover varieties, Kenland and Marathon. d) The two samples of red clover.

b) The three strains of Rhizobium, 3D0K1, 3DOK5, and 3DOK7

A crop scientist is conducting research with a drought resistant corn hybrid. She is interested in determining if using fertilizer X will increase yield. She prepares 28 single acre plots and randomly assigns 14 to have normal soil while the other 14 are planted with fertilizer X. The resulting average yield for each group of 14 plots was recorded. Which of the following is NOT correct? a) The explanatory variable is whether the corn plants had fertilizer X or not. b) This is best described as an observational study. c) This is a completely randomized design. d) The response variable is the average yield for each group of 14 plots.

b) This is best described as an observational study

Confidence intervals are better to report than just the sample statistic. a) True, because confidence intervals give a probability about the sample statistic. b) True, because confidence intervals provide a range of values that we believe will include the population parameter. c) False, because confidence intervals have a probability of being wrong. d) False, because sample statistics are closer to the population parameter than an interval.

b) True, because confidence intervals provide a range of values that we believe will include the population parameter

The null and alternative hypotheses are written in terms of the population parameter of interest. a) True, because we calculate the parameter when we take a sample, therefore we know its value. b) True, because the goal of inference is to draw conclusions about the population parameter, even if we can never calculate it. c) False, because the null and alternative hypotheses should be written in terms of a test statistic (for example, 𝐻0 ∶ 𝑦̅ = 𝑦̅0 𝑣𝑠. 𝐻𝑎: 𝑦̅ > 𝑦̅0). d) False, because we can never calculate the parameter, so we cannot make a conclusion about something we do not know.

b) True, because the goal of inference is to draw conclusions about the population parameter, even if we can never calculate it

Suppose we conduct a study on how many hours of wakefulness result from drinking one cup of coffee, and we find that a 95% confidence interval for the mean hours of wakefulness after one cup of coffee is (2.8, 3.4). How would we interpret this confidence interval? a) There is a 95% probability that the true mean hours of wakefulness after one cup of coffee is in (2.8, 3.4). b) We are 95% confident that the true mean hours of wakefulness after one cup of coffee is in (2.8, 3.4). c) Coffee makes people stay awake longer. d) If we take another random sample, there is a 95% probability the sample mean will be in (2.8, 3.4).

b) We are 95% confident that the true mean hours of wakefulness after one cup of coffee is in (2.8, 3.4)

We have conducted a survey where we took a random sample of 400 subjects. We would like to create an 85% confidence interval for a population proportion. Which of the following would be the appropriate confidence coefficient (i.e. multiplier, critical value) for this confidence level? a) 1.72 b) 1.04 c) 1.44 d) It is impossible to tell without knowing the sample proportion.

c) 1.44

The biologist wants your recommendation on how to randomize the treatment to the red clover samples, but cautions that the two varieties may respond differently to the treatment. What would you suggest? a) A completely randomized design. b) A randomized block design, blocking by strain of Rhizobium. c) A randomized block design, blocking by variety of clover. d) A matched pairs design by matching the variety of clover.

c) A randomized block design, blocking by variety of clover

Which of the following statements is FALSE regarding matched-pair designs: a) Matched-pair designs are examples of block-design b) The degrees of freedom for a paired differences test is equal to the number of paired observations minus one. c) Each pair must have two unique subjects d) An advantage of matched-pair designs is that they control for the effects of confounding variables

c) Each pair must have two unique subjects

If p-value is greater than the significance level, we a) Reject the null hypothesis, because it is unlikely to observe the sample statistic assuming the null hypothesis is true. b) Accept the null hypothesis, because it is NOT unlikely to observe the sample statistic assuming the null hypothesis is true. c) Fail to reject the null hypothesis, because it is NOT unlikely to observe the sample statistic assuming the null hypothesis is true. d) Fail to reject the null hypothesis, because it is unlikely to observe the sample statistic assuming the null hypothesis is true.

c) Fail to reject the null hypothesis, because it is NOT unlikely to observe the sample statistic assuming the null hypothesis is true

Medical researchers have concluded that the effect of a new drug is statistically significant based on the result of a hypothesis test. Suppose the test has been carried out correctly, and all the assumptions were satisfied. Which of the following statements is true? a) The null hypothesis must be true. b) The alternative hypothesis must be true. c) It is possible that the new drug is actually not effective. d) None of the above.

c) It is possible that the new drug is actually not effective

Which of the following is true about p-values? a) The population proportion is equal to p-value under the null hypothesis b) The probability that the null hypothesis is true increases as p-value increases c) P-value is the probability of observing a test statistic as or more extreme than the one we observed, assuming the null hypothesis is true d) None of the above

c) P-value is the probability of observing a test statistic as or more extreme than the one we observed, assuming the null hypothesis is true

Which of the following is NOT true of 95% confidence intervals? (select one) a) In repeated random sampling, in about 95% of random samples, the confidence interval computed from the sample will contain the population parameter. b) In repeated random sampling, in about 5% of random samples, the confidence interval computed from the sample will not contain the population parameter. c) A 95% confidence interval is equivalent to a two-sided hypothesis test that uses a significance level of 0.05. d) After we take a random sample and compute a confidence interval from that sample, there is a 95% probability that the true parameter lies in our computed confidence interval.

d) After we take a random sample and compute a confidence interval from that sample, there is a 95% probability that the true parameter lies in our computer confidence interval

A hypothesis is conducted to test the following hypotheses: 𝐻0: 𝜇 = 10 𝑣𝑠. 𝐻𝑎: 𝜇 ≠ 10 A sample of size 50 is randomly selected and a p-value is calculated to be 0.048. Which of the following is correct? a) The area under the null distribution that is as or more extreme than the observed sample statistic is 0.048. b) If a 95% confidence interval for the mean is calculated, the interval will not include 10. c) 0.048 is calculated assuming the null hypothesis 𝐻0 is true. d) All of them are correct.

d) All of them are correct

Pedro collected a random sample and was able to obtain a 95% confidence interval: (0.546, 0.798). Which of the following would result in a narrower confidence interval? a) Choosing a smaller confidence level b) Increasing the sample size c) Guessing the parameter d) Both A and B

d) Both A and B

Jerry created a 94% confidence interval (0.467,0.695) for a population proportion (all relevant assumptions are satisfied). His boss dismisses Jerry's result and says, "If you are not 100% confident in your sample yet, you should probably try to collect a new one." Is the boss's statement true or false? a) False, because confidence interval based on a new sample will lead to the same result. b) True, because higher confidence level usually indicates better quality of the sample. c) False, because Jerry's confidence interval already corrects for sampling bias. d) False, because 94% refers to the percent of intervals obtained from repeated sampling that contain the population proportion.

d) False, because 94% refers to the percent of intervals obtained from repeated sampling that contain the population proportion

Is the following statement true or false? The null and alternative hypotheses are written in terms of the sample statistic. a) True, because the null and alternative hypotheses should be written in terms of a test statistic (for example H0: 𝒑̂ = 0.5 vs Ha: 𝒑̂ > 0.5) because we found the statistic not the population parameter. b) False, because we calculate the parameter when we take a sample, therefore we know its values. c) True, because we can never calculate the parameter, so we must design the hypothesis test around the sample statistic. d) False, because the goal of inference is to draw conclusions about the population parameter, so we design the hypothesis around an "assumed" value.

d) False, because the goal of inference is to draw conclusions about the population parameter, so we design the hypothesis around an "assumed" value

A researcher calculates a 95% confidence interval for a proportion as (.67, .98). Their boss reads their report and says, "There is a 95% probability that the true population proportion is between 0.67 and 0.98." Is the boss's statement true or false? a) True: confidence level reflects the probability of a specific confidence interval containing the population proportion. b) True: confidence level reflects the proportion of random samples that will result in a confidence interval containing the population proportion. c) False: confidence level reflects the probability of a specific confidence interval containing the population proportion. d) False: confidence level reflects the proportion of random samples that will result in a confidence interval containing the population proportion.

d) False: confidence level reflects the proportion of random samples that will result in a confidence interval containing the population proportion

When is it ok to find a 95% confidence interval instead of conducting an 𝛼 = 0.05 level hypothesis test? a) It is always acceptable to do this. b) Only when the hypothesized value of the parameter is not in the confidence interval. c) Only when doing a test for a single population mean or a single population proportion. d) For any hypothesis test with a two-sided alternative hypothesis.

d) For any hypothesis test with a two-sided alternative hypothesis

Samir is trying to test the effectiveness of a new pain relief drug for patients who fractured their backs. He initially split 400 patients into two groups randomly, with one group receiving the placebo and one receiving the new drug. However, he saw his friend was in the placebo group and decided to swap one patient receiving the drug with his friend. He then hires one of the scientists who created the drug to record results. Which of the following experimental problems does Samir NOT have? a) Failure to randomize b) Not having an unbiased evaluator c) A and B d) Hawthorne Effect

d) Hawthorne Effect

Why is it important to use a representative sample? a) So that the p-value can be significant. b) So that the sampling distribution becomes exactly the same as the population distribution. c) So that the standard deviation can become larger in the sampling distribution than the population distribution. d) So that any inference done is trustworthy with respect to the population.

d) So that any inference done is trustworthy with respect to the population

A biologist would like to analyze the nitrogen content of two specific varieties of red clover plants (Kenland and Marathon) exposed to three strains of Rhizobium: 3D0K1, 3DOK5, and 3DOK7. In preparation for this study, the biologist planted 36 samples of one variety of red clover (Kenland) and 42 samples of the other (Marathon). What would you use for the explanatory and response variables? a) The explanatory variable is the nitrogen content. The response variable is the Rhizobium, with the three levels 3D0K1, 3DOK5, and 3DOK7. b) The explanatory variable is the Rhizobium, with the three levels 3D0K1, 3DOK5, and 3DOK7. The response variable is the variety of clover, with the two levels Kenland and Marathon. c) The explanatory variable is the variety of clover, with the two levels Kenland and Marathon. The response variable is the Rhizobium, with the three levels 3D0K1, 3DOK5, and 3DOK7. d) The explanatory variable is the Rhizobium, with the three levels 3D0K1, 3DOK5, and 3DOK7. The response variable is the nitrogen content.

d) The explanatory variable is the Rhizobium, with the three levels 3D0K1, 3DOK5, and 3DOK7. The response variable is the nitrogen content

In 2009, withering of a large number of crops were ultimately traced to contamination by the pesticide of a certain brand. The manufacturer now claims that the pesticide is safe, but before it can be released, it must be tested. The treatments would be ordinary-size portions of pesticide: the new one from the company (the test pesticide) and one that was certified safe (perhaps the ones that has been used in the past with no issues). The response would be a crop scientist's assessment of the health of crops harvested. Identify the confounding variable. a) The new pesticide from the company b) The certified safe pesticide c) The crop scientist's assessment d) The weather conditions under which the crops were sown and harvested

d) The weather conditions under which the crops were sown and harvested

We are able to perform statistical inference because a) the p-value is significant. b) the sampling distribution is the same as the population distribution. c) the standard deviation is larger in sampling distributions than the population distribution. d) the sample statistics have a predictable distribution.

d) the sample statistics have a predictable distribution