8. exercises_ci

Introductory Statistics Explained (1.11) Exercises Confidence Intervals © 2023, 2024 Jeremy Balka Chapter 8: Confidence intervals v1.11 W24 Draft J.B.’s strongly suggested exercises: 1 , 2 , 4 , 5 , 7 , 8 , 9 , 10 , 11 , 13 , 14 , 24 , 26 , 27 , 29 , 31 , 32 NB The section titles and numbers are not yet synced up with the text. 1 Introduction 2 Interval Estimation of μ when σ is Known 1. Suppose we are sampling from a normally distributed population and we wish to calculate a confi- dence interval for the population mean μ . Find the appropriate z value and margin of error in the following situations. (a) n = 22 , σ = 5 , 95% confidence level. (b) n = 10 , σ = 10 , 90% confidence level. (c) n = 18 , σ = 15 , 99% confidence level. (d) n = 18 , σ = 15 , 63.2% confidence level. (e) n = 50 , σ = 35 , 21.2% confidence level. 2. Suppose we draw a random sample of n observations from a normally distributed population, and we wish to construct a confidence interval for the population mean μ . Suppose that the population standard deviation σ is known. Give the confidence level for each of the following intervals. (a) ¯ X ± 1 . 21 σ p n (b) ¯ X ± 1 . 35 σ p n (c) ¯ X ± 1 . 57 σ p n (d) ¯ X ± 2 . 45 σ p n (e) ¯ X ± 3 . 15 σ p n 3. Which of the following statements are true? (You should be able to explain why a statement is true or why a statement is false.) (a) In statistical inference, we often construct confidence intervals for statistics. (b) A confidence interval is a range of plausible values for a parameter. 1 Margin 2 on a 1 0.6321 2 o.su qnormlo.su o.at margin oa 3 is prorm 1.21 pnorm 1 1.21 0.773 CI 77.31 prorm 1.3s pnorm 1.3s 0.823 I 82.31 i

(c) The confidence level of an interval is determined by sample data. (d) The point estimate of μ lies at the midpoint of the confidence interval for μ . (e) The confidence interval procedures of this section assume a normally distributed population, but this assumption becomes less important as the sample size increases. 2.1 Interpretation of the Interval 4. Researchers are interested in estimating the population mean body mass index (BMI) for female students at a university. (A person’s BMI is their weight in kilograms divided by the square of their height in metres.) Measurements on 200 female student volunteers at this university resulted in an average BMI of 25.7 kg/m 2 . (For the purposes of this question, assume that the population standard deviation is 6.0 kg/m 2 . (This value is likely not far from reality.) Also assume that the volunteers can be thought of as a random sample of female students from this university. This assumption may or may not be reasonable, depending on how the volunteers were obtained.) (a) What is a 95% confidence interval for the population mean? (b) What is an appropriate interpretation of this interval? (c) What is a 99% confidence interval for the population mean? (d) What is an appropriate interpretation of this interval? 5. A study by ? investigated physical characteristics of the ears of Caucasian Italians. In one part of the study, a sample of 66 females between the ages of 18 and 30 years old revealed a mean right ear length of 61.9 mm with a standard deviation of 4.1 mm. The resulting 95% confidence interval for μ was found to be (60.91, 62.89). Assuming the 66 females in the sample can be thought of as a random sample of Caucasian Italian females between the ages of 18 and 30, give an appropriate interpretation of this confidence interval in the context of the situation at hand. 6. Suppose that at a certain university, a random sample of 50 graduating students yielded an average student loan debt of $28,500, with an associated 95% confidence interval for μ of (26,500, 30,500). Which, if any, of the following options are correct interpretations of this interval? (a) 95% of graduating students at this university have student loan debt between $26,500 and $30,500. (b) We can be 95% confident that the average student loan debt of the 50 students in the sample lies between $26,500 and $30,500. (c) We can be 95% confident that the true mean student loan debt of graduating students at this university lies between $26,500 and $30,500. (d) In repeated sampling, 95% of the 95% confidence intervals would contain $28,500. 2.2 What Factors A ff ect the Margin of Error? 7. All else being equal, what is the e ff ect of the following on the margin of error of a confidence interval for μ ? (a) An increase in the sample size. (b) An increase in the variance. (c) An increase in the sample mean. (d) An increase in the confidence level. E iii no IT inresses 2 a a margin

35 40 45 50 55 60 Hand Grip Strength (kgf) -2 -1 0 1 2 35 40 45 50 55 60 Theoretical Quantiles Sample Quantiles Figure 1: Plots of the grip strengths of 32 male student volunteers. 3.3 Assumptions of the One-Sample t Procedures 12. Suppose we intend to use the t procedure to construct a confidence interval for the population mean μ . If the population is not normally distributed, what are the implications? 13. What types of violation of the normality assumption are very problematic for the t procedure? What types of violation of the normality assumption are not a big problem? 4 Determining the Minimum Sample Size n 14. Suppose we are about to sample from a normally distributed population, and we wish to estimate the population mean μ . Calculate the minimum sample size required in the following scenarios. (a) σ = 10 , and we wish to estimate μ to within 5 with 90% confidence. (b) σ = 10 , and we wish to estimate μ to within 5 with 95% confidence. (c) σ = 30 , and we wish to estimate μ to within 5 with 95% confidence. (d) σ = 10 , and we wish to estimate μ to within 0.1 with 99% confidence. (e) σ 2 = 16 , and we wish to estimate μ to within 1 with 95% confidence. 5 Chapter Exercises 5.1 Basic Calculations 15. Suppose we are about to randomly sample 16 values from a population where μ = 5 and σ = 10 . What is the value of σ ¯ X ? 16. A random sample of 16 independent observations had a mean of 20.0 and a standard deviation of 2.0. What is the value of SE ( ¯ X ) ? 17. Suppose the following sets of values represent samples drawn from normally distributed populations. For each scenario, calculate a 95% confidence interval for the population mean μ . (a) 3, 5, 12. (b) - 4 , 8, 23. (c) 2012, 2010, 2054, 2032. (d) - 52 , - 48 , - 48 , - 46 , - 42 .

5.2 Concepts 18. A person is interested in estimating the average weight (in kg) of adult males of a certain breed of dog. They draw a sample of adult males of this species, measure their weights, and find that the 95% confidence interval for the true mean weight is (12 . 2 , 14 . 9) . Of the following options, which one is the best interpretation of this interval? (Assume that the sample can be thought of as a simple random sample of adult males of this breed.) (a) In repeated sampling, 95% of the time μ will lie between 12.2 kg and 14.9 kg. (b) In repeated sampling, 95% of the sample mean weights will lie between 12.2 kg and 14.9 kg. (c) In repeated sampling, 95% of the 95% confidence intervals will be (12 . 2 , 14 . 9) . (d) We can be 95% confident that the population mean weight of all adult males of this breed lies between 12.2 kg and 14.9 kg. (e) 95% of adult males of this breed weigh between 12.2 kg and 14.9 kg. 19. Suppose we repeatedly draw random and independent samples from a normally distributed popula- tion and for each sample we calculate a 95% confidence interval for the population mean μ . In 100 of such intervals: (a) What is the distribution of the number of intervals that contain μ ? (b) What is the probability that at least one interval does not contain μ ? 20. Suppose that a researcher is interested in estimating the average survival time of rats after they have been infected with a certain type of virus. The researcher infects 100 rats with the virus and then measures the time until death. The average time until death is found to be 1.4 months. (For the purposes of this question, assume that σ is known to be 1.0 months.) (a) Survival times are often skewed to the right. If that is the case here, would inference methods based on the normal distribution be inappropriate? (b) What is the the value of the standard deviation of the sampling distribution of the sample mean? (c) What is a 92% confidence interval for the mean μ ? (d) What is the true value of μ ? (e) What does μ represent in this situation? 21. True or false? t ↵ / 2 tends toward z ↵ / 2 as the degrees of freedom increase. 22. (Warning: This is a very challenging question!) In a (not very helpful) video on YouTube, a statistics instructor at a university talks about inter- preting a confidence interval for the population mean, and he states that if we were to repeatedly sample from the population, 95% of the sample means would fall within the 95% confidence interval. This is simply untrue. A correct interpretation is that in repeated sampling, 95% of the calculated intervals would capture the true value of the population mean μ . This question was motivated by that YouTube video. Suppose that we are about to draw a simple random sample of n observations from a normally distributed population, then calculate a 95% confidence interval for μ . (Suppose that σ is known, so we will calculate the interval using the formula: ¯ X ± z . 025 σ p n .) If we were to draw another, independent sample of the same size from this population, what is the probability that the sample mean of this second sample falls within the 95% confidence interval for μ found in the first sample? (The probability is not 0.95, as you may guess from the fact that this question is being asked.)

15 20 25 30 35 40 BMI (a) Boxplot. -2 -1 0 1 2 15 20 25 30 35 40 Theoretical Quantiles Sample Quantiles (b) Normal quantile-quantile plot. Figure 2: BMI for 94 boys in rural Texas. (d) Give an appropriate interpretation of the 95% confidence interval in the context of the situation at hand. To what population do your conclusions apply? Comment on any biases that might be present. 27. Franklin et al. (2012) 2 investigated various aspects of tandem running in ants. Tandem running is a form of recruitment in which one ant with knowledge of the location of a food source or new nest site leads another ant to that location. (Optional: Watch an example of tandem running here: (1:53) ( http://www.youtube.com/watch?v=X2C7Sy2oPik )) Tandem running can be thought of as a form of teaching and learning. The study investigated various factors associated with tandem running, and investigated whether the age of the ant and the experience of the ant had any e ff ect on tandem running. Ants ( Temnothorax albipennis ) were categorized into 4 categories: Young and inexperienced (YI), young and experienced (YE), old and inexperienced (OI), old and experienced (OE). One aspect of the study investigated the speed of tandem running for 51 tandem runs in which a YE ant was leading a YI ant. The mean speed (mm/s) of the run was recorded. Figure 3 illustrates the speed of the runs. For these 51 runs, the sample mean was 1.65 mm/s and the sample standard deviation was 0.62 mm/s. 0.5 1.0 1.5 2.0 2.5 Speed (mm/s) (a) Boxplot. -2 -1 0 1 2 0.5 1.0 1.5 2.0 2.5 Theoretical Quantiles Sample Quantiles (b) Normal quantile-quantile plot. Figure 3: Speed of 51 tandem runs. (a) Do these plots show a violation of the normality assumption? If there is a violation, is it a serious problem in this situation? (Can we still use the t procedures?) 2 The data used here is simulated data based on their Figure 2, with similar results and conclusions.

(b) What is the point estimate of the population mean μ ? What is the value of the standard error of the sample mean? (c) What is a 95% confidence interval for the population mean μ ? (d) Give an appropriate interpretation of the 95% confidence interval, in the context of the situation at hand. To what population do your conclusions apply? Comment on any biases that might be present. 28. The Athens Collection is a collection of 225 skeletal remains, housed at the University of Athens in Greece. The remains were collected from cemeteries near Athens, from people who lived mainly in the second half of the 20th century. A study 3 investigated the diameter of molar teeth of individuals in this collection. Of 24 upper M1 molars from adult males, the mean crown diameter was found to be 10.38 mm with a standard deviation of 0.63 mm. (a) Suppose we wish to construct a 95% confidence interval for the mean crown diameter of upper M1 molars in adult males in Greece. What assumptions are necessary for the t confidence interval procedure to be valid? Are the assumptions likely to be satisfied in this scenario? (b) What is the point estimate of the population mean μ ? What is the value of the standard error of the sample mean? (c) Assuming normality, construct a 95% confidence interval for the population mean. (d) Give an appropriate interpretation of the 95% confidence interval, in the context of the situation at hand. To what population do your conclusions apply? Comment on any biases that might be present. 29. Larviposition is the laying of larvae (the eggs hatch inside the female), and it occurs in a small number of insect species. ? investigated various aspects of larviposition of the blowfly Calliphora varifrons . In one aspect of the study, female fecundity was investigated by trapping female C. varifrons blowflies in a field, dissecting them, and counting the number of live larvae they had inside. Of the 49 females that carried live larvae, the mean number of larvae was 33.4, with a standard deviation of 7.0. (a) Suppose we wish to construct a 95% confidence interval for the mean number of live larvae carried by female C. varifrons blowflies in this field. What assumptions are necessary for the t confidence interval procedure to be valid? Are the assumptions likely to be satisfied in this scenario? (b) Although the full data was not reported in the original journal article, the authors did report that the number of live larvae varied between 20 and 44 for the 49 females in the study. What does this tell us about the validity of the t procedure in this scenario? (c) What is the point estimate of the population mean μ ? What is the value of the standard error of the sample mean? (d) Assuming normality, construct a 95% confidence interval for the population mean. (e) Give an appropriate interpretation of the 95% confidence interval, in the context of the situation at hand. To what population do your conclusions apply? Comment on any biases that might be present. 30. In another aspect of the study first discussed in Question 29 , ? presented 42 female C. varifrons blowflies with a cube of fresh beef liver, and measured the rate of larviposition. The rate of deposition (number of larvae laid per second) had an average of 0.46, with a standard deviation of 0.32. (a) Suppose we wish to construct a 95% confidence interval for the true mean rate of larvae deposi- tion for female C. varifrons blowflies under the conditions of this experiment. What assumptions are necessary for the t confidence interval procedure to be valid? Are the assumptions likely to be satisfied in this scenario? 3 .

33. ? investigated various aspects of homicides in Denmark. One aspect of the study involved investi- gating characteristics of intimate partner homicide (a person murdered their intimate partner). In 36 intimate partner homicides over the years 1983–2007, the average age of the perpetrator was 43.8 years and the standard deviation was 15.5 years. (a) Suppose we wish to construct a 95% confidence interval for the true (theoretical) mean age of the perpetrators of intimate partner homicides in Denmark over these years. What assumptions are necessary for the t confidence interval procedure to be valid? Are the assumptions likely to be satisfied in this scenario? (b) What is the point estimate of the population mean μ ? What is the value of the standard error of the sample mean? (c) Assuming normality, construct a 95% confidence interval for the population mean. (d) Give an appropriate interpretation of the 95% confidence interval, in the context of the situation at hand. To what population do your conclusions apply? Comment on any biases that might be present. 5.4 Extra Practice Questions 34. A new airline is interested in estimating the mean flying time for 747 flights from Toronto (YYZ) to Vancouver (YVR). A sample of 15 flights yielded a mean time of 295.6 minutes and a sample standard deviation of 10.2 minutes. The airline wishes to use the data to construct a 95% confidence interval for the population mean flight time. They (wisely) think that they should check the normality assumption before using a procedure based on the assumption of a normally distributed population. They plot a normal quantile-quantile plot of the 15 flight times, and the results are illustrated in Figure 4 . -1 0 1 280 285 290 295 300 305 310 Normal Q-Q Plot Theoretical Quantiles Sample Quantiles Figure 4: Normal QQ plot for the 15 flight times in Question 34 . (a) Does this plot give any indication that the t procedures should not be used here? (b) Assuming that the sample of flight times can be thought of as a simple random sample from the population of flight times, and assuming that the flight times are approximately normally distributed, calculate a 95% confidence interval for the population mean flight time.

(c) Suppose that the interval in 34b was found to be (288.1, 296.1). This is not the correct interval but assume that it is for the purposes of this question. Of the following options, which one is the best interpretation of that interval? i. 95% of YYZ–YVR flights take between 288.1 and 296.1 minutes. ii. We can be 95% confident that the population mean flight time from YYZ to YVR for these types of flights lies between 288.1 and 296.1 minutes. iii. We can be 95% confident that the sample mean flight time of the 15 flights lies between 288.1 and 296.1 minutes. iv. A randomly selected YYZ–YVR flight time has a 95% chance of taking between 288.1 and 296.1 minutes. v. All of the above. 35. Suppose you wish to estimate the average arsenic level in the soil of playgrounds of elementary schools in a very large school district. You randomly sample schools from this district, find that the sample mean arsenic level is 14.60 ppm with a 95% confidence interval for μ of (14 . 1 , 15 . 1) . Of the following options, which one is the most appropriate interpretation of that interval? (a) In repeated sampling, 95% of the sample means will fall between 14.1 and 15.1 ppm. (b) We can be 95% confident that the mean arsenic level of the schools in the sample lies between 14.1 and 15.1 ppm. (c) In repeated sampling, 95% of the time the confidence interval for μ will be (14.1, 15.1). (d) We can be 95% confident that the population mean arsenic level of the soils of schools in this district lies between 14.1 and 15.1 ppm. (e) In repeated sampling, μ will change from sample to sample. 95% of the time μ will lie between 14.1 and 15.1 ppm, and 5% of the time it will be outside of this interval. 36. Executives at a large company wish to investigate the average wait time for phone calls that are received at a call centre. The wait times (in minutes) for a random sample of 600 phone calls to the call centre are recorded, entered into the statistical software R, and the following output is found. One Sample t-test data: calltime t = 50.0812, df = 599, p-value < 2.2e-16 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: 2.884044 3.119472 Give an appropriate interpretation of the confidence interval found in the output. 37. A car manufacturer is investigating the fuel consumption of two types of engine design. As part of the study, 8 engines of Type A are placed into a certain type of car and driven under real-world conditions. The cars are driven under similar conditions and the fuel consumption (litres/100 km) is recorded. The eight cars had a mean fuel consumption of 6.8 l/100k with a standard deviation of 0.32 l/100k. (a) If we had access to the data, how should we go about investigating whether the fuel consumption values are approximately normally distributed? (b) Suppose that it is reasonable to assume that the fuel consumption values are approximately normally distributed. What is a 95% confidence interval for the mean fuel consumption for this type of engine? (c) Give an appropriate interpretation of the 95% interval.

(a) What is a 95% confidence interval for the population mean? (b) The correct interval 95% interval is (9.01, 9.39). Which one of the following is the most appro- priate interpretation of this interval? i. We can be 95% confident that the population mean calcium level of women in this area lies between 9.01 and 9.39 mg/dL. ii. We can be 95% confident that the sample mean calcium level of the 18 women in the sample lies between 9.01 and 9.39 mg/dL. iii. In repeated sampling, 95% of the sample means would lie between 9.01 and 9.39 mg/dL. iv. We are more than 95% confident that the population mean lies within the interval, since 9.2 lies within the interval. v. Since confidence intervals are two-sided, we can only be 90% confident that the population mean calcium level of women in this area lies between 9.01 and 9.39 mg/dL. 42. A researcher drew a random sample from a normally distributed population and found a sample mean of 61.1. For this particular population, the population standard deviation was known to the researcher but the population mean was unknown. The researcher found a 95% confidence interval for the population mean of (60.4, 61.8). (a) What is the point estimate of μ ? (b) What is the margin of error? (c) Calculate a 93% confidence interval for the population mean.