Assignment3_Winter 2024 V6

STAT 151 STAT 151 Assignment 3 Due date : refer to the Course Outline Purposes This assignment has two parts. The first part assesses your knowledge of properties of normal density curves, calculating probabilities related to normal distributions, and finding quantiles of normal distributions. The first part also assesses your understanding of the distribution of the sample mean (including the mean, standard deviation, and shape) and the central limit theorem. The second part assess your skills in using R commander to find probabilities and quantiles of normal distributions. Instructions For every assignment in this course, you are required to complete the questions or tasks in Part A by hand. This means that to do any calculation or drawing, you will NOT use R commander or any computer application. That is, you are meant to do the calculations manually with a non- programmable scientific calculator and use a pen or pencil to draw figures or build a distribution table on paper (or on an iPad/tablet). Then you will submit a photo of your written solution using the appropriate submission box on the corresponding Crowdmark submission page. Before you complete Part B using R commander, you should read and practice the R commander steps by following the related examples in the Lab Manual and the Demos, which you can download via a link in the Course Content folder on mêskanâs. Where appropriate, units should be included in your answer. Show your calculations fully and provide a concluding sentence to your problems. Part A 1. Assume that a random variable under consideration has a density curve. Answer the following questions. a) Suppose that the density curve has an area of 0.361 to the left of 13 and an area of 0.428 to the right of 15. What is the size of the area between 13 and 15? (2 marks) b) Determine the z -score that marks the 80 th percentile of the standard normal distribution. Also determine the z-score that marks the upper 0.6% of the standard normal distribution. (4 marks) 2. A pharmaceutical company manufactures bottles of a certain medicine. The ml in the bottles follows a normal distribution with a mean of 100 ml and a standard deviation of 2 ml. 1

STAT 151 a) Find the percentage of medicine bottles that contain more than 104.5 ml of the medicine. (3 marks) b) Find the percentage of medicine bottles containing between 98 ml and 102 ml. (3 marks) c) Determine the 1% percentile of the distribution of ml for the medicine bottles. (3 marks) d) Above what ml are the top 2% of mls among the medicine bottles? (3 marks) e) If a medicine bottle contains no more than 94 ml of the medicine, it is rejected by the quality control board. That is, a bottle is rejected by the quality control department if it contains less than 94 ml. What percentage of bottles are rejected? (3 marks) f) A medicine bottle is rejected by the quality control department if it contains less than 94 ml. Compute the probability of at least one rejection among a random sample of 10 bottles. (5 marks) 3. Suppose we are studying a population with mean μ = 50 and standard deviation σ = 10 . Answer the following questions. a) What are the parameters (the mean and standard deviation) of the sampling distribution of the same mean for samples of size n = 100 ? (2 marks) b) Suppose that the distribution of the population is right-skewed. What is the shape of the sampling distribution of the sample mean in a)? Explain. (2 marks) c) If the sample size n changes, do the parameters of the sampling distribution of the sample mean also change? Explain your answer fully, considering how they change if they do. (3 marks) d) Suppose we collect a sample of size n = 20 from this population. Under what condition is the sampling distribution of the sample mean normally distributed? (2 marks) 4. Let X denote the amount of time a randomly selected customer spends on hold with some insurance company. Suppose X follows a distribution with a mean of 8 hours and a variance of 16 hours. The population density curve for X is shown below. a) Is the population approximately normally distributed? Explain. (2 marks) b) If you randomly pick sixteen customers, describe the sampling distribution of the average time spent on hold; provide the mean, standard deviation, and shape. (3 marks) 2

STAT 151 b. The file “GRADESINTROSTATS” gives introductory statistics grades for all 31 students in the class mentioned above. It includes assignment, lab quiz, final lab exam, lecture midterm and lecture final grades. Use R commander to assess the shape of the assignment 5 sample data. Does the sample data provide evidence to suggest the student grades on assignment 5 are not normally distributed? Are there any apparent outliers? (5 marks) c. Do your answers to part a) and part b) seem to agree? Why or why not? (3 marks) d. The data found in the file “ELECTRICITYCHARGESONEBEDAPARTMENT” gives monthly electricity costs for 12 one-bedroom apartments in a certain building in your city. Use R commander to assess the shape of the sample data. Is there evidence that monthly costs are not normally distributed? Are there any apparent outliers? (5 marks) 4. Use R commander to explore the distribution of the sample mean X when the variable of interest X follows a normal distribution. The data file “SAMPLEMEANNORMAL80” contains three columns: : (1) Column 1: 1,000 observations from a normal distribution with mean 80 and standard deviation 10. A histogram of this column approximates the shape of the population distribution. (2) Column 2: sample means of 1,000 samples of size n=6 generated from a normal distribution with mean 80 and standard deviation 10. A histogram of this column approximates the distribution of the sample mean X for samples of size n = 6 . (3) Column 3: sample means of 1,000 samples of size n=36 generated from a normal distribution with mean 80 and standard deviation 10. A histogram of this column approximates the distribution of the sample mean X for samples of size n = 36 . a. Find the sample mean and sample standard deviation for each column (called the empirical results) and complete the following table. (6 marks) Mean Standard Deviation Variable Theoretical Empirical Theoretical Empirical Shape of theoretical X distribution X (population) μ = 80 79.689 σ = 10 9.557 Normal X with n = 6 μ X = μ = ¿ σ X = σ √ n = ¿ X with n = 36 μ X = μ = σ X = σ √ n = ¿ b. Draw a histogram for each column, and comment on the centre, spread (variation), and shape of the X distributions. (6 marks) 4

STAT 151 5. Use R commander to explore the distribution of the sample mean X when the variable of interest X follows an extremely skewed distribution. The data file “SAMPLEMEANEXPONENTIAL4” contains three columns: (1) Column 1 (x). 1,000 observations from an exponential distribution with mean 4. A histogram of this column approximates the shape of the population distribution. (2) Column 2 (xbarn5): sample means of 1,000 samples of size n=5 generated from an exponential distribution with mean 4. A histogram of this column approximates the distribution of the sample mean X for samples of size n = 5 . (3) Column 3 (xbarn64): sample means of 1,000 samples of size n=64 generated from an exponential distribution with mean 4. A histogram of this column approximates the distribution of the sample mean X with sample size n = ¿ 64 a. Find the sample mean and sample standard deviation for each column (called the empirical results) and complete the following table. Please note that the standard deviation of an exponential distribution is the same as its mean. (6 marks) Mean Standard Deviation Variable Theoretical Empirical Theoretical Empirical Shape of theoretical X distribution X (population) μ = 4 3.961 σ = 4 3.692 Right skewed X with n = 5 μ X = μ σ X = σ √ n = ¿ X with n = 64 μ X = μ σ X = σ √ n = ¿ b. Draw a histogram for each column, and comment on the centre, spread (variation), and shape of the distributions. (6 marks) 6. Refer to Question 4 of Part A, in which we let X denote the amount of time a randomly selected customer spends on hold with some insurance company. Suppose X follows a distribution with a mean of 8 hours and a variance of 16 hours. a. If you randomly pick 121 customers, find the probability that the average time spent on hold is within 1 hour of the population mean. Compare the output with the result you obtained in Question 4 (d) of Part A. (4 marks) b. If you randomly pick 100 customers, find the probability that the average time spent on hold is at most 7 hours. (3 marks) c. If you randomly pick 100 customers, find the probability that the average time spent on hold is at least 7.5 hours. (3 marks) 5