STAT work
STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 Chapter covered: 8 Show your work for full credit! Problem 1. Opinion Dynamics Corporation conducted a survey of 1000 adult Americans 18 years of age or older and asked,” Are you currently on some form of a low-carbohydrate diet?”. Of the 1,000 randomly surveyed, 120 indicated that they were on a low-carb diet. PART 1 (a) Find the point estimate for the population proportion of adult Americans 18 years of age or older who are currently on some form of a low-carbohydrate diet. Specify its value and statistical notation for this estimate. Round your answer to 2 decimal places. (b) Find the standard error of this statistic . Round your answer to 3 decimal places. (c) What is the value of z-multiplier zα/2 for a 97% confidence interval? Round your answer to 2 decimal places. Use four different ways to get this value STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (d) Find the margin of error for a 97% confidence interval. PART 2 (a) Use your answers from Problem 1 Part 1 and construct a 97% confidence interval. Interpret. STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (b) Based on your confidence interval, can we conclude that more than 5% of all adult Americans 18 years of age or older are currently on some form of a low-carbohydrate diet? Explain. Part 3 (a) Does the information provided in Problem 1 satisfy conditions (assumptions) associated with confidence intervals for p? State the assumptions and determine whether each is met or not. (Hint: See lecture notes on page 106. There are two or three assumptions.) (b) Use the statistical software R to construct the 98% confidence interval of the population proportion of adult Americans 18 years or older who are currently on some form of a low-carbohydrate diet. STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (c) What is the effect of increasing the level of confidence on the width of the interval and on the margin of error and on the accuracy of the estimate? (Hint: Use results of the Confidence intervals for the 97% and 98% to give your conclusions) STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 Problem 2. The drug Lipitor is meant to lower cholesterol levels. A researcher wishes to estimate the proportion of patients who report a headache as a side effect among those receiving 10-mg doses of Lipitor daily. What sample size should be obtained if the researcher wants an estimate within 1.2 percentage points of the true proportion with 90 % confidence if (a) The Researcher uses the 2003 estimate of 5.4% obtained from the American Community Survey. (b) The Researcher does not use any prior estimates. (c) Why is the minimum sample size in question 2 larger than the one in question 1. STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 Problem 3. A researcher is interested in determining the mean age (in weeks) at which babies first crawled. Based on a survey of mothers conducted by Essential Baby, the following data was obtained. 51 31 43 36 40 25 47 37 55 27 39 28 Treat the sample as a simple random sample of all the ages (in weeks) at which babies first crawl from a population that is approximately normally distributed. PART 1 (a) Find the sample mean and the sample standard deviation and the number of subjects in the studies. This can be done by using the statistical software R or algebra. The c() command is used to enter your data by hand in R. An example using c() is on p.6 of the lecture notes. (b) What is the point estimate of the population mean age? Specify its value and statistical notation for this estimate. STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (c) Find the standard error of this estimate. Explain what this number means. (d) What is the value of t-multiplier tα/2,n−1 for a 96% confidence interval? Round your answer to 3 decimal places. Use four different ways to get this value (e) Find the margin of error for a 96% confidence interval. PART 2 (a) Find the 96 % confidence interval of the mean age at which babies first crawl. Interpret the interval in context. STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (b) Can we conclude that the population mean age at which babies first crawl is greater than 50 weeks? Explain. (c) Find the t-value such that the area in the right tail is 0.15 with 29 degrees of freedom (d) Find the t-value such that the area to the left of this t-value is 0.025 with 20 degrees of freedom. STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 PART 3 (a) State assumptions of the confidence interval for the population mean age at which babies first crawl. Determine whether each assumption is met based on the Problem 3 description. Explain. Hint:(Draw a Q-Q plot a boxplot to verify that there are no outliers, and a histogram. Submit all three plots) STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (b) (Multiple Choice Question) The histogram from the previous question is : (i) a population distribution (ii) a data distribution (sample distribution) (iii) a sampling distribution Choose one and Briefly explain. (c) Use the statistical software R to construct a 96% confidence interval for the population mean age at which babies first crawl. (HINT: Use the t.test() command) STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (d) The following interprets the 96% confidence interval for the population mean age at which babies first crawl using the sample data given from Problem 3 parts 2) and 3) (31.80 weeks and 44.70 weeks). Which of the following is/are correct? (i) In the entire population, 96% of the sample mean ages at which babies first crawl do not fall between 31.80 weeks and 44.70 weeks. (ii) We are 96% confident that the population mean age at which babies first crawl falls between 31.80 weeks and 44.70 weeks. (iii) If we repeatedly sample 12 babies randomly and construct 96% confidence intervals, then in the long run, on average 96% of those intervals contain the population mean age at which babies first crawl (iv) There is a 4% probability that the population mean age at which babies first crawl is between 31.80 weeks and 44.70 weeks. Problem 4. A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x̄, is found to be 13.4, and the sample standard deviation, s, is found to be 16.6. (a) Construct a 95% confidence interval for µ if the sample size, n , is 15. (b) Find the margin of error for a 95% confidence interval for µ if the sample size, n, is 29. How does increasing the sample size affect the margin of error and the confidence interval? You do not need to find or construct a new confidence interval STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024 (c) Construct a 99% confidence interval for µ if the sample size, is n=15. Compare the results to those obtained in part (a). How does increasing the level of confidence affect the size of the margin of error, E, and of the confidence interval? What is the width of this 99 % confidence interval. Are there some values in the 99% confidence interval that are not in the 95% confidence interval? You do not need to find or construct a new confidence interval (d) Is the critical value tα/2,n−1 a quantile or a probability? Use the pt() command in R and write the link between this quantile and α/2 or 1 − α/2 STAT 3011 Homework 5 (Due: March, 17, 2024) Spring 2024
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