Math 227 Exam 1 Review (1.1 - 2.6) PDF

Math 227 Exam 1 Review (1.1 - 2.6) Multiple Choice/ Fill-In the Blank: 1) Complete the table below by filling in the correct notation or description of the notation being used. Notation Description Sample proportion μ ρ Sample mean difference r...

Math 227 Exam 1 Review (1.1 - 2.6) Multiple Choice/ Fill-In the Blank: 1) Complete the table below by filling in the correct notation or description of the notation being used. Notation Description Sample proportion μ ρ Sample mean difference r Population proportion 2) The graphic below shows a sampling distribution of means for groups of size n = 30. Identify the statement that best describes what each dot represents. 2 E2Review.nb a) The population mean. b) The distance between the population and sample means. c) The sample mean. d) The sample size. 3) Identify the statement below that best describes the relationship between confidence level and confidence interval width. a) As the confidence level increases, the width of the interval increases. b) As the confidence level increases, the width of the interval decreases. c) As the confidence level increases, the width may increase or decrease depending on the population parameter being estimated. d) As the confidence level increases, the width stays the same.  E2Review.nb 3 4) Identify the statement below that best describes the relationship between the p-value and the null hypothesis. a) If the p-value is low, we are not likely to reject the null. b) If the p-value is low, we are likely to reject the null. c) If the p-value is low, we are likely to accept the alternative. d) If the p-value is low, we are not likely to reject the alternative. 5) A student constructs a 99% confidence interval for a population mean. Identify the state- ment that best describes the meaning of “99% confident”. a) There is a 99% probability that the confidence interval contains the population parameter. b) There is a 99% probability that the sample statistic is equal to the population parameter. c) The student is 99% confident the sample statistic is equal to the population parameter. d) About 99/100 confidence intervals generated will contain the population parameter. Definitions and Explanations: 6) Describe the process of generating a boot-strapped sampling distribution for a proportion for groups of size n. 7) Define Margin of Error and explain how it is related to standard error for a 95% confidence interval. 8) Describe the shape and center of a sampling distribution of a sample statistic after repeat- edly drawing samples of size n from the population. 9) Explain the difference between standard deviation and standard error. 10) A 99.9% confidence interval is constructed for a difference in proportions - -0.04 < P < 0.06. Explain whether or not there is sufficient evidence to support the claim that there is a “real” differ- ence in proportions. 4 E2Review.nb Skills: Use the information below to answer questions 11 - 15. There are 24 students enrolled in an introductory statistics class at a small university. As an in-class exercise the students were asked how many hours of television they watch each week. Their responses, broken down by gender, are summarized in the provided table. Assume that the students enrolled in the statistics class are representative of all students at the university. 11) If the parameter of interest is the difference in means, find a point estimate of the parameter based on the available data. 12) If the parameter of interest is the difference in means, use technology to construct a boot- strap distribution with at least 1,000 samples and estimate the standard error. 13) Construct a 95% confidence interval for the difference in the mean number of hours spent watching television for males and females at this university. Round the margin of error to two decimal places. 14) Suppose another class does the same in class exercise and gets a 95% confidence interval of -0.86 to 5.34 for the difference in the mean number of hours spent watching television for males and females at this university. Interpret this 95% confidence interval in the context of this data situation. 15) You wish to provide a 90% confidence interval for the difference in the mean number of hours spent watching television for males and females at this university based on a bootstrap distribution. Which percentiles would you use?  E2Review.nb 5 Use the information below to answer questions 16 - 20. November 6, 2012 was election day. Many of the major television networks aired coverage of the incoming election results during the primetime hours. The provided table displays the amount of time (in minutes) spent watching election coverage for a random sample of 25 U.S. adults. 16) Use the data from the sample to estimate the parameter of interest. Report your answer with two decimal places. 17) Use technology to construct a bootstrap distribution with at least 1,000 samples and estimate the standard error. 18) Use the estimate of the standard error to construct a 95% confidence interval for the mean amount of time (in minutes) U.S. adults spent watching election coverage on election night. Use three decimal places in your answer. 19) Interpret the 95% confidence interval found in (18). 20) Describe what would happen to the confidence interval if we increased the sample size from 25 adults to 50 adults. 6 E2Review.nb Use the information below to answer questions 21 - 23. 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. Count- ing 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. 21) Write a null and alternative hypothesis. 22) Use technology with 1000 samples to find the p-value. 23) Using a significance level of 5%, describe how we should interpret the null and alternative hypothesis. Use the information below to answer questions 24 - 26 The owner of a small pet supply store wants to open a second store in another city, but he only wants to do so if more than one-third of the city’s households have pets (otherwise there won’t be enough business). He samples 150 of the households and finds that 64 have pets. 24) Write a null and alternative hypothesis. 25) Use technology with 1000 samples to find the p-value. 26) Using a significance level of 1%, describe how we should interpret the null and alternative hypothesis.  E2Review.nb 7 Answers: 1) Notation Description  p Sample proportion μ Population mean ρ Population correlation coefficient x1 - x2 Sample mean difference r Sample correlation coefficient P Population proportion 2) C 3) A 4) B 5) D 6) Take a sample of size n from the population, resample with replacement from the sample n-times, find the proportion for each sample, use the sample proportions to generate a sampling distribution. 7) The margin of error is the distance from the sample statistic to the upper/lower boundaries of the confidence interval; for a 95% confidence interval, the margin of error is roughly 2 standard errors. 8) The distribution will be bell-shaped with a center targeting the population parameter. 9) The standard deviation is roughly the average distance from each data point to the mean, whereas the standard error is roughly the average distance from each sample statistic to the mean of sample statistics. 10) No, there is not sufficient evidence to support the claim there is a real difference in proportions since the confidence interval contains 0. 11) 2.09 12) SE = 1.511 13) -0.93 to 5.11 14) We are 95% confident that the difference in mean number of hours of TV for males and females at this university is between -0.86 and 5.34 hours. 15) 5th and 95th percentiles. 16) x(bar)=80.44 minutes 17) SE = 8.769 18) 62.902 to 97.978 minutes 19) Based on the sample of 25 adults, yielding a sample mean of 80.44 minutes of election coverage, we are 95% confident the population mean is between 62.902 and 97.978 minutes. 20) The width of the confidence interval would decrease. 8 E2Review.nb 21) Hypotheses: H0 : μ = 130 versus H1 : μ < 130 22) The sample statistic is 119.037 years with a p-value of 0.114. 23) Because 0.114 > 0.05, we would not reject the null hypothesis and thus we have no evi- dence to conclude that the average age of the driftwood on the beach is less than 130 years. 24) Hypotheses: H0 : p = 1/3 versus H1 : p > 1/3 25) The sample statistics is 0.426 with a p-value of approximately 0.004. 26) Conclusion should be consistent with p-value, though it should convey that the sample provides pretty strong evidence to reject the null hypothesis (since 0.004 < 0.01 and conclude that there is strong evidence that more than 1/3 of the households in this city have pets.