1: Health Care workers who use latex gloves with glove powder on a daily basis are particularly susceptible to developing a latex allergy. Each in a sample of 48 hospital employees who were diagnosed with a latex allergy based on a skin-prick test reported on their exposure to latex gloves. Summary statistics for the number of latex gloves used per week are x = 19.6 and s = 12.2. Complete parts (a) - (d). a) Give a point estimate for the average number of latex gloves used per week by all health care workers with a latex allergy. b) Form 95% confidence interval for the average number of latex gloves used per week by all health care workers with a latex allergy. c) Give a practical interpretation of the interval, part (b). d) The conditions required for the interval, part (b) to be valid are shown below. Answer to question (d) is: .I. The sample selected was randomly selected from the target population .......... .ii. The sample size is sufficiently large, that is, n>30. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 2: The white wood material used for the roof of an ancient temple is imported from a certain country. The wooden roof must withstand as much as 100 centimeters of snow in the winter. Architects at a university conducted a study to estimate the mean bending strength of the white wood roof. A sample of 25 pieces of the imported wood were tested and yielded the statistics x = 74.2 and s = 10.8 on breaking strength of the white wood with a 99% confidence interval. Interpret the result. a) Find the 99% confidence interval for the true mean breaking strength of the white wood. (Round to two decimal places as needed). ( ), ( ) :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 3: A group of researchers wants to estimate the true mean skidding distance along a new road in a certain forest. The skidding distances (in meters) were measured at 20 randomly selected road sites. These values are given in the accompanying table. Complete parts a through d. a) Estimate the true mean skidding distance for the road with 99% confidence interval. Round answer to one decimal place Answer: ( ), ( ) b) Give a practical interpretation of the interval, part (a) Answer to (b): In general, we have a certain level of confidence that the population mean falls within the computed confidence interval. In this case, the population mean is the mean of the skidding distance for the whole road. c) What conditions are required for the inference, part b, to be valid? Are these conditions reasonably satisfied? Answer to (c): There are two conditions for the small sample confidence interval. The sample must be random and the population distribution must be approximately normal. The problem indicates that the sample is random. If the sample distribution is approximately normal, then it is reasonable to assume that the population distribution is approximately normal. Use a histogram of the sample to determine if it is approximately normal. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 4: In sociology, a personal network is defined as the people with whom you make frequent contact. A research program used a stratified random sample of men and women born between 1908 and 1937 to gauge the size of the personal network of older adults. Each adult in the sample was asked to \"please name the people you have frequent contact with and who are also important to you.\" Based on the number of people named, the personal network size for each adult was determined. The responses of 2,817 adults in this sample yielded statistics on network size that is, the mean number of people named person was 14.2, with a standard deviation of 9.9 Complete parts a through c. a) Form a 95% confidence interval for the mean personal network size of all older adults. (Round to the nearest hundredths as needed) Answer: ( ), ( ) b) Give a practical interpretation of the interval found in part (a). (Round to the nearest hundredths as needed) Answer: ( ), ( ) c) The personal network size of all older adults is known to have a highly skewed distribution. Does this impact the validity of the inference, found in part (b)? Answer to part C: A statement of confidence reflects the confidence in the estimation process rather than in the particular interval that is calculated from the sample data, consider how this affects the validity of the confidence interval. Recall that the Central Limit Theorem states that date will be approximately normally distributed for samples of size n>30 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 5: A newspaper reported that 60% of people say that some coffee shops are overpriced. The source of this information was a telephone survey of 200 adults. a) Identify the population of interest in this study Answer to (a): A population is a set of units (usually people, objects, transactions, or events) that is of interest in a study. The population of interest for this study is adults. b) Identify the sample for the study Answer to (b): A sample is a subset of the units of a population. The sample for this study is the 200 adults that were surveyed c) Identify the parameter of interest in the study Answer to (c): A parameter is a numerical descriptive measure of a population. The parameter of interest, or target parameter, is the unknown population parameter that a study is interested in estimating. The parameter of the interest for this study is "p" the population proportion of adults who say that some coffee shops are overpriced. d) Find and interpret a 95% confidence interval for the parameter of interest Answer: The 95% confidence interval for "p" is approximately ( ), ( ) :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 6: A random sample of 1000 satellite radio subscribers was asked, \"Do you have a satellite radio receiver in your car?\" The survey found that 200 subscribes did, in fact, have a satellite receiver in their car. a) From the sample, calculate an estimate of the true proportion of satellite radio subscribers who have a satellite radio receiver in their car. Answer: ** Type an integer or the decimal for the answer. ^p = b) Form a 95% confidence interval for the estimate, part (a) Answer: 95% confidence interval is: ( ), ( ) c) Give an interpretation of interval, part (b) Answer: What is the confidence Interval?? ( ), ( ) Remember: A confidence interval is an interval of values used to estimate the true value of a population parameter. In general, the confidence level for an interval refers to the confidence that the interval contains the true value of the parameter. Use this information to interpret the confidence interval. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 7: In 2006, a survey of 100 adults in a region found that 50% had access to a high-speed Internet connection at home. a) Give a point estimate of the true percentage of adults in the region that had access to a high-speed Internet connection at home. Answer: Type in integer or decimal form (round by 2 decimals) ^p = b) Construct a 95% confidence interval for the estimate, part (a), and interpret the result. Answer: 95% of confidence interval for (p) is: ( ), ( ) c) In 2005, a similar survey in the region found that 36% had access to a high speed Internet connection at home. Is there evidence to conclude that the percentage has increased since 2005? Explain. Hint: **COMPARE the confidence interval found in art (b) to the percentage given in part (c). Decide if this provides evidence that the percentage has increased.** Answer: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 8. A gigantic warehouse stores approximately 50 million empty aluminum beer and soda cans. Recently, a fire occurred at the warehouse. The smoke from the fire contaminated many of the cans with black spot, rendering them unusable. A statistician was hired by the insurance company to estimate p, the true proportion of cans in the warehouse that were contaminated by their fire. How many aluminum cans, should be randomly sampled to estimate p to within 0.09 with 95% confidence? Answer: The statistician should sample ( 0.09 with 95% confidence. ) .... Cans to estimate the proportion that was contaminated by the fire to within :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 9. According to an article the bottled water you are drinking may contain more bacteria and other potentially carcinogenic chemicals than are allowed by state and federal regulations. O the more than 1400 bottles studied, nearly one-third exceeded government levels. Suppose that a department wants an updated estimate of the population proportion of bottled water that violates at least one government standard. Determine the sample size (number of bottles) needed to estimate this proportion to within +/- 0.01 with 99% confidence. Answer: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 10: It costs you $12 to draw a sample of size n=1 and measure the attribute of interest. You have a budget of $3000. Complete parts (a) and (b) below. a) Do you have sufficient funds to estimate the population means for the attribute of interest with a 95% confidence interval 4 units in width? Assume Standard Deviation Sigma = 16. Answer:...?? Part A: 1) Yes, because the required sample size is ( ) ... OR ... 2) No, because the required sample size is ( ) Part B: If you used 90% confidence level, would your answer to part (a) change? Explain