probability of normal continuous distributions

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MTH 34200 – HW 08 (20 points) – Due Tuesday, 10/23 Name_______________________________________ 1. A sample of 16 adult red-necked wallabies in a certain region is collected and weighed. The mean weight of the wallabies in the sample is 15.6 kg. The standard deviation of the weights of the wallabies in the sample was 2.3 kg. Find a 99% confidence interval for the true mean weight of the population of wallabies in this region. 2. A city official would like to estimate the mean ACT score for high school seniors in her city by drawing a sample of seniors and calculating the mean score for the sample. She has reason to believe that the scores are normally distributed and that the standard deviation for scores obtained by students in her city is the same as the nationwide standard deviation during that year, which was 4.7. Assuming that the official would like to be 95% confident in her estimation with a margin of error of 0.5 points or less, what is the smallest same size that she should use? 3. Researchers would like to investigate whether males and females earn comparable wages in a certain industry. Samples of 18 men working in the industry and 23 women working in the industry are collected. The salaries within the sample of men had a mean of $45,000 and a standard deviation of $3250. The salaries within the sample of women had a mean of $43,750 and a standard deviation of $3000. Assume that the salaries are normally distributed for both populations, and that the population standard deviations are equal. Construct a 95% confidence interval for the amount by which the true mean salary for men in this industry is greater than the true mean salary for women in the industry. 4. A lightbulb factory manufactures lightbulbs using two different types of filaments, which we will call Type A and Type B. An engineer at the factory is studying the lifetimes of bulbs with the two types of filaments. Let 𝑋1 be the lifetime of a randomly selected bulb with a Type A filament and let 𝑋2 be the lifetime of a randomly selected bulb with a Type B filament. Both variables are measured in days. It is known from prior studies that 𝑉𝑎𝑟[𝑋1 ] = 0.8 ⋅ 𝑉𝑎𝑟[𝑋2 ]. Assume that 𝑋1 and 𝑋2 are normally distributed. A sample of 18 bulbs with Type A filament is collected. The lifetime of bulbs in this sample has a mean of 132 days and a standard deviation of 14 days. A sample of 14 bulbs with Type B filament is collected. The lifetime of bulbs in this sample has a mean of 114 days and a standard deviation of 16 days. Find a 99% confidence interval for 𝜇1 − 𝜇2 . MTH 34200 – HW 09 (20 points) – Due Thursday, 10/25 Name_______________________________________ 1. An economist is interested in estimating the mean salary of full-time workers in a certain metropolitan area. She collects a sample of 329 such workers. The annual salary of workers in the sample had a mean of $37,420 and a standard deviation of $4350. Find an approximate 95% confidence interval for the true mean annual salary of full-time workers in the area. Show your work, and round your answers to the nearest dollar. 2. A pollster has conducted a survey to determine the level of support for a ballot initiative in an upcoming city election. Of the 1743 residents surveyed, 959 indicated that they supported the initiative. Find an approximate 99% confidence interval for the true proportion of residents who support the ballot initiative. Show your work, and round your answers to four decimal places. 3. A recent survey asked participants about their political affiliations, as well as if they watch any of the three major cable news networks (CNN, Fox News, and MSNBC) at least once a week. Of those polled, 1688 identified themselves as Democrats and 1532 identified themselves as Republicans. The following table details how many members of each group stated that they watch a given network at least once a week. Democrats Republicans CNN 1065 359 Fox News 567 1216 MSNBC 730 217 Find an approximate 99% confidence interval for the difference in the proportion of Democrats who watch CNN on a weekly basis and the proportion of Republicans who watch CNN on a weekly basis. Show your work, and round your answers to four decimal places. 4. A biologists collects a sample of 42 adult yellow-bellied marmots living on a certain mountain in Colorado. The weight of the marmots in the sample had a mean of 8.8 lbs and a standard deviation of 0.72 lbs. Assume that the weights are normally distributed. Find a 95% confidence interval for the population variance of the weight of adult marmots living on the mountain. Show your work, and round your answers to four decimal places. ...
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