MATH220 Test 2 Fall Name. Section

Transcription

1 MATH220 Test 2 Fall 2014 Name Section This test has problems which are worth 100 points Show your steps in each problem to receive full or partial credit Note only writing down the final answer without showing the necessary steps will not receive full credit Formulas you may need: x x =, n s2 = z = x µ z = x µ / n ˆp ± z ˆp(1 ˆp) α/2 s x ± t α/2 n x ± z α/2 n n (x x) 2 n 1 n = ( z α/2 m )2, n = ˆp(1 ˆp)( z α/2 m )2 Note that you are not supposed to ask the instructor whether your solutions are correct or not or how to solve a problem during the test My work complies with JMU honor code Your signature: 1

2 1 (6 pts) (a) Find the z scores that bound the middle 82% of the area under the standard normal curve z 1 = 134 z 2 = 134 (b) Let a be a positive number such that the area to the right of z = a under the standard normal curve is b What is the area between z = a and z = a? The area is 1 2b (c) Match each of the three confidence intervals based on the same data with its confidence level (90%, 95%, or 99%): Put a confidence level after each interval (56, 144) 99% (72, 128) 90% (66, 134) 95% 2 (25 pts) Assume the heights of adult male giraffe follow a normal distribution with µ = 15 feet and = 16 feet (a) If a population of giraffes is moved to a park where the minimum height of leaves is 13 feet, what proportion will not be able to reach their food? P (x < 13) = P (z < ) = P (z < 125) = (b) Find the proportion of giraffes between 14 and 17 feel tall P (14 x 17) = P ( ) = = (c) Find a height that separates the top 10% of the giraffe heights from the rest p = 090, z = 128, x = µ + z = = feet (d) If a random sample of 10 giraffes are chosen, find the probability that the mean height of the 10 giraffes is below 13 feet P ( x < 13) = P (z < / ) = P (z < 395) <

3 3 (10 pts) The maximum weight an elevator can hold safely is 8120 pounds An alarm will be triggered if the maximum safe weight is exceeded According to a national report, the weights of adult US men have mean 194 pounds and standard deviation 68 pounds (a) If 40 people are on the elevator and their total weight is 8120 pounds What is their average weight? The average weight is 8120/40 = 203 pounds (b) If a random sample of 40 men ride the elevator, what is the probability that the alarm will be triggered? Let x indicate the average weight of a random sample of 40 men The alarm will be triggered if x > 203 P ( x > 203) = P (z > / ) = P (z > 084) = (20 pts) An industrial plant claims to discharge no more than 1000 gallons of wastewater per hour, on average, into a lake An environmental action group decides to monitor this plant A random sample of 5 hours is selected over a certain period The observations are: 2000,1000,3000,2000, 2000 (a) Find the sample mean x and the sample standard deviation s x = 2000, s = ( ) 2 +( ) 2 +( ) 2 +( ) 2 +( ) 2 = (b) Construct a 90% confidence interval for the mean discharge of wastewater per hour by this plant 3

4 x ± t 005 s/ n = 2000 ± / 5 = (13258, 26742) gallons (c) Based on the confidence interval, is the claim made by the plant plausible? We are 90% confident that the mean discharge of wastewater is between 1326 and 2674 gallons, so the claim made by the plant is not plausible 5 (25 pts) Flu shots are recommended for all health care providers, but fewer than half get vaccinated in a typical flu season This season was different A study of a nationally representative group of 1200 health care providers found that 56% of them had received a flu shot by mid January (a) Find a 90% confidence interval for the proportion who had received a flu shot by mid January among all health care providers 056 ± /1200 = 056 ± 0024 = (0536, 0584) (b) What is the margin of error of your confidence interval in (a)? The margin of error is 0024 (c) Suppose another survey is planned and it is desired that the margin of error of a 95% confidence interval not exceed 001, what sample size is needed to achieve this? n = ˆp(1 ˆp)( z 0025 m )2 = (196/001) 2 = (14 pts) According to the National Health Statistics Reports, a sample of 360 one-year-old baby boys in US had a mean weight of 255 pounds Assume the population standard deviation is = 53 pounds (a) Construct a 95% confidence interval for the mean weight of all one-year-old baby boys in US x ± z 0025 n = 255 ± = 255 ± 055 = (2495, 2605) pounds 4

5 (b) Based on your confidence interval, is it likely that the mean weight of all one-year-old baby boys is less than 28 pounds? We are 95% confident that the population mean is between 2495 and 2605 pounds, so it is very likely that it is less than 28 pounds 5