Confidence Intervals for Mean Weight and Mileage: Calculation and Interpretation, Assignments of Statics

Download Confidence Intervals for Mean Weight and Mileage: Calculation and Interpretation and more Assignments Statics in PDF only on Docsity! Homework 4 - solutions 1) Suppose you oversee a manufacturing line that produces steel washers. You have a machine that packs the washers in boxes that you label as 50 lbs. You know that the machine has a standard deviation of 2.5 lbs. You take a simple random sample of 75 boxes and find lbsX 5.52 . Calculate and interpret a 99% confidence interval for the true mean weight of the boxes of washers.   )53.245,51.755( )75/5.2(58.25.52 2   nzx  We are 99% confident that the true mean weight of the boxes of washers is between 51.755 lbs and 53.245 lbs. 2) An engineer is concerned about the weight of a particular bike frame (because of competitive cyclists who care about the weight of their bikes). She gathers a sample of 40 bike frames and weighs each bike frame. Suppose the mean weight is 4.25 lbs and the sample standard deviation is 0.5 lb. a) What assumptions motivate the development of the large-n confidence interval formula? The observations are assumed to be iid random variables. b) Give and interpret a 90% confidence interval for the average weight of a bike frame.   )38.4,12.4( )40/5.0(645.125.4 lbslbs nszx   We are 90% confident that the average weight of the bike frame is between 4.12 and 4.38 lbs. c) Give and interpret a 98% confidence interval for the average weight of a bike frame.   )43.4,07.4( )40/5.0(33.225.4 lbslbs nszx   We are 98% confident that the average weight of a particular bike frame is between 4.07 and 4.43 lbs. 3) An engineering student wonders how much he should pay for a textbook for an undergraduate statistics course. Consequently, the engineering student randomly selected 40 students who were enrolled in the statistics course last year and recorded how much each person paid for the textbook (rounded to the nearest $5). Below are the recorded amounts. Amount(dollars) Frequency 45.00 2 50.00 4 55.00 6 60.00 12 65.00 8 70.00 4 75.00 4 a) Give and interpret a 95% confidence interval for the mean price of the textbook. 94.7$ 61$   s x   )46.63$,54.58($ )40/94.7(96.161  nszx We are 95% confident that the average amount paid for the statistics textbook last year by an undergraduate student is between $58.54 and $63.46. b) Determine the size of the next sample such that a 95% confidence interval, based on information from the sample, has a width of $4. Use any previous information if necessary. 61 55.60 4 )94.7)(96.1(2 2 2 2                n n w z n  c) Use the five-step significance-testing format to assess the strength of the evidence collected to test whether students paid on average more or less than the manufacturer’s retail price of $64 for the textbook during the past year. 017.0]39.2[2]39.2|[| 39.2 4094.7 646164 64$: 64$:0         ZPZP ns x Z H H a   There is strong evidence to suggest that the average amount paid for the Stat 305 last year is different than the manufacturer’s retail price. 4) A person from the DNR wants to know the average weight of bucks (male deer) that have been killed by hunters this fall. The person went to four meat processing plants and obtained a random sample of 25 bucks from each plant. The mean and variance of each sample have been recorded below. 7) (data from Vardeman & Jobe, Prob 13, pp 430-431) M. Murphy recorded the mileages he obtained while commuting to school in his nine- year-old economy car. He kept track of the mileage for ten different full tanks of fuel, involving gasoline of two different octanes. Below are the data. 87 Octane: 26.43, 27.61, 28.71, 28.94, 29.30 90 Octane: 30.57, 30.91, 31.21, 31.77, 32.86 Assuming that the mileages are normally distributed, a) give and interpret a 90% confidence interval for the average mileage for the 87 octane.   )32.29,08.27( )5/1725.1(132.2198.28   nstx We are 90% confident that the true average mileage for the 87 octane fuel is between 27.08 and 29.32 miles. b) What value gives the center of your confidence interval? 198.28x gives us the center of the confidence interval. c) What three values affect the width of your confidence interval? For each of these, determine whether the value would need to be increased or decreased if we wanted to decrease the width of our confidence interval (holding all else constant). Our sample size, n, would need to be increased for our interval width to be decreased. Our distributional constant, t, would need to decrease for our interval width to be decreased. (Note: This is equivalent to saying we need to decrease our confidence level) The sample standard deviation, s, would need to decrease if our interval width were to decrease. d) What is meant by “90% confident”? If we were to repeat to process of taking a sample and constructing a 90% confidence interval over and over again, we would expect approximately 90% of the intervals to contain the true mean. e) use the five-step significance-testing format and assess the strength of the evidence provided by the data that the average mileage for the 90 octane differs from 30 mpg. 05.002.0 652.3 58963.0 30464.31 30: 30:0         p ns x T H H a    There is evidence to suggest that the average mileage for 90 octane is not equal to 30 mpg. f) Based on your answer to part e), if you were to create a 90% confidence interval for the average mileage for the 90 octane, would it contain 30 mpg? What about a 99% confidence interval? All values in a 90% confidence interval, when set as the mean under the null hypothesis, would result in a p-value of greater than 0.1. Similarly, all values in a 99% confidence interval, when set as the mean under the null hypothesis, would result in a p-value of greater than 0.01. Since our p-value in part e) was 05.002.0  p , the value of 30 mpg would NOT be contained in the 90% confidence interval, but would be contained in the 99% confidence interval.