Homework 6
Show all appropriate work
1) A 95% confidence interval for the average lifetime of a cellphone was (1.89, 2.78) years.
Which of the following are statistically valid statements?
______ 95% of all cellphones will last between 1.89 and 2.78 years
___X__ We are 95% confident the average lifetime for cell phones is between 1.89 and 2.78 years
______ 95% of the time the average cell phone lifetime is between 1.89 and 2.78 years
______ There is a 95% probability the average lifetime for cellphones is between 1.89 and 2.78 years
______ The average lifetime for cellphones is between 1.89 and 2.78 years 95% of the time
______ A new confidence interval has a 95% probability of being 1.89 to 2.78 years
______ 95% of all studies would have a confidence interval between 1.89 and 2.78
___X___ There is a 95% probability the next confidence interval will capture the true population average
______ There is a 95% probability that the next cellphone will last between 1.89 and 2.78 years
___X___ If we did 100 confidence intervals about 95 of them would capture the true average
___X___ 95% of confidence intervals done this way will correctly capture the true average
______ If we did many similar studies, 95% would have a sample mean between 1.89 and 2.78 years
___X___ 95% of the time the sample average will be within a margin of error of the true average
___X___ The true average lifetime of a cellphone is between 1.89 and 2.78 years with 95% confidence
___X___ The probability that the true average is between 1.89 and 2.78 years is either 0 or 1
______ It is completely impossible to ever actually understand what confidence really means
2)
The wind on any random day in Laramie is normally distributed with a standard deviation of 5.1
mph. A sample of 25 random days in Laramie had an average of 17mph. Find a 92% confidence
interval to capture the true average wind speed in Laramie.
(15.215, 18.785)
3) I think the homework is supposed to take 4 hours a week. I randomly select 40 students and
measure how long it takes them to do a random homework. I get an average of 3.25 with a
standard deviation of 2.4 hours. Test at the 5% significance level if the homework takes the
right amount of time.
H0: mu=4
HA: mu ne 4
Alpha = 0.05
Z=(3.25-4)/(2.4/sqrt(40)) = -1.976
Df=39
p-value = 2*.0239 = .0478
Reject
The homework is different from 4 hours on average
4) It just so happens that the average weight of an engineering student is 185 pounds. Dr. Ogden
doesn’t know this, so he decides to randomly sample many engineering students and he gets
the following 95% confidence interval: (167.22, 197.30). What is the probability that his 95%
confidence interval captured the true population average? 1
5) You are planning on applying to the Niart Station as an engineer. They are a massive company
that hires millions of engineers around the world. You know the standard deviation for the
salaries is $12 k. To find the average you sample 15 random employees and get an average
salary of $72 k. Find an 80% confidence interval for the true average salary of Niart employees.
The sample size is not large enough to do this problem
6) Ever wondered how much force it takes to punch through a sheetrock wall? MissBusters did a
study by assuming the distribution was normal. They randomly selected 5 walls and found it
took an average of 3.22 Newtons with a standard deviation of 0.81 Newtons to break through a
standard sheetrock wall. What is the 99% confidence interval for the force needed to break
through the wall?
3.22 +- 2.576*.81/sqrt(5) = (2.286,4.153)
7) You are going to estimate how long a person pushes a button by randomly selecting people and
asking them to push a button (which times how long the button is pressed). You practiced on
the people at work and found a standard deviation of about 0.92 seconds. You want to get a
98% confidence interval that is only 0.2 in width. How many people do you need to have in your
study? 460
8) The 95% confidence interval for the average lifetime of light bulbs is (166, 198) hours.
Which of the following is correct?
A) 95% of all light bulbs will last between 166 hours and 198 hours
B) There is a 95% probability the true average lifetime is between 166 and 198 hours
C) If we did 100 samples, about 95 of them would correctly capture the true average
D) The true average is between 166 and 198 about 95% of the time
E) We are 95% confident the next light bulb will last between 166 and 198 hours
9) Two civil engineers are arguing about the tensile strength of spider silk. Ann is convinced the
tensile strength is 1000 MPa, while Bob is certain it is 1100 MPa. Fortunately both of them
agree the standard deviation is 200 Mpa. To settle the argument Ann decides to take 40
random strands of spider silk and test the average strength. She will test with α=0.01 whether
the spider silk is significantly larger than 1000. Bob is frustrated because he thinks her test does
not have the power to show her average is wrong. Calculate the power for Bob.
Xcritical = 1073.566, power = 0.7967
10) The airport security team does random checks of people waiting in the security line. They
assume the person is not carrying any weapons, but to gather data they scan them with x-rays,
pat them down and watch their body language. If they decide the person is a threat he/she will
go into the next room for an interrogation. What alpha would you suggest they use, and why?
Two possible answers (note H0: the person is not a threat)
Small alpha because we all hate being delayed when we’re trying to get on a plane
Large alpha because we don’t want a threat on our plane
11) A study plans to investigate the average time a 16 year old spends playing video games each
week. From studies of similar age groups it is believed the standard deviation should be near
8.1 hours. The study wants a 90% confidence interval to have a margin of error of 0.5 (the
nearest half hour). How many 16 year olds are they going to need to sample?711
12) Your company requires drug tests. They test if the average drug levels equal zero. If you fail
you’re fired. Your friend tells you “Man! I’m really worried about a Type I Error here, but dude
I like totally don’t worry about a Type II Error at all.”
Does your friend do drugs? (explain how you know)
H0: no drugs. Type 1 error means you are fired for drugs when you don’t do any drugs.
No, your friend does not do drugs.
13) You are the captain on a luxury yacht in the middle of the Caribbean when you notice a small
boat bobbing along in your general direction. You are obligated to assume they are castaways
and go pick them up, unless you have reason to suspect they might be pirates.
a. What is a confident captain?
Knows when there are castaways that should be helped
b. What is a powerful captain?
Knows when thar be pirates Aaaargh!
14) The distribution for the length of a person’s tongue is normally distributed. A random sample of
4 people had an average of 6.3 inches with a standard deviation of 1.2 inches. Find a 95%
confidence interval for the true average length of a tongue.
(5.124, 7.476)
15) I have read enough papers on the subject of human iron levels that I believe the distribution is
normal with standard deviation of iron level of 1.42. If a sample of 12 men had an average iron
level of 1.783, what would the 99% confidence interval be?
Z=2.58: (0.725, 2.841)
16) Z0 = 1.645
Xcritical = 11.139
Za = -1.2427
Power=0.8925