1. It's electric!
Here are 6 measurements of the electrical conductivity of a liquid: 5.32, 4.88, 5.10, 4.73, 5.15, 4.75.
The liquid is supposed to have a conductivity of 5.
The measurements above are an SRS from the population. The population has sd of 0.2
This liquid is used to help insulate circuits. As such, we need to make sure that it has an average conductivity of 5--if it does not, we cannot use it for the intended use. Run a hypothesis test to see if we can use this liquid for it's intended purpose.
2. Grapes of Wrath (alternately-- Brine and Prejudice)
Sulfur compounds cause "off-odors" in wine, so winemakers want to know the odor threshold--the lowest amount of sulfur a human nose can detect. For DMS (dimethyl sulfide), trained wine testers can detect it at levels of about 25 micrograms per liter. We want to see if the threshold for untrained noses is less sensitive. Here are the odor thresholds for 10 untrained students:
31 31 43 36 23 34 32 30 20 24
sd = 7
a. Give a 95% confidence interval for odor threshold of the students
b. run a hypothesis test checking to see if students have a higher threshold than experts.
3. Hemoglobin
Hemoglobin counts are measured in grams per 100 milliliters of blood. For adult females, the population is relatively normal with an mu of 14 and sigma of 2.5.
A woman has been tested regularly for high levels of hemoglobin. Here are her results:
15 18 16 19 14 12 14 17 15 11
a. Give a 95% confidence interval for the patient's hemoglobin levels.
b. Do a hypothesis test, looking to see if the woman's level of hemoglobin could be considered high at the 90% level.
4. Hail damage.
You work for an insurance company and are put in charge of determining rates for hail insurance--specifically wheat crops. Believe it or not, hail damage is a fairly large problem for crops--nationally, 11% of all wheat crops are destroyed by hail (I know, right?) You are going through some claims for a county in Colorado and get the following claims for percentage of destruction:
15, 8, 9, 11, 12, 20, 14, 11, 7, 10, 24, 20, 13, 9, 12, 5
Your job is to determine if there is more crop damage in this county than the normal average--if there is you are going to raise the rates for insurance for the following years. Assume at the national sd is 5%.
a. Give a null and alternative hypothesis.
b. Check for relative normality in the data before proceeding further. If there are any points that you feel you should take out, be VERY sure to explain why, as it may make people question your methods later on as you are reporting your findings.
c. In order to validate your raising rates, you need to be 99% confident that the damage here is more than the national average. Should you raise your rates? Explain.
5. Sun Spots
So . . . sun spots. We've looked for them for generations, and it has been believed that the mean number of sunspots in an average four week period is 41, with a standard deviation of 35. We've been looking at the numbers over the past four years (plus a little):
12.5 14.1 37.6 48.3 67.3 70.0 43.8 56.5 59.7 24.0 12.0 27.4 53.5 73.9 104.0 54.6 4.4 177.3 70.1 54.0 28.0 13.0 6.5 134.7 114 72.7 81.2 24.1 20.4 13.3 9.4 25.7 47.8 50.0 45.3 61.0 39.0 12.0 7.2 11.3
Why do we care? Well, it appears that when the average number of sunspots over a longish period of time is above the mean, it can produce times of general warming on the Earth. My question is: do we have enough evidence here to claim that we might be in for a time of warming on the Earth due to fluctuations in the sun? Give a null and alternative hypothesis, give an appropriate level of significance, etc. etc.
6. Normal Ladies (all the normal ladies)
Here are the IQ test scores of 31 seventh-grade girls in a Midwest school district. This is an SRS over the whole district. The sd of IQ scores is known at 15 for this population.
114 100 104 89 102 91 114 114 103 105 108 130 120 132 111 128 118 119 86 72 111 103 74 112 107 103 98 96 112 112 93
I am curious as to whether these girls could be considered average. State a Ho, a Ha, and go through the proceedings to determine whether or not these girls should be considered average in terms of IQ. (in case you are unaware, 100 is considered average)
7. Corny Data
In the United States, the mean yield of corn has been 120 bushels per acre. This year, 40 farmers gave their yields, with a mean of 123.8. The average yield has sd of 10 bushels.
a. Assume we want to prove that this sample shows the mean is higher than 120 this year. Write a null and alternative hypothesis.
b. Can you conclude that the mean this year is higher than 120? Show your numbers and explain your reasoning.
c. Give a 99% confidence interval for the mean yield this year.
8. Avalanche
Question Four:
Avalanches come in different types an styles, but a common one is a 'slab avalanche'. In Canada they have been extensively studied--they have a thickness in cm that is roughly N(67, 10.5). Scientists in Colorado are interested to see whether or not avalanches have similar characteristics to those in Canada--if they do, the research done in Canada can be used to explain certain snow phenomena in Colorado. If not, they will have to pay millions of dollars for their own research.
The scientists mark the thickness of the set of avalanches below:
59 51 76 38 65 54 49 62 68 55 64 67 63 74 65 79
a) Run specific hypothesis testing, and let the scientists know whether or not they can use the research from Canada.
---stop here. you do not need to do questions 9 and 10. this is as far as you need to go. to go further would be folly.---
9. Eye Grease
Athletes performing in bright sunlight often smear black eye grease under their eyes to reduce glare. Does the eye grease work? In one study, 16 students took a test of sensitivity to contrast after 3 hours facing into bright sun: once with eye grease and once without. Here are the differences in sensitivity, with eye grease minus without eye grease:
0.07 0.64 -0.12 -0.05 -0.18 0.14 -0.16 0.03 0.05 0.02 0.43 0.24 -0.11 0.28 0.05 0.29
Run a hypothesis test to see whether or not we think the eye grease helps minimize sensitivity.
Give a 99% confidence interval as to the efficacy of the eye grease.
10. M. Night Sham (a review question of important importance)
the .csv below is a list of the movies made by M. Night Shyamalan.
Create a plot of the movies comparing year and IMDB rating (from 0 to 10):
find a line of best fit
determine if a linear equation is appropriate (use correlation AND residual plot)
make a fantastic plot with all the goodies
which point has the largest residual?
If M. Night were to make a movie today, what do we expect the IMBD score to be?
Currently the worst movie on IMDB (http://www.imdb.com/chart/bottom) is "The Hottie and the Nottie" from 2008. It scored a 1.8. When do we expect M. Night to make a movie that bad?
Side note: there is actually a movie on there that is rated worse than "The Hottie and the Nottie", but after looking at the reviews, it is primarily because of how it portrays an aspect of history--and not for the actual merits of the film. That being said, if you prefer to use that movies score (1.4), please do.
In what year would N. Night have made a perfect movie? (score 10). Is this a reasonable answer?