Transcript AP Statistics - University Academy

AP Statistics
Chapter 9
Scavenger Hunt Review
Power
your goal and objective is to be able to REJECT the null hypothesis ... in
the case of a simple experiment perhaps ... to show that the new
method works better than the old method.
"power" is being able to REJECT the null when the treatment really
makes a difference.
So, we want to design the study and set the study conditions
and implement the study in such a way as to give yourself a good
chance of being able to REJECT the null where there really is an effect.
I think this is the best place to focus on that 2 by 2 table of:
Reality .... Null is true ... Null is false
What you do ... Retain the null ... Reject the null
What we hope to do in the study is to REJECT the null when the reality
is that the null is false.
What impacts the power of your test?
1) The effect size: the greater the improvement, the more likely
you are to notice. If your new treatment raises the cure rate to
90%, that should be pretty obvious. But if it only improves relief to
65%, you're probably not going to spot that. Larger effect = more
power. (And less chance of overlooking the improvement = Type II
error.)
2) The sample size: the more people who try your treatment, the
better your chances of detecting an improvement. Testing the new
drug on 1000 people makes you more likely to spot an effect than
if you had only 20 people try it. Larger sample = more power.
3) Your convinceability (if that's a word): the weaker the evidence
you are willing to accept, the more easily you'll notice an
improvement. If you are willing to believe marginal evidence, you'll
spot a higher cure rate more easily than if you insist on compelling
proof. Easier to convince = more power. (But it also makes you
more vulnerable to thinking the new treatment is better when it
really isn't = Type I error.)
Increasing Power
• (if you increase alpha, increase sample size, decrease sigma is
in our book, but I've always wondered how you can change
sigma, and having an alternative hypothesis farther away from
the null hypothesis.)
Directions
• Print slides and mount on the walls in your
room, but in a random order.
• Assign students to start on a different
problem.
• Answer to previous problem is at top of the
next slide.
0.0423
• State the null hypothesis and alternative hypothesis
• During the winter months, the temperatures at the
Colorado cabin owned by the Starnes family can stay
well below freezing for weeks at a time. To prevent the
pipes from freezing, Mrs. Starnes sets the thermostat at
50o F. The manufacturer claims that the thermostat
allows variation in home temperature of 3o F. Mrs.
Starnes suspects that the manufacturer is overstating
how ell the thermostat works.
Null: Sigma=3
Alternative: Sigma>3
• A Gallup Poll report on a national survery
of 1028 teenagers revealed that 72% of
teens said they rarely or never argue with
their friends. Yvonne wonders whether
this national result would be true in her
large high school, so she surveys a
random sample of 150 students at her
school. She finds that 96 students seldom
or rarely argue with friends. What is the
test statistic and p-value?
-2.18, .0292
• A state’s DMV claims that 60% of teens
pass their driving test on the first attempt.
An investigative reporter examines an
SRS of the DMV records for 125 teens; 86
of them passed the test on their first try. Is
this good evidence that the DMV’s claim is
incorrect? Do we accept or reject the null
at a .05 level?
Reject
• How well materials conduct heat matters when designing
houses, for example. Conductivity is measured in terms
of watts of heat power transmitted per square meter of
surface per degree Celsius of temperature difference on
the two sides of the material. In these units, glass has
conductivity about 1. The NIST provides exact data on
properties of materials. Here are measurements of the
heat conductivity of 11 randomly selected pieces of a
particular type of glass: 1.11, 1.07, 1.11, 1.07, 1.12,
1.08, 1.08, 1.18, 1.18, 1.18,1.12. Is there convincing
evidence that the conductivity of this type of glass is
greater than 1. What is the p-value? Do we accept or
reject the null?
0, Reject
• For students with special preparation, SAT Math scores
in recent years have varied Normally with mean of 518.
One hundred students go through a rigorous training
program designed to raise their SAT Math scores by
improving their mathematics skills. Carry out a test for
the null=518, the alternative being greater than 518. If
the sample has a mean of 536.7 and a std. dev. Of 114,
is the result significant at the 5% level?
No
• Radon is a colorless, odorless gas that is naturally released by
rocks and soils and may concentrate in tightly closed houses.
Because radon is slightly radioactive, there is some concern
that it may be a health hazard. Radon detectors are sold to
homeowners worried about this risk, but the detectors may be
inaccurate. University researches placed a random sample of
11 detectors in a chamber where they were exposed to 105
picocuries per liter of radon over 3 days. A graph of the radon
readings from the 11 detectors show no strong skewness or
outliers. The minitab output shows the results of a one-sample
t interval. Is there significant evidence at the 10% level that the
mean reading differs from the true value 105?
n  11, mean  104.82, stdev  9.54,
SEmean  2.88,90%CI :  99.61,110.03
Stop Here
• Stop Here
0.8351
• A promoter knows that 23% of males
enjoy watching boxing matches. In a
random sample of 125 men, what is the
distribution, mean and standard deviation
of the sample?
Approx normal with
mean = 0.23, std dev = 0.38
• A promoter knows that only 12% of
females enjoy watching boxing matches.
In a random sample of 125 women, what
is the probability that more than 10% of
the females enjoy watching boxing
matches?
0.7548
• The average number of missed school
days for students in public schools is 8.5
with a standard deviation of 4.1. In a
sample of 200 public school students,
what is the probability that the average
number of days missed is less than 8
days?
Answers
• Adult women & time for themselves:
– Approx normal, μ= .47, σ = .016
– Np = 1025*.47 = 481.75
– Nq = 1025 *.53 = 543.25
• Internet access fees
– Approx normal with μ= 28, σ = 0.447
– P(x-bar > 29) = 0.0126
Answers
• Mail order company
– Approx normal with μ= .9, σ = .03
– Np = .9 * 100 = 90, nq = .1 * 100 = 10
– P(p-hat ≤ .86) = 0.0912
• Age of cars
– Approx normal, mean = 4, std dev = 0.11
Answers
• Car Dealership service
– Approx normal with mean = 0.61, std dev =
0.049
– Np = 61, nq = 39
• College teacher incomes
– P(x-bar ≥ 40,000) = 0.228 with std dev = 2500
Answers
• Paper strength
– Std dev = .36
– P(24.5 ≤ x – bar ≤ 25.5) = .8351
• Males & boxing matches
– Approx normal with mean = .23, std dev = 0.38
– Np = .23 * 125 = 28.75
– Nq = .77 * 125 = 96.25
Answers
• Females & boxing matches
– Approx normal with mean = .12, std dev =
.029
– Np = 125 * .12 = 15
– Nq = 125 *.88 = 110
– P(p=hat ≥ 0.1) = .7548
• Missed school days
– Std dev = .29
– P(x-bar < 8) = .0423