Transcript File

Chapter 10 Review Day
Test next
class!!!
Section 10.1—Comparing Two
Proportions
• Choose independent SRSs of size n1 from Population 1 with
proportion of successes p1 and of size n2 from Population 2 with
proportion of successes p2. The sampling distribution of 𝑝1 - 𝑝2
has the following properties:
– SHAPE—approximately Normal if the samples are large enough that n1p1,
n1(1-p1), n2p2, and n2(1-p2) are all at least 10.
– CENTER—the mean is p1 – p2.
– SPREAD—As long as each sample is no more than 10% of its population (the
10% condition), then the standard deviation is
𝑝1(1−𝑝1)
𝑛1
+
𝑝2(1−𝑝2)
𝑛2
Confidence Intervals and Tests
• Confidence intervals and tests to compare the proportions p1 and
p2 of successes for two populations or treatments are based on the
difference 𝑝1 - 𝑝2 between the sample proportions.
• When the Random, Normal, and Independent conditions are met,
we can use two-sample z procedures to estimate and test claims
about p1 – p2.
• We can use a two-sample z interval for p1 – p2 to gain an
approximate level C confidence interval.
• Significance tests of Ho: p1-p2 = 0 use the POOLED (combined)
sample proportion 𝒑𝐂. Use a two-sample z test for p1 – p2 with
P values calculated from the standard Normal distribution.
Section 10.2—Comparing Two
Means
• Choose independent SRSs of size n1 from Population 1 and size
n2 from Population 2. The sampling distribution of 𝑥 1 - 𝑥 2 has the
following properties:
– SHAPE—Normal if both population distributions are Normal; approximately
Normal otherwise if both samples are large enough (n≥30) by the CLT.
– CENTER—the mean is μ1 – μ2.
– SPREAD—As long as each sample is no more than 10% of its population
(10% condition), its standard deviation is
𝜎21
𝑛1
+
𝜎22
.
𝑛2
Confidence Intervals and Tests
• Confidence intervals and tests for the difference of the means of
two populations or the mean responses to two treatments μ1 and
μ2 are based on the difference 𝑥 1 - 𝑥 2 between the sample means.
• If we somehow know the population standard deviations σ1 and
σ2, we can use a z statistic and the standard Normal distribution
to perform calculations.
• Since we almost never know the population standard deviations
in practice, we use the two-sample z statistic.
• This statistic does NOT have exactly a t distribution. There are
two options for using a t distribution to approximate the
distribution of the two-sample t statistic—technology or
conservative (using the smaller of the two degrees of freedom).
Using technology will give us a smaller/narrower interval and
smaller P-values.
Confidence Intervals and Tests
•
•
•
•
•
When the Random, Normal, and Independent conditions are met, we can
use two-sample t procedures to estimate and test claims about μ1 – μ2.
We can find an approximate level C confidence interval for μ1 – μ2 where
t* is the critical value for confidence level C for the t distribution with df
from either option 1 or option 2. This is called the two-sample t
interval for μ1 – μ2.
To test Ho: μ1 – μ2 = hypothesized value, use a two-sample t test for
μ1 – μ2. P-values are calculated using the t distribution with df from
option 1 or option 2.
The two-sample t procedures are QUITE ROBUST against
departures from Normality, especially when both sample/group
sizes are large.
DON’T use two-sample t procedures to compare means for
PAIRED DATA!!!
For each of the following, determine:
1)Would you use a one-sample t,
two-sample t, or paired t method?
2)Would you perform a hypothesis
test or find a confidence interval?
1. Random samples of 50 men and 50
women are asked to imagine buying
a birthday present for their best
friend. We want to estimate the
difference in how much they are
willing to spend.
Two-sample t confidence interval
2. Mothers of twins were
surveyed and asked how
often in the past month
strangers had asked
whether the twins were
identical.
One sample t test or interval
3. Are parents equally strict with
boys and girls? In a random
sample of families, researchers
asked a brother and sister from
each family to rate how strict
their parents were.
Paired-t test
4.Forty-eight overweight subjects are randomly
assigned to either aerobic or stretching exercise
programs. They are weighed at the beginning
and at the end of the experiment to see how
much weight they lost.
a) We want to estimate the mean amount of
weight lost by those doing aerobic exercise.
One sample t interval
b) We want to know which program is more
effective at reducing weight.
Two sample t test
5. A National Cancer Institute study published in 1991
examined the incidence of cancer in dogs. Of 827 dogs
whose owners used the weed killer 2-4-D on their lawns
or gardens, 473 were found to have cancer. Only 19 of
the 130 dogs that had not been exposed to this herbicide
had cancer. Construct a 95% confidence interval for the
difference in pets’ cancer risk.
5. STATE-PLAN-DO-CONCLUDE!
Check conditions!
We will perform a 2-Prop Z-interval
(0.35633, .49525)
We are 95% confident that the
interval from 36%-50% captures
the true difference in the rate of
cancer in pets exposed to 2-4-D.
6. Wegman’s (a food market chain) has developed a new
store-brand brownie mix. Before they start selling the mix
they want to compare how well people like their brownies to
brownies made from a popular national brand mix. In order
to see if there was any difference in consumer opinion,
Wegman’s asked 124 shoppers to participate in a taste test.
Each was given a brownie to try. Subjects were not told
which kind of brownie they got – that was determined
randomly. 58% of the 62 shoppers who tasted a Wegman’s
brownie said they liked it well enough to buy the mix,
compared to 66% of the others who said they would be
willing to buy the national brand. Does this result indicate
that consumer interest in the Wegman’s mix is lower than for
the national brand?
6. STATE-PLAN-DO-CONCLUDE!!
• State hypotheses! Check conditions!
• We will perform a 2-Prop Z test, (lower tail),
z=-0.926, p-value=0.177.
• Since p > α (0.177 > 0.05) we fail to reject the
Ho. There is not enough convincing evidence
of a difference in consumer opinion.
7. How quickly do synthetic fabrics such as
polyester decay in landfills?
A researcher buried polyester strips in the soil for
different lengths of time, then dug up the strips and
measured the force required to break them. Breaking
strength is easy to measure and is a good indicator of
decay. Lower strength means the fabric has decayed.
For one part of the study, the researcher buried 10
strips of polyester fabric in well-drained soil in the
summer. The strips were randomly assigned to two
groups: 5 of them were buried for 2 weeks and the other
5 were buried for 16 weeks. Here are the breaking
strengths in pounds:
Do the data give good evidence that the polyester
decays more in 16 weeks then in 2 weeks? Carry out
an appropriate test to help answer the question.
Group 1 (2
weeks):
118
126
126
120
129
Group 2 (16
weeks):
124
98
110
140
110
7. STATE-PLAN-DO-CONCLUDE!!
STATE: We want to perform a test at the α = 0.05 significance level of Ho:
μ1 – μ2 = 0 versus Ha: μ1 – μ2 >0, where μ1 is the actual mean breaking
strength at 2 weeks and μ2 is the actual mean breaking strength at 16 wks.
PLAN: Use a two-sample t test for the difference in the means if the
conditions are satisfied.
 Random: This is a randomized comparative experiment.
 Normal: Since n1 and n2 are both less than 30, we examine the data in a
graph. Do a dot plot to show that neither group displays strong
skewness or outliers.
 Independent: Due to random assignment, these two groups of pieces of
cloth can be considered as independent. Also, knowing one piece of
cloth’s breaking strength gives no information about the breaking
strength of another piece of cloth.
DO: From the data, n1 = 5, 𝑥 1 =123.8, S1 = 4.60,
n2 = 5, 𝑥2 = 116.4, S2 = 16.09. Using the conservative
df = 4, the test statistic is t = 0.989, and the P-value is
P( t > 0.989) = 0.1893.
CONCLUDE: Since the P-value is greater than 0.05, we fail to
reject the Ho. We do not have enough evidence to conclude
that there is a difference in the actual mean breaking
strength of polyester fabric that is buried for 2 weeks
and fabric that is buried for 16 weeks.
HOMEWORK!
• Read and review Chapter 10!
• To be collected on test day:
--This review sheet, completed
– Chapter 10 Review exercises, p. 661-664, for extra
credit
– Finish Webassign!