confidence intervals and hypothesis tests with proportions
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Confidence Intervals and Hypothesis tests with Proportions. What happens to your confidence as the interval gets smaller?. Your confidence level decreases with smaller intervals. %. %. %. %. Confidence level. Is the success rate of the method used to construct the interval - PowerPoint PPT PresentationTRANSCRIPT
Confidence Intervals and Hypothesis tests
with Proportions
What happens to your confidence as the interval gets smaller?
Your confidence level decreases with smaller intervals
%%
%%
Confidence level• Is the success rate of the method
used to construct the interval
• Using this method, ____% of the time the intervals constructed will contain the true population parameter
• Found from the confidence level• The upper z-score with probability p lying to
its right under the standard normal curve
Confidence level tail area z*.05 1.645.025 1.96.005 2.576
Critical value (z*)
.05
z*=1.645
.025
z*=1.96
.005
z*=2.57690%95%99%
Confidence interval for a population proportion:
npp 1p̂ *z
Statistic + Critical value × Standard deviation of the statistic
Margin of error
npp ˆ1ˆ
But do we know the population proportion?
What are the steps for performing a confidence interval?
1.) Assumptions• SRS of context• Approximate Normal distribution because
np > 10 & n(1-p) > 10• Population is at least 10n
2.) Calculations
3.) Conclusion
We are ________% confident that the true proportion context is between ______ and ______.
• As the confidence level increases, do the intervals generally get wider or more narrow? Explain.
• As the sample size increases, do the intervals generally get wider or more narrow? Explain.
• When 100 confidence intervals are generated, why are they all different?
• If the confidence level selected is 90%, about how many of 100 intervals will cover the true percentage of
orange balls? Will exactly this number of intervals cover the true percentage each time 100 intervals are
created? Explain.
A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost.
Assumptions:•Have an SRS of adults•np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44 Since both are greater than 10, the distribution can be approximated by a normal curve•Population of adults is at least 10,120.
41,.35.1012)62(.38.96.138.1*ˆ
nppzP
We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.
Step 1: check assumptions!
Step 2: make calculations
Step 3: conclusion in context
The manager of the dairy section of a large supermarket took a random sample of 250 egg cartons and found that 40 cartons had at least one broken egg. Find a 90% confidence interval for
the true proportion of egg cartons with at least one broken egg.
Assumptions:•Have an SRS of egg cartons•np =250(.16) = 40 & n(1-p) = 250(.84) = 210 Since both are greater than 10, the distribution can be approximated by a normal curve•Population of cartons is at least 2500.
198,.122.250)84(.16.645.116.
We are 90% confident that the true proportion of egg cartons with at least one broken egg is between 12.2% and 19.8%.
Step 1: check assumptions!
Step 2: make calculations
Step 3: conclusion in context
Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?
To find sample size:
However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!
nppzm 1*
Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?
60125.600
25.96.104.
5.5.96.104.
5.5.96.104.
1*
2
nn
n
n
nppzm
Use p-hat = .5
Divide by 1.96
Square both sides
Round up on sample size
What are hypothesis tests?
Calculations that tell us if the sample statistics (p-hat) occurs by random chance or not OR . . . if it is statistically significantIs it . . .
–a random occurrence due to natural variation?
–an occurrence due to some other reason?
Statistically significant means that it is NOT a random chance
occurrence!
Is it one of the sample
proportions that are likely to
occur?Is it one that isn’t likely to
occur?
These calculations (called the test statistic) will tell
us how many standard deviations a sample
proportion is from the population proportion!
Steps:1) Assumptions2) Hypothesis statements &
define parameters3) Calculations4) Conclusion, in context
Assumptions for z-test:
• Have an SRS of context• Distribution is (approximately)
normal because both np > 10 and n(1-p) > 10
• Population is at least 10n
YEA –These are the same
assumptions as confidence intervals!!
How to write hypothesis statements• Null hypothesis – is the statement
(claim) being tested; this is a statement of “no effect” or “no difference”
• Alternative hypothesis – is the statement that we suspect is true
H0:
Ha:
How to write hypotheses:Null hypothesis H0: parameter = hypothesized value
Alternative hypothesis Ha: parameter > hypothesized value Ha: parameter < hypothesized value Ha: parameter = hypothesized value
Facts to remember about hypotheses:• Hypotheses ALWAYS refer to
populations (use parameters – never statistics)
• The alternative hypothesis should be what you are trying to prove!
• ALWAYS define your parameter in context!
Activity: For each pair of hypotheses, indicate which are not legitimate & explain why
1H1H e)6H4H d)1H1H c)
123H123H b)15H15H a)
a0
a0
a0
a0
a0
.ˆ:;.ˆ:
.:;.:.:;.:
:;::;:
pppp
xx
Must use parameter (population) x is a statistics
(sample)
is the population proportion!Must use same
number as H0!P-hat is a statistic – Not a parameter!
Must be NOT equal!
P-value -• Assuming H0 is true, the
probability that the statistic would have a value as extreme or more than what is actually observed
Notice that this is a conditional probability
The statistic is our p-hat!
Why not find the probability that the p-hat equals a certain value?
Remember that in continuous distributions, we cannot find
probabilities of a single value!
P-values -• Assuming H0 is true, the
probability that the statistic would have a value as extreme or more than what is actually observedIn other words . . . What
is the probability of getting values more (or
less) than our p-hat? p̂p̂
We can use normalcdf to find this probability.
Level of significance - • Is the amount of evidence
necessary before we begin to doubt that the null hypothesis is true
• Is the probability that we will reject the null hypothesis, assuming that it is true
• Denoted by a–Can be any value–Usual values: 0.1, 0.05, 0.01–Most common is 0.05
Statistically significant –• Our statistic (p-hat) is statistically
significant if the p-value is as small or smaller than the level of significance (a).
Decisions:• If p-value < a, “reject” the null
hypothesis at the a level.• If p-value > a, “fail to reject” the null
hypothesis at the a level.
Our “guilty” verdict.Our “not guilty” verdict.
Remember that the verdict is never “innocent” – so we can never decide
that the null is true!
Facts about p-values:• ALWAYS make the decision about
the null hypothesis!• Large p-values show support for
the null hypothesis, but never that it is true!
• Small p-values show support that the null is not true.
• Double the p-value for two-tail (≠) tests
• Never accept the null hypothesis!
Never “accept” the null hypothesis!Never “accept” the null
hypothesis!
Never “accept” the null hypothesis!
Calculating p-values• For z-test statistic (z) ––Use normalcdf(lb,ub) to find the probability of the test statistic or more extreme
–Remember the standard normal curve is comprised of z’s where = 0 and s = 1
We will see how to compute this value
tomorrow.
Since we are in the standard normal curve,
we do not need , s here.
Writing Conclusions:1) A statement of the decision
being made (reject or fail to reject H0) & why (linkage)
2) A statement of the results in context. (state in terms of Ha)
AND
“Since the p-value < (>) a, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha.” Be sure to write Ha in
context (words)!
Formula for hypothesis test:statistic - parameterTest statistic
SD of parameter
z npppp
1ˆ p̂ pp ˆ
Example 5: A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract?
Assumptions:•Have an SRS of people•np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater than 10, this distribution is approximately normal.•Population of people is at least 4000.
H0: p = .2 where p is the true proportion of people whoHa: p > .2 heard the ad
05.1056.25.1
400)8(.2.2.225.
avaluepz
Since the p-value > a, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than .2. The company will not renew their advertising contract with the radio station.
Use the parameter in the null hypothesis to check assumptions!
Use the parameter in the null hypothesis to calculate standard
deviation!
Calculate the appropriate confidence interval for the above problem.CI = (.19066,.25934)How do the results from the confidence interval compare to the results of the hypothesis test?The confidence interval contains the parameter of .2 thus providing no evidence that more than 20% had heard the ad.
Two-Sample Proportions
Inference
Assumptions:• Two, independent SRS’s from
populations ( or randomly assigned treatments)
• Populations at least 10n• Normal approximation for both
1 1
1 1
101 10
n pn p
2 2
2 2
101 10
n pn p
Formula for confidence interval:
statistic of SD valuecritical statisticCI
21 ˆˆ pp *z 2
22
1
11 ˆ1ˆˆ1ˆn
ppnpp
Note: use p-hat when p is not known
Standard error!
Margin of error!
Example 1: At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of 316 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 259 had no visible scars after treatment. Another random sample of 419 patients with minor burns received no plasma compress treatment. For this group, it was found that 94 had no visible scars after treatment. What is the shape & standard error of the sampling distribution of the difference in the proportions of people with no visible scars between the two groups?
Since n1p1=259, n1(1-p1)=57, n2p2=94, n2(1-p2)=325 and all > 10, then the distribution of difference
in proportions is approximately normal.
0296.0419
)78(.22.316
)18(.82...
ES
Example 1: At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of 316 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 259 had no visible scars after treatment. Another random sample of 419 patients with minor burns received no plasma compress treatment. For this group, it was found that 94 had no visible scars after treatment. What is a 95% confidence interval of the difference in proportion of people who had no visible scars between the plasma compress treatment & control group?
Assumptions:•Have 2 independent randomly assigned treatment groups•Both distributions are approximately normal since n1p1=259, n1(1-p1)=57, n2p2=94, n2(1-p2)=325 and all > 5•Population of burn patients is at least 7350.
Since these are all burn patients, we can add 316 + 419 = 735.
If not the same – you MUST list separately.
654.,537.41978.22.
31618.82.96.122.82.
11*ˆˆ2
22
1
1121
n
ppnppzpp
We are 95% confident that the true the difference in proportion of people who had no visible scars between the plasma compress treatment & control group is between 53.7% and 65.4%
Example 2: Suppose that researchers want to estimate the difference in proportions of people who are against the death penalty in Texas & in California. If the two sample sizes are the same, what size sample is needed to be within 2% of the true difference at 90% confidence?
nn)5(.5.)5(.5.645.102. n
25.25.645.102.
Since both n’s are the same size, you have common denominators – so add!
n = 3383
Hypothesis statements:
H0: p1 - p2 = 0
Ha: p1 - p2 > 0Ha: p1 - p2 < 0Ha: p1 - p2 ≠ 0
Be sure to define
both p1 & p2!
H0: p1 = p2
Ha: p1 > p2 Ha: p1 < p2 Ha: p1 ≠ p2
Since we assume that the population proportions are equal in the null hypothesis, the variances are equal.
Therefore, we pool the variances! 21
21ˆnnxxp
Formula for Hypothesis test:
statistic of SDparameter - statisticstatisticTest
z
21
212111ˆ1ˆ
ˆˆ
nnpp
pppp
Usually p1 – p2 =0
Example 4: A forest in Oregon has an infestation of spruce moths. In an effort to control the moth, one area has been regularly sprayed from airplanes. In this area, a random sample of 495 spruce trees showed that 81 had been killed by moths. A second nearby area receives no treatment. In this area, a random sample of 518 spruce trees showed that 92 had been killed by the moth. Do these data indicate that the proportion of spruce trees killed by the moth is different for these areas?
Assumptions:•Have 2 independent SRS of spruce trees•Both distributions are approximately normal since n1p1=81, n1(1-p1)=414, n2p2=92, n2(1-p2)=426 and all > 10•Population of spruce trees is at least 10,130.H0: p1=p2 where p1 is the true proportion of trees killed by moths Ha: p1≠p2 in the treated area p2 is the true proportion of trees killed by moths in the untreated area
59.0
5181
495183.17.18.16.
11ˆ1ˆ
ˆˆ
21
21
nnpp
ppz P-value = 0.5547a = 0.05
Since p-value > a, I fail to reject H0. There is not sufficient evidence to suggest that the proportion of spruce trees killed by the moth is different for these areas