Stat 217 – Day 15 Statistical Inference (Topics 17 and 18)
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Transcript of Stat 217 – Day 15 Statistical Inference (Topics 17 and 18)
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Stat 217 – Day 15Statistical Inference (Topics 17 and 18)
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Previously – Central Limit Theorem If taking random samples
from a population with proportion , and the sample size is large, then the sampling distribution of the sample proportions will follow a normal distribution with mean equal to the population proportion and standard deviation equal to
n
)1(
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Activity 15-1 (p. 295)
Probability of sample proportion at least .25 = .156
In many, many random samples of American adults, about 16% of samples have > .25p̂
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Activity 15-1
(g) What if sample size is 400 instead of 100?
A sample proportion .25 or larger is even more unlikely to come from a larger sample.
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Activity 15-1
(i) What about the population size?
DOESN’T MATTER!
As long as it is much, much bigger than the size of the sample
(j) Virginia, = .209, probability of sample proportion exceeding .25?
SAME!
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Preliminary Questions (p. 333) Which tire would you pick?
Left front Right front
Left rear Right rear
(driver)
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Activity 17-1
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1. Define parameter
Identify the symbol and specify the (unknown) parameter in words Make sure the type of number (mean, proportion),
the population of interest and variable are clear
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2. Stating hypotheses
Ho: Ho-hum hypothesis Ho: parameter = hypothesized value
Ha: Aha! Hypothesis Ha: parameter <,>, ≠ hypothesized value
Good practice: always state in symbols and in words
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Stating Hypotheses
Direction of alternative hypothesis depends on research question
Lab 1: Do infants prefer the helper?
(a) Ho: = .5
(b) Ha: > .5
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3. Checking technical conditions Sample size condition Randomness condition
If sample size condition is not met? If randomness condition is not met?
Good practice: Include shaded and labeled sketch of the relevant sampling distribution
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4. Test statistic
Standardize the observed statistic comparing it to the hypothesized value, dividing by the standard deviation of the statistic (amount of sampling variability)
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5. p-value
Row-Row-Row your boat
It is key to knowWhat p-value means --
It’s the chance (with the null)you obtain data that’s
At least that extreme!
NOT the chance the null hypothesis is true!!
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6. Stating conclusions
If you ask Ben a question he either says “no” or he says nothing.
So if he says nothing, does this “prove” he agrees?
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Activity 17-1: Flat Tires (p. 334)(a) If nothing special, people will pick the right
front tire 25% of the time (proportion .25)
(b) parameter, (c) > .25
(d) 73(.25)=18.25>10; 73(.75)=54.75)>10
.051
.25
=34/73=.466p̂
24.4051.
25.466.
z
Less than a .02% chance would get a sample proportion this large if = .25 Strong evidence for > .25
2. H0: = .25
1. Let represent the population proportion that would pick right front
Ha: > .25
4. Test statistic
Table II: probability above < .0002 5. p-value
6. Reject H0
3. Technical conditions
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In conclusion
6. We have strong evidence from this sample proportion, that more than 25% of the population would choose the right front tire in this situation Caution: This was not a random sample. We may
consider this sample representative of Stat 217 students in general at Cal Poly
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Test of Significance (p. 338)
1) Define population parameter in words2) State 2 competing claims about parameter
null hypothesis H0: parameter = valuealternative hypothesis Ha: parameter < > or ≠ value
3) Check “technical conditions” for procedure4) Calculate test statistic (assuming H0 true)
comparing what observed to what conjectured in H0
5) Calculate p-value (see Ha for “more extreme”)how often see sample data this extreme when H0 true
6) Make a decision to either reject or fail to reject H0
Is p-value small?State conclusion in context
Compare p-value to “level of significance, ”
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So what next?
If reject .25 as a plausible value for the , the proportion of all CP students who would pick the right front, next question might be what are plausible values for ?!
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What about .3?
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What about .4?
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Observations
So we fail to reject .25 and .3 as a plausible value for Another reminder that when you fail to reject the
null, that doesn’t mean you have proven the parameter has that value.
but we reject .4 as a plausible value What is the set of all plausible values?
Which values of the parameter will we fail to reject based on our observed sample statistic?
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The main idea
Plausible values of the population parameter are values that are not “too far” from the observed sample statistic
Say within 2 standard deviations When CLT applies, 95% of sample proportions
should fall within 2 standard deviations of the population proportion
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Activity 16-1 (p. 312)
(a) Observational units Youths
(b) Variable Whether or not a television in their room
(c) Parameter or statistic Statistic, p-hat
(d) Want to know proportion of all American youth that have a TV in their room(e) Can’t determine it exactly (not a census)(f) But should be in the ball park (large random sample)
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Activity 16-1
To estimate the standard deviation, use the sample proportion Standard error
(h) (.68(1-.68)/2032) = .0103 If we use p-hat (sample proportion) in place
of (population proportion), we obtain an approximate confidence interval for .68 – 2(.0103) to .68 + 2(.0103) (.659, .701)
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Ta dah!
I am 95% confident that between 65.9% and 70.1% of all American youths have a television set in their bedroom
Why am I allowed to say this? Empirical rule Normal distribution Central Limit Theorem Random sample and n > 10 and n(1-)>10
Ok, use p-hat here too…
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Test of Significance Calculator
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To Turn in, with partner
(n) boys or girls CI (p. 318) Calculation and interpretation (I’m 95% confident
that…)
For Tuesday Activities 16-3, 16-6 (with technology)