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Statistical Inference for Population Proportions - Prof. Corlis E. Robe, Exams of Statistics

East Tennessee State University (ETSU)Statistics
Prof. Corlis E. Robe

The concepts of inferring population proportions from sample data, including calculating confidence intervals and performing hypothesis tests. It covers the estimation of population proportions, standard errors, and the use of z-scores for inference. The document also includes examples and instructions for calculating confidence intervals and testing hypotheses.

Typology: Exams

Pre 2010

Uploaded on 08/13/2009

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koofers-user-l3j 🇺🇸

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ROBE Math 1530 Spring
2005
Ch. 18, p. 469 “Inference about a population proportion”
What proportion of voters really voted for W. in the 2004 presidential election?
This wants a confidence interval for the proportion of W. votes.
Is the proportion of students who fail Math 1530 really only 12% as Professor Robe
claims?
We need to run a hypothesis test for the proportion of Math 1530 students
who fail.
H0 : p = 12%, Ha : p > 12%.
A population proportion p is a parameter just like a population mean .
Example p. 250 #2.
Just like a sample mean
x
estimates a population mean , a sample proportion
p
ˆ
estimates a population proportion p.
p. 470
p
ˆ
= count of successes in a sample / sample size n
example p. 470 #1.
If the study of religious practices were repeated – would they get those same
127 students in the sample?
Would they get the same count 107 who pray?
Sooo -
p
ˆ
is a variable. A variable from a random sample is a random
variable,
so probability applies, if we ask the question right.
Across lots and lots of samples, we know that
x
will balance out around μ – the
mean of
x
’s sampling distribution is μ – and we know what the distribution of those
x
’s will be. What about
p
ˆ
?
pf3
pf4
pf5

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ROBE Math 1530 Spring 2005 Ch. 18, p. 469 “Inference about a population proportion” What proportion of voters really voted for W. in the 2004 presidential election? This wants a confidence interval for the proportion of W. votes. Is the proportion of students who fail Math 1530 really only 12% as Professor Robe claims? We need to run a hypothesis test for the proportion of Math 1530 students who fail. H 0 : p = 12%, Ha : p > 12%. A population proportion p is a parameter just like a population mean . Example p. 250 #2. Just like a sample mean x estimates a population mean , a sample proportion p ˆ estimates a population proportion p. p. 470 p ˆ^ = count of successes in a sample / sample size n example p. 470 #1. If the study of religious practices were repeated – would they get those same 127 students in the sample? Would they get the same count 107 who pray? Sooo - p ˆ^ is a variable. A variable from a random sample is a random variable , so probability applies, if we ask the question right. Across lots and lots of samples, we know that x will balance out around μ – the mean of x ’s sampling distribution is μ – and we know what the distribution of those x ’s will be. What about p ˆ^?

Box p. 471 (plus some) Choose an SRS of size n from a large population that contains a proportion p of ‘successes.’ Let p ˆ^ be the sample proportion of successes p ˆ^ = count of successes / n. Δ As n increases, the sampling distribution of p ˆ^ becomes approximately Normal – The rule of thumb is that both np and n(1-p) > 10; most statisticians prefer that we also have n > 400 Δ μ ( p ˆ^ ) = p provided the samples are SRS’s. Δ σ ( p ˆ^ ) = n p( 1  p ) provided n is less than 10% of the population. Example p. 472 #3a) b) Given a population’s proportion p, σ ( p ˆ^ ) = n p( 1  p ) is the standard deviation of p ˆ^ , i.e., the standard deviation of p ˆ^ ’s sampling distribution. In the event that the actual p is not available, we estimate with the sample’s proportion p ˆ^ and n pˆ ( 1  pˆ )

= SE ( p ˆ^ ) is called the standard error of p ˆ

(p. 474) Example p. 470 # p is the proportion of college students who pray, p ˆ^ = 107/127 = 0. and SE( p ˆ^ ) =

“We can be 95% confident that the true proportion of Ohio voters who voted for Kerry is between 0.45 and 0.61” This is much too ‘vague’ an estimate for p p. 482 It is possible to choose n in order to get a desired margin of error margin = z* n pˆ ( 1  pˆ )

can be rearranged to give n = p^ p

m z *        1 2 where p* is p ˆ^ from an old poll or the most conservative case p* = 0.5. example continued. Suppose a Kerry pollster wants a new poll result with a new confidence interval that has 95% confidence but also with a margin of error of 3% or less. How big a sample will she need? She’ll use the old poll’s p ˆ^ = 0.53 for p*. n 2       ( ) ( ) p. 483 Significance Tests for a proportion The TV news reports that (only) 70% of college students voted in the most recent Presidential election. You suspect that the proportion is much lower among ETSU students because of rainy, cold weather on election day. (Drum roll) This calls for a Hypothesis Test !! the population is all ETSU students the parameter is p = the proportion who voted H 0 : p = 70% Ha : p < 70% Next we need a sample, we calculate the sample’s p ˆ^ , and we get the sample’s z test statistic z =   n p p pˆ p   1 And calculate the P-value Say we take an SRS of n = 500 ETSU students and find X = 310 for p ˆ^ = 310 / 500 = 0.62.

Then z =   n p p pˆ p   1 = and our P-value = Prob( p ˆ^ < 0.62) = Prob( z < ) = and our sample gives evidence that