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Sect. 8.1: Inference for a Population Proportion
Previously, we have been making inferences about the population mean.
Now, we will be concerned about estimating the proportion, p , for some population that consists of "successes" and "failures" as in Ch. 5.
The_______________________ of interest is the population proportion, p , of “successes”.
The ___________________ we will be using to estimate p is the sample proportion, p ˆ where
n
X
pˆ^ n
numberofsuccesses(1's)inthesample
For _____________sample sizes, the Binomial distribution must be used for inference about p.
We will assume___________ sample size and use the _______________________________.
Recall from Chapter 5: If a SRS of size n is chosen from a population with proportion p of “successes”, then
As the sample size n increases, the sampling distribution becomes approximately _________
The mean of the sampling distribution is ______ , therefore it is ____________________.
The standard deviation of the sampling distribution is
Confidence Intervals: Note that the standard deviation of pˆ^ involves the _____________ we are trying to estimate, so
we use instead the _____________________ of pˆ^ , (i.e. the estimate of this standard deviation)
which has the form:
S.E.
Confidence intervals for p.
A C = 1- confidence interval for p has the form pˆ
where _______ is the upper critical value for the standard normal distribution. Use Table___
to get this value. Use this interval for confidence level above _______%, only when both ____
and ___________ are __________________.
For somewhat ____________ Moore and McCabe point out that this method can be
___________. They propose an alternative method that calculates an estimate of p as though
____ additional observations had been obtained and _______ of them were "successes." This
method –the Plus Four estimate moves the estimate closer to ______
Plus Four Confidence intervals for p.
Estimate p by
X
p n
A C = 1- confidence interval for p has the form
p p p z n
where z * is the upper ___________ critical value for the standard normal distribution. Use Table D to get this value.
Use this interval for confidence level above _______%, only when _____________. Hypothesis Testing and Confidence Intervals
The hypothesis testing involving sample proportions is another version of the ________-sample z test. When we are testing the null hypothesis H 0 : p = p 0 , we use ______ in place of __________ when calculating the standard deviation in our test statistic, i.e. we standardize assuming p 0 is the
true mean. Z =
The P-value is calculated based on the form of H a :
H a : p > p
P is P(Z > z)
H a : p < p
P is P(Z < z)
H a : p p
P is 2P(Z > |z|)
Replace z with the observed value of the test statistic.
This method should be used only when the ____________ is large enough such that the
______ number of success and the ______________ number of failures is _______________
or more.
Necessary Sample Size
Recall the margin of error for the large sample confidence interval is
* ˆ^ (1^ ˆ)
p p m z z n A certain sample size is necessary to guarantee a particular margin of error. Since this is decided prior to data collection, we need to guess the value of pˆ^. Call this guess p *. Two ways
of guessing are
using information from similar studies or pilot studies or past experience.
using p * = _____. The margin of error is ____________ when
pˆ (^) = _________ so that our guess will be conservative. That is, any other value for the
sample proportion pˆ will yield a __________________ margin of error than planned.
A level C confidence interval for a population proportion p will have margin of error approximately equal to m when the sample size is 2
= *(1 *) z n p p
where p * is some guessed value for the sample proportion pˆ.
Example PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The ability to taste PTC is inherited. About 75% of Italians can taste PTC. You want to estimate the proportion of Americans with at least one Italian grandparent who can taste PTC. How large a sample is necessary to test in order to estimate the proportion of PTC tasters within 0. with 95% confidence?
Example : Last year's survey of the sta135 class had 502 responses to a question "did you eat breakfast today?". Let's take that group of 502 as the population. I asked a stat program to draw 10 random samples, each of size 25. Here is a list of the individuals included in each sample:
No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.
No. 10 24 19 21 9 22 16 14 79 1 27 27 32 71 26 33 35 19 112 10 51 52 51 72 62 77 54 20 135 24 97 61 114 95 99 79 78 23 159 30 98 85 115 98 108 125 81 24 170 34 106 123 122 110 139 130 150 39 206 92 131 138 126 191 173 139 154 52 216 103 141 169 134 201 177 173 160 70 231 114 172 176 141 225 184 174 200 92 232 119 182 230 198 227 230 196 235 103 258 163 197 231 219 232 249 210 246 133 272 167 217 244 242 253 250 230 261 144 286 201 219 295 249 265 253 236 297 152 291 204 236 324 309 286 262 256 303 161 301 249 283 349 315 310 267 269 313 166 325 332 288 381 331 313 302 319 340 231 336 340 294 445 339 349 311 323 357 266 353 353 316 453 351 360 321 353 361 288 360 375 325 455 384 395 338 355 383 311 383 429 326 459 391 404 362 372 396 328 387 431 339 474 417 443 385 384 399 333 424 432 363 480 434 467 439 396 423 349 466 437 404 488 439 477 441 397 451 377 484 449 456 491 448 479 444 420 457 432 498 455 462 496 492 482 488 473 476 456 500 480 478 I have marked some of the population members who appeared in more than one sample. With half the population covered in the 10 samples, it is not surprising that there would be some repetitions and some members never hit. No two samples are identical. Since ______% of the students answered yes, the parameter p is known to be ______. Recall: For a 90% interval, z* = 1.64. Since p is _________, a sample size of________ is large enough to use the normal distribution as the sampling distribution of pˆ. However, n = ____ is __________ large enough to use the traditional CI, according to M&M's rules.
The estimated standard deviations and MOE's are shown in the following table:
X 12 6 14 12 15 17 16 10 16 12 pˆ 0.48 0.24 0.56 0.48 0.60 0.68 0.64 0.40 0.64 0. p~^ 0.48 0.28 0.55 0.48 0.59 0.66 0.62 0.41 0.62 0. SE(tr) 0.10 0.09 0.10 0.10 0.10 0.09 0.10 0.10 0.10 0. M(90%) 0.16 0.14 0.16 0.16 0.16 0.15 0.16 0.16 0.16 0. SE(+4) 0.09 0.08 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0. M(90% +4) 0.15 0.14 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.
Since n ___________, we can use the Plus Four approach:
p
So the CI is
0.944(0.056) 0.944 * 0.944 *0. 54
z z A 95% interval, (z* = ____),is (0.883,________).
We would use (0.883, ______). The traditional method gives, (0.941,_____), or (0.941, ____), shorter, but ________________.
Example - It's Wednesday November 3, 2004 - do you know who your president is? Last surveys before the 2004 election for some major national polls
Poll date Bush Kerry Nader Other ended Zogby 11/2/04 49.4% 49.1% --- --- Gallup 10/31/04 49% 49% 0.5% 0.5% Gallup 10/24/04 51% 46% 1% --- Pew 10/30/04 51% 48% 1% --- Harris 11/1/04 49% 48% 2% 1% TIPP 11/1/04 50.1% 48.0% 1.1% 0.8% Tarrance (R) 11/1/04 51.2% 47.8% 0.5% 0.5% Lake (D) 11/1/04 48.6% 50.7% --- ---
Actual Popular Vote: 50.75% 48.30% 0.36% 0.59% Why did the polls differ from each other?
Why did they differ from themselves a few days before?
Gallup had about 1600 "likely voters" in their final poll but only about 700 in their polls done in early October. Why did they increase the numbers?
Gallup reported a ±3% margin of error for their final pre-election poll.
Thus for Bush they made the confidence statement 49% ± 3%
How did they get it?
For 95% confidence, MOE is: n
p(1-p) 2
For the Gallup poll, this is:
Would a Plus Four C.I. be different?
What does this mean?
The sample size is_________ and p is_______________. They use the ________________ approximation to get the MOE.
________________ of the time pˆ^ must be____________ the interval that goes from p __________ to
p ____________, i.e. the probability is __________ that the interval that goes from pˆ^ _________ to
pˆ^ ________________ will contain p.
