Chapter 19 – Confidence Intervals for Proportions. Next few chapters. What percentage of adults own smartphones ? What is the average SAT score of Baltimore County students? Do a higher percentage of women vote for Democrats than men?
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What percentage of adults own smartphones?
What is the average SAT score of Baltimore County students?
Do a higher percentage of women vote for Democrats than men?
Do cars who use a fuel additive get better fuel efficiency?
Is it true that 30% of our students work part-time?
Does the average American eat more than 4 meals out per week?
For the remainder of the semester we are going to focus on confidence intervals and hypothesis tests
Confidence Interval: range of values we predict the true population statistic is within
Hypothesis Test: determine whether or not a claim made about a population statistic is valid
We use a sample to make our prediction about the population
Since each sample we take will give us a slightly different estimate, we have to understand the random sampling variation we’ve been studying
We can never be precise about our estimate, but we can put it within a range of values we feel confident about
We saw in the last chapter that when we use a sample to estimate a proportion that the proportion estimates were distributed Normally with:
We know our estimate is just an estimate, but want to know how good an estimate it is
We can use the standard deviation of our sampling distribution model and our proportion estimate to find the Standard Error:
We can use this error to get a sense for how confident we are that our estimate is correct
It is not a mistake we made, but a way to measure the random sampling variation, and since we don’t have the population proportion, we can’t know the s.d.
Sample was from 2,016 adults, 423 of which had tattoos.
Since this is just one sample, let’s look at the sampling distribution model like we did last chapter:
A random sample of 168 students were asked about their digital music library. Overall, out of 117,709 songs, 23.1% were legal. Construct a 95% confidence interval for the fraction of legal digital music.
Technically, a 95% confidence interval means that 95% of all samples of the same given size will include the true population proportion.
This represents confidence intervals of 20 simulated
samples for the sea fans
infected from the example
in the text.
You can see that most of theconfidence intervals includethe true proportion.
Figure from DeVeaux, Intro to Stats
If you were going to guess someone’s height, would you be more likely to be right with a wider or smaller range for your guess?
The larger the margin of error you have, the more likely your prediction is to be correct.
The more precise we want to be, the less confident we can be that we are correct.
Estimate ± ME
While 2 is a good estimate for a 95% confidence interval, using the Normal probability table, we can see that z*= 1.96 is more accurate.
What would be the critical value for a 92% confidence interval?
Use Table in Appendix D to find appropriate z-score.
Using our earlier example involving tattoos, what would the margin of error be for a 92% confidence interval?
Now we also know for a 92% confidence interval, we use z* = 1.75
ME = 1.75(.009) = .016 (ME for 95% CI: .018)
When conditions are met
Confidence Interval =
Make sure you can interpret your confidence interval.
A Gallup poll shows that 62% of Americans would amend the Constitution to use the popular vote for Presidential elections instead of the electoral vote. They used a random sample of 1,005 adults aged 18+
Verify that the conditions were met.
Construct a 95% confidence interval.
Interpret your interval.
If we find from a pilot study that 32% of Math 153 students are full-time students, how many students would we have to sample to estimate the proportion of Math 153 full-time students to within 7% with 90% confidence?