AP Stats Chap 7 Terminology (Vocab)

AP Stats – Chap 19 Confidence Intervals for Proportions

In many cases, we will not know p, the true proportion of success for the entire population, but we will know the proportion of success for our sample. This is denoted p . Also…the proportion of failure of our sample will be q .

Therefore, we can’t find the SD of the sample, but we can find the Standard Error of the Sample using the formula…

Suppose I want to know what proportion of teenagers typically goes to the movies on a Friday night.

Suppose I take an SRS of 25 teenagers and calculate the sample proportion to be pˆ = 0.40 .

The sample proportion _____ is an unbiased estimator of the unknown population proportion _____, so I would estimate the population proportion to be approximately _____. However, using a different sample would have given a different sample proportion, so I must consider the amount of variation in the sampling model for _____.

Based on one sample, it would _____ be correct to conclude that 40% of all teenagers typically go to the movies on a Friday night.

But don’t despair!… based on my one sample, I can come up with an ______that **may** contain the true proportion of teenagers who typically go to the movies on a Friday night.

Not only will I tell you what that interval is, but I will also tell you how ______I am that the true proportion falls somewhere in that interval.

Since we don’t know p , we cannot calculate the standard deviation of the sampling model. We can, pqˆ ˆ however, use _____ to estimate the value of p and calculate the standard error instead. n

So the standard error for the sampling model for the proportion of teenagers who typically go to the movies on a Friday night is:

According to the 68-95-99.7 Rule, _____ of all possible samples of size 25 will produce a statistic pˆ that is within _____ standard errors of the mean of our sampling model.

This means that, in our example, 95% of the pˆ ’s will be between ______and ______.

So the distance between the actual _____ value and the statistic _____ will usually (95% of the time) be less than or equal to ______. Therefore, in 95% of our samples, the interval between pˆ - 0.196 and pˆ + 0.196 will contain the parameter p .

We say that the ______is 0.196

For our sample of 25 teenagers, pˆ = 0.40 . Because the margin of error is 0.196, then we are 95% confident that the true population proportion lies somewhere in the interval ______, or ______.

The interval [0.21, 0.59] is called a ______because we are 95% ______that the true proportion of teenagers who typically go to the movies on a Friday night is between about _____ and _____.

Example: “We are 95% confident that between 42.1% and 61.7% of Las Redes sea fans are infected.”

This type of statement is a

They will also known as

Confidence Intervals

The green line is…

The red dots are…

The blue segments are…

The CLT says…

Okay… maybe you’re not satisfied with the interval we constructed. Is it too wide? Would you prefer a more precise conclusion? One way of changing the length of the interval is to change the ______.

So how do we construct 90% confidence intervals? 99% confidence intervals? C% confidence intervals?

Since the sampling model of the sample proportion pˆ is ______, we can use normal calculations to construct confidence intervals.

 For a 95% confidence interval, we want the interval corresponding to the ______95% of the normal curve.  For a 90% confidence interval, we want the interval corresponding to the ______90% of the normal curve.

 And so on…

If we are using the standard normal curve, we want to find the interval using ______.

Suppose we want to find a 90% confidence interval for a standard normal curve. If the middle 90% lies within our interval, then the remaining ______lies ______our interval. Because the curve is symmetric, there is ______below the interval and ______above the interval. Find the ______with area 5% below and 5% above.

Margin of Error (ME) Critical Values

These z-values are denoted ______. Because they come from the standard normal curve, they are centered at mean _____.

______is called the ______, with probability p lying to its ______under the standard normal curve.

To find the upper critical p value, we find the complement of C and divide it in half, or find:

For a 95% confidence interval, we want the z-values with upper p critical value ______.

For a 99% confidence interval, we want the z-values with upper p critical value ______.

The extent of the blue segment on either side of p is called the margin of error.

ME = where z* = the z-value (the critical value) associated with the specific confidence interval you want.

For 90% confidence interval, the critical value is

Remember that z-values tell us how many ______we are above or below the mean. To construct a 95% confidence interval, we want to find the values ______standard deviations below the mean and 1.96 standard deviations above the mean, or:

pqˆ ˆ Using our sample data, this is pˆ 1.96 , assuming the population is at least ______times as large n as the sample size, ______.

In general, to construct a level C confidence interval using our sample data, we want to find:

pqˆ ˆ The margin of error is z * . Note that the margin of error is a positive ______. It is not an n interval.

We would like ______confidence and a ______margin of error.

A higher confidence level means a higher percentage of all samples produce a statistic close to the true value of the parameter. Therefore we want a ______level of confidence.

A smaller margin of error allows us to get closer to the true value of the parameter (length of the interval is small), so we want a ______margin of error.

So how do we reduce the margin of error?

▪ ______the confidence level (by decreasing the value of z*)

▪ ______the standard deviation

▪ ______the sample size. To cut the margin of error in half, increase the sample size by ______times the previous size.

You can have ______confidence and a ______margin of error if you choose the right sample size. To determine the sample size n that will yield a confidence interval for a population proportion with a specified margin of error m, set the expression for the margin of error to be equal to m and solve for n. Always round n up to the next greatest integer.

CAUTION!!

These methods only apply to certain situations. In order to construct a level C confidence interval using pqˆ ˆ the formula pˆ z* , for example, the data must come from a ______sample. Also, we n want to eliminate (if possible) any ______.

The margin of error only covers random sampling errors. Things like ______, ______, and ______can cause additional errors.

If you are asked to create a confidence interval, you should PANIC!!

P ______

A ______Independence:

Randomization Condition:

10% Condition:

Success/failure condition:

N ______: If all of the necessary assumptions and conditions havce been met, we may proceed with the ______.

I ______:

^ ^ ^ p q p� z n

C ______: We are ______% confident that the true proportion (percentage) of ______is between ______and ______.

AP Stats – Chap 19 Highlights This chapter provided your first formal exposure to statistical inference by introducing you to confidence intervals, a widely used technique. You learned how to construct a confidence interval for a population proportion (known as a “parameter”). You also examined how to interpret both the resulting interval and also what the confidence level means.

For example, using a 95% C-level gives you...  95% confidence that the interval contains the actual value of the unknown population parameter (the population proportion), and  statistical support that 95& of all intervals generated by this procedure – in the long run – will succeed in “capturing” the unknown population value (this is the CLT).

You also investigated the effects of sample size and confidence level on the interval and its margin of error. The ideal confidence interval is very narrow with a very high C-level. But there’s a trade-off: using a higher confidence level produces a wider interval if all else remains the same. One solution is to use a larger sample. The larger sample produces a narrower interval for the same confidence level.

You also learned how to plan ahead (working backwards) by determining, before collecting any data, the sample size you would need to achieve a certain margin of error for a given confidence level.

The general form of all confidence intervals is Estimate +/- Margin of Error. And Margin of Error = (Critical Value) x (Standard Error of the Estimate).

骣    * pq In symbols, this looks like p z 琪 . ( )琪 n 桫 The margin of error is affected by several factors, primarily  a higher confidence level produces a greater margin of error (a wider interval)  a larger sample size produces a smaller margin of error (a narrower interval)

Common C-levels are 90%, 95%, 98%, and 99%. The phrase “95% confidence” means that if you were to take a large number of random samples and use the same confidence interval procedure on each sample, then – in the long run – 95% of those intervals would succeed in capturing the actual parameter value. Note that this is NOT the same as saying there is a 95% probability that the parameter is inside the calculated interval. It’s technical in the wording, but the difference is IMPORTANT.

Always check the conditions before applying this 1-Propotion z-Interval.  The sample must be large enough to guarantee np and nq are each greater than or equal to ten  If the sample was selected randomly from the population, you’re good to go. If not, can it be safely assumed?  Is it safe to assume that the measurements you’re collecting are independent from each other?  The sample must be less than 10% of the population.

How ‘bout an example? 

New Medicine An experiment finds that 27% of 53 subjects report improvement after using a new medicine.

1. Check the conditions and assumptions!

2. State/calculate all needed values.

3. Create a 95% confidence interval for the actual cure rate.

4. Interpret the confidence interval in this context using all relevant terms in your explanation.

5. This interval is too wide. Make it narrower…90% confidence.

6. What are the advantages and disadvantages of this narrower interval?

7. Explain what the phrase “90% confidence” means in this context.

8. What sample size would we need in a follow-up study if we want a margin of error of only 3% with 98% confidence?

How ‘bout another one? 

Female Workers The countries of Europe report that 46% of the labor force is female. The United Nations wonders is the percentage of females in the labor force is the same in the United States. Representatives from the United Stated Department of Labor plan to check a random sample of over 10,000 employment records on file to estimate a percentage of females in the US labor force.

9. The representatives from the Department of Labor want to estimate a percentage of females in the US labor force to within +/- 5%, with 90% confidence. How many employment records should they sample?

10. They actually select a random sample of 525 employment records, and find that 229 of the people are females. Create the confidence interval.

11. Interpret the confidence interval in this context.

12. Explain what 90% confidence means in this context.

13. Should the representatives from the Department of Labor conclude that the percentage of females in their labor force is lower than Europe’s rate of 46%? Explain.

Final one!  Private Schools Virginia’s Department of Education reports that 12% of the high school students in the Commonwealth attend private high schools. The local university wonders if the percentage is the same in their applicant pool for the upcoming fall semester. Admissions officers plan to check a random sample of the over 10,000 applicants on file to estimate the percentage of students applying for admission who attend private schools.

14. The admissions officers want to estimate the true percentage of private school applicants within +/- 4%, with 95% confidence. How many application should they sample?

15. They actually select a random sample of 450 applications, and find that 46 of those students attend private schools. Create the confidence interval.

16. Interpret the confidence interval in this context.

17. Explain what 95% confidence means in this context.

18. Should the admissions officers conclude that the percentage of private school students in their applicant pool is lower than the statewide enrollment rate of 12%? Defend your answer.