Confidence Intervals: Estimating Population Proportion p by using 𝑝 Section 8.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Lesson 6.1.1

Objectives Conditions For estimating p, why we use 𝒑 Standard deviation -Standard Error Calculating Confidence Intervals for p statistic ± (critical value) · (standard deviation of statistic) How do we find the critical Value One Sample Z interval for a Population Proportion Formula, by hand --Calculator efficiency Conditions of Constructing Confidence Intervals Putting it all together: The 4 steps to ALWAYS cover. Choosing the sample size

So… In 8.1, we saw that a confidence interval can be used to estimate an unknown population parameter. We are often interested in estimating the proportion p of some outcome in the population. Some examples: What proportion of US adults are unemployed right now? What proportion of high school students cheated on a test? What proportion of pine trees in a national park are infested with beetles? What proportion of a company’s lap top batteries last as long as the company claims? This section shows us how to construct and interpret a confidence interval for a population proportion.

Why We can Use 𝒑 In practice, of course, we don’t know the value of our population proportion (p). If we did, we wouldn’t need to construct a confidence interval for it! So we use 𝒑 to estimate for p. In large samples, 𝒑 will be close to p. So we can replace p with 𝒑 ! Lets look at our normal conditions and standard deviation of p, and replace it with 𝒑 .

Normal Conditions and Standard Deviation Pop Proportion  Sample Proportion np≥10  n 𝒑 ≥ 10 n(1-p)≥10  n(1- 𝒑 )≥10 σ 𝒑 = 𝑝(1−𝑝) 𝑛  σ 𝒑 = 𝒑 (1− 𝒑 ) 𝑛 Again, since we don’t know the value of p we replace it with 𝒑 . When the standard deviation of a statistic is estimated from data, the result is called the standard error. Since we are replacing a value, this creates some error. We are ok with this.

Standard Error 𝑆𝐸 𝑝 𝑆𝐸 𝒑 = 𝒑 (1− 𝒑 ) 𝑛 Since we don’t know the value of p we replace it with 𝒑 . When the standard deviation of a statistic is estimated from data, the result is called the standard error. Since we are replacing a value, this creates some error. It turns out that our Standard Deviation IS our Standard Error 𝑆𝐸 𝒑 = 𝒑 (1− 𝒑 ) 𝑛 Don’t confuse this with Margin of Error….that is coming later.

Conditions for Constructing A Confidence Interval Random: The data should come from a well-designed random sample or randomized experiment. Normal: n 𝒑 ≥ 10 AND n(1- 𝒑 )≥10 Independent: The procedure of calculating confidence intervals assume that individual observations are independent and the 10% condition is met. 𝑛≤ 1 10 𝑁 OR 10𝑛≤𝑁 Run through these conditions before constructing an interval!!!

What if 1 condition is violated? CANT GO ON! Poorly designed experiment Start over Reflect…

Check Your Understanding In each of the following settings, check whether the conditions for calculating a confidence interval for the population proportion p are met. Conditions: Random, Normal, Independent 1. An AP Statistics class at a large high school conducts a survey. They ask the first 100 students to arrive at school one morning whether or not they slept at least 8 hours the night before. Only 17 students say “Yes.” 2. A quality control inspector takes a random sample of 25 bags of potato chips from the thousands of bags filled in an hour. Of the bags selected, 3 had too much salt. 1. Not random, 1st condition violated. 2. normal conditions not satisfied (25)(3/25) is not > 10 Lesson 6.1.1

Calculating a Confidence interval for p. statistic ± (critical value) · (standard deviation of statistic) 𝒑 ± 𝑧 ∗ 𝒑 (1− 𝒑 ) 𝑛 How do I find the critical value 𝑧 ∗ ? To find a level C (say…95%?) confidence interval, we need to catch the central area C under the standard Normal curve.

How do I find the critical value 𝑧 ∗ ? Finding the critical value for a 95% confidence interval, we look at the area under the curve that’s NOT being accounted for. 100%-95% =5% / 2 tails = 2.5% each tail, now turn to Table A and find the Z score that corresponds to an area .025 to the left

Example: PROBLEM: Use Table A to find the critical value z* for an 80% confidence interval. Assume that the Normal condition is met.

Example Cont… SOLUTION: For an 80% confidence level, we need to catch the central 80% of the standard Normal distribution. In catching the central 80%, we leave out 20%, or 10% in each tail. So the desired critical value z* is the point with area 0.1 to its right under the standard Normal curve. The table shows the details in picture form: Search the body of Table a to find the point −z* with area 0.1 to its left. The closest entry is z = −1.28. (See the excerpt from Table A above.) So the critical value we want is z* = 1.28.

Calculator to find my Critical Value! We are AP, and AP is pressed for time! How to find the critical value on your calc: 2nd>Vars (distr) >3: invNorm( >Enter > fill in the following invNorm(area to left, 0,1) Example: invNorm(.05,0,1) .05 = area to the left of the curve, 0 and 1 are just to state it’s a standard Normal curve. The Calculator makes Table A obsolete!

Putting it all Together: A Four-Step Process State: What parameter do you want to estimate, and at what confidence level? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, preform calculations. Conclude: Interpret your interval in the context of the problem. AP expects ALL FOUR, do not skip step 1!

One Sample Z Interval for a Population Proportion Choose a SRS of size n from a large population that contains an unknown proportion p of successes. An approximate level C confidence interval for p is : 𝒑 ± 𝑧 ∗ 𝒑 (1− 𝒑 ) 𝑛 Where 𝑧 ∗ is the critical value for the standard Normal curve with area C between −𝑧 ∗ 𝑎𝑛𝑑 𝑧 ∗ . CHECK CONDITIONS! Use this interval only when the numbers of successes and failures in the sample are both at least 10 and the population is at last 10 times as large as the sample.

See Handout for article: Teens Say Sex Can Wait Example: See Handout for article: Teens Say Sex Can Wait

Statistics/List editor>2nd>F2 (F7) > 5:1-propZint Calculator! We are AP, and AP is pressed for time! How to construct an confidence interval for population proportion: TI 83/84 STAT> TESTS> A:1-PropZInt> enter information in X, n, and C-level> highlight Calculate> Enter TI 89 Statistics/List editor>2nd>F2 (F7) > 5:1-propZint

Statistics/List editor>2nd>F2 (F7) > 5:1-propZint Calculator Practice: We’ll demonstrate using the example of n = 439 teens surveyed, X = 246 said they thought that young people should go to college. Use your calculator to calc a 95% confidence interval: TI 83/84 STAT> TESTS> A:1-PropZInt> enter information in X, n, and C-level> highlight Calculate> Enter TI 89 Statistics/List editor>2nd>F2 (F7) > 5:1-propZint

AP Warning! AP EXAM TIP You may use your calculator to compute a confidence interval on the AP exam. But there’s a risk involved. If you just give the calculator answer with no work, you’ll get either full credit for the “Do” step (if the interval is correct) or no credit (if it’s wrong). We recommend showing the calculation with the appropriate formula and then checking with your calculator. If you opt for the calculator-only method, be sure to name the procedure ( One-prop z interval) and to give the interval ( 0.514 to 0.606) and state all the information used for that interval (sample size….ect) YOU DO NOT HAVE PERMISSION TO DO THIS YET!

AP Warning! AP EXAM TIP Many students use the 1propZint feature to correctly calculate the confidence interval and then try to “show their work” with an incorrect formula. This will result in a loss of credit because the two attempts are considered “parallel solutions” and students are scored on the worse response. If students want to include a formula in their response, they should make sure it produces the same results as the calculator. If the results aren’t the same, students should choose one of the answers and cross the other out. Also if students chose to include a formula they should skip the symbolic formula and start with numbers substituted in. in some cases, students choose the right formula and use the correct values, but include an incorrect symbol (e.g. p instead of 𝒑 ) and lose credit.

Choosing the Sample Size To determine the sample size n that will yield a level C confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n: 𝑧 ∗ 𝒑 (1− 𝒑 ) 𝑛 ≤𝑀𝐸 Where 𝒑 is a guessed value for the sample proportion. The margin of error will always be less than or equal to ME if you take the guess of 𝒑 to be 0.5.

Choosing the Sample Size Example A company has received complaints about its customer service. The managers intend to hire a consultant to carry out a survey of customers. Before contacting the consultant the company president wants some idea of the sample size that she will be required to pay for. One critical question is the degree of satisfaction with the company's customer service, measured on a 5 – point scale. The president wants to estimate the proportion p of customers who are satisfied (that is, who choose either “somewhat satisfied” or “very satisfied,” the two highest levels on the 5 – point scale). She decides that she wants the estimate to be within 3% (0.03) at a 95% confidence level. How large a sample is needed? 𝑧 ∗ 𝒑 (1− 𝒑 ) 𝑛 ≤𝑀𝐸

Choosing the Sample Size Example …She decides that she wants the estimate to be within 3% (0.03) at a 95% confidence level. How large a sample is needed? 𝑧 ∗ 𝒑 (1− 𝒑 ) 𝑛 ≤𝑀𝐸

Objectives Conditions For estimating p, why we use 𝒑 Standard deviation -Standard Error Calculating Confidence Intervals for p statistic ± (critical value) · (standard deviation of statistic) How do we find the critical Value One Sample Z interval for a Population Proportion Formula, by hand --Calculator efficiency Conditions of Constructing Confidence Intervals Putting it all together: The 4 steps to ALWAYS cover. Choosing the sample size

Continue working on Ch. 8 Reading Guide Homework 8.2 Homework Worksheet Continue working on Ch. 8 Reading Guide