Inference for Proportions One Sample

Confidence Intervals One Sample Proportions

Rate your confidence Name my age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading pool, will make basket? Shooting the ball at a large trash can, will make basket? Shooting the ball at a carnival, will make basket?

What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval.

Point Estimate singleUse a single statistic based on sample data to estimate a population parameter Simplest approach variationBut not always very precise due to variation in the sampling distribution

Confidence intervals Are used to estimate the unknown population parameter Formula: estimate + margin of error

Margin of error Shows how accurate we believe our estimate is more preciseThe smaller the margin of error, the more precise our estimate of the true parameter Formula:

Assumptions: SRS Normal distribution n > 10 & n(1- ) > 10 Population is at least 10n

Formula for Confidence interval: Normal curve Note: For confidence intervals, we DO NOT know p – so we MUST substitute p-hat for p  in both the SD & when checking assumptions.

Found from the confidence level The upper z-score with probability p lying to its right under the standard normal curve Confidence level Tail Area Z* Critical value (z*).05 z*= z*= z*= % 95% 99%

Confidence level Is the success rate of the method used to construct the interval Using this method, ____% of the time the intervals constructed will contain the true population parameter

What does it mean to be 95% confident? 95% chance that  is contained in the confidence interval The probability that the interval contains  is 95% The method used to construct the interval will produce intervals that contain  95% of the time.

A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost.

Assumptions: Have an SRS of adults n =1012(.38) = & n(1- ) = 1012(.62) = Since both are greater than 10, the distribution can be approximated by a normal curve Population of adults is at least 10,1012. We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%. Step 1: check assumptions! Step 2: make calculations Step 3: conclusion in context

Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? To find sample size: However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!

What p-hat (p) do you use when trying to find the sample size for a given margin of error?.1(.9) =.09.2(.8) =.16.3(.7) =.21.4(.6) =.24.5(.5) =.25 By using.5 for p-hat, we are using the worst-case scenario and using the largest SD in our calculations.

Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? Use p-hat =.5 Divide by 1.96 Square both sides Round up on sample size

Hypothesis Tests Hypothesis Tests One Sample Proportions

Example 1: Julie and Megan wonder if head and tails are equally likely if a penny is spun. They spin pennies 40 times and get 17 heads. Should they reject the standard that pennies land heads 50% of the time? How can I tell if pennies really land heads 50% of the time? What is their sample proportion? expect unlikely But how do I know if this is one that I expect to happen or is it one that is unlikely to happen? Hypothesis test will help me decide!

What are hypothesis tests? Calculations that tell us if a value occurs by random chance or not – if it is statistically significant Is it... –a random occurrence due to variation? –a biased occurrence due to some other reason?

Nature of hypothesis tests - First begin by supposing the “effect” is NOT present Next, see if data provides evidence against the supposition Example: murder trial How does a murder trial work? First - assume that the person is innocent must Then – must have sufficient evidence to prove guilty

Steps: 1)Assumptions 2)Hypothesis statements & define parameters 3)Calculations 4)Conclusion, in context Notice the steps are the same except we add hypothesis statements – which you will learn today

Assumptions for z-test: Have an SRS from a binomial distribution Distribution is (approximately) normal YES YES – These are the same assumptions as confidence intervals!! Use the hypothesized parameter in the null hypothesis to check assumptions!

Example 1: Julie and Megan wonder if head and tails are equally likely if a penny is spun. They spin pennies 40 times and get 17 heads. Should they reject the standard that pennies land 50% of the time? Are the assumptions met? Binomial Random Sample 40(.5) >10 and 40(1-.5) >10 Infinate amount of spins > 10(40)

Writing Hypothesis statements: Null hypothesis – is the statement being tested; this is a statement of “no effect” or “no difference” Alternative hypothesis – is the statement that we suspect is true H0:H0: Ha:Ha:

The form: Null hypothesis H 0 : parameter = hypothesized value Alternative hypothesis H a : parameter = hypothesized value H a : parameter > hypothesized value H a : parameter < hypothesized value

Example 1 Contd.: Julie and Megan wonder if head and tails are equally likely if a penny is spun. They spin pennies 40 times and get 17 heads. Should they reject the standard that pennies land 50% of the time? State the hypotheses : Where p is the true proportion of heads H 0 :  =.5 H a :  ≠.5

Example 2: A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company ’ s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract? State the hypotheses : Where  is the true proportion that heard the ad. H 0 :  =.2 H a :  >.2

Formula for hypothesis test:

Example 1 Contd. Test Statistics for Julie and Megan’s Data

P-values - The probability that the test statistic would have a value as extreme or more than what is actually observed

Level of significance - Is the amount of evidence necessary before we begin to doubt that the null hypothesis is true Is the probability that we will reject the null hypothesis, assuming that it is true Denoted by α –Can be any value –Usual values: 0.1, 0.05, 0.01 –Most common is 0.05

Statistically significant – as smallsmallerThe p-value is as small or smaller than the level of significance ( α ) fail to rejectIf p > α, “fail to reject” the null hypothesis at the  level. rejectIf p < α, “reject” the null hypothesis at the  level.

Facts about p-values: ALWAYS make decision about the null hypothesis! Large p-values show support for the null hypothesis, but never that it is true! Small p-values show support that the null is not true. Double the p-value for two-tail (=) tests Never acceptNever accept the null hypothesis!

Never “accept” the null hypothesis!

At an α  level of.05, would you reject or fail to reject H 0 for the given p-values? a).03 b).15 c).45 d).023 Reject Fail to reject

Writing Conclusions: 1)A statement of the decision being made (reject or fail to reject H 0 ) & why (linkage) 2)A statement of the results in context. (state in terms of H a ) AND

“Since the p-value ) α, I reject (fail to reject) the H 0. I do (do not) have statistically significant evidence to suggest that H a.” Be sure to write H a in context (words)!

Example 1 Contd. The Decision P-Value =.342 Compare the P-Value to the Alpha Level.342 >.05 Since the P-Value is greater than the alpha level I fail to reject that spinning a penny lands heads 50% of the time. I do not have statistically significant evidence to suggest that spinning a penny is anything other than fair.

What? You and Jeff Spun your pennies and got 10 heads out of 40 spins? Well that not what Meg and I got. So what now?

You Decide Joe and Jeff decide to test the same hypothesis but gather their own evidence. They spin pennies 40 times and get 10 heads. Should they reject the standard that pennies land heads 50% of the time?

But we DID reject! We DID NOT reject! Who is Correct? BOTH OF THEM!!! Conclusion are based off of your data. It is important however to discuss possible ERRORS that could have been made.

Errors in Hypothesis Tests Every time you make a decision there is a possibility that an error occurred.

H o is TrueH o is False RejectType I ErrorCorrect Fail to RejectCorrectType II Error ERRORS Murder Trial Revisited Actually Innocent Actually Guilty Decision GuiltyType I ErrorCorrect Decision Not GuiltyCorrectType II Error

Type I Error When you reject a null hypothesis when it is actually true. Denoted by alpha (α) -the level of significance of a test

Type II Error When you fail to reject the null hypothesis when it is false Denoted by beta (β)

Example 2 Revisited: A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company ’ s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract?

Assumptions: Have an SRS of people np = 400(.2) = 80 & n(1-p) = 400(.8) = Since both are greater than 10, this distribution is approximately normal. Population of people is at least H 0 : p =.2where p is the true proportion of people who H a : p >.2heard the ad Since the p-value >α, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than.2. Use the parameter in the null hypothesis to check assumptions! Use the parameter in the null hypothesis to calculate standard deviation!

What type of error could the radio station have made? Type IType II OR

Two-Sample Proportions Inference

Sampling Distributions for the difference in proportions When tossing pennies, the probability of the coin landing on heads is 0.5. However, when spinning the coin, the probability of the coin landing on heads is 0.4. Let’s investigate. Looking at the sampling distribution of the difference in sample proportions: What is the mean of the difference in sample proportions (flip - spin)? What is the standard deviation of the difference in sample proportions (flip - spin)? Can the sampling distribution of difference in sample proportions (flip - spin) be approximated by a normal distribution? Yes, since n 1 p 1 =12.5, n 1 (1-p 1 )=12.5, n 2 p 2 =10, n 2 (1- p 2 )=15 –so all are at least 5)

Assumptions: TwoindependentTwo, independent SRS ’ s from populations Populations at least 10n Normal approximation for both

Formula for confidence interval: Note: use p-hat when p is not known Standard error! Margin of error!

Example 1: At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of 316 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 259 had no visible scars after treatment. Another random sample of 419 patients with minor burns received no plasma compress treatment. For this group, it was found that 94 had no visible scars after treatment. What is the shape & standard error of the sampling distribution of the difference in the proportions of people with visible scars between the two groups? Since n 1 p 1 =259, n 1 (1-p 1 )=57, n 2 p 2 =94, n 2 (1- p 2 )=325 and all > 5, then the distribution of difference in proportions is approximately normal.

Example 1: At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of 316 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 259 had no visible scars after treatment. Another random sample of 419 patients with minor burns received no plasma compress treatment. For this group, it was found that 94 had no visible scars after treatment. What is a 95% confidence interval of the difference in proportion of people who had no visible scars between the plasma compress treatment & control group?

Assumptions: Have 2 independent SRS of burn patients Both distributions are approximately normal since n 1 p 1 =259, n 1 (1-p 1 )=57, n 2 p 2 =94, n 2 (1-p 2 )=325 and all > 5 Population of burn patients is at least Since these are all burn patients, we can add = 735. If not the same – you MUST list separately. We are 95% confident that the true difference in the proportion of people who had no visible scars between the plasma compress treatment & control group is between 53.7% and 65.4%

Example 2: Suppose that researchers want to estimate the difference in proportions of people who are against the death penalty in Texas & in California. If the two sample sizes are the same, what size sample is needed to be within 2% of the true difference at 90% confidence? Since both n ’ s are the same size, you have common denominators – so add! n = 3383

Example 3: Researchers comparing the effectiveness of two pain medications randomly selected a group of patients who had been complaining of a certain kind of joint pain. They randomly divided these people into two groups, and then administered the painkillers. Of the 112 people in the group who received medication A, 84 said this pain reliever was effective. Of the 108 people in the other group, 66 reported that pain reliever B was effective. (BVD, p. 435) a) Construct separate 95% confidence intervals for the proportion of people who reported that the pain reliever was effective. Based on these intervals how do the proportions of people who reported pain relieve with medication A or medication B compare? b) Construct a 95% confidence interval for the difference in the proportions of people who may find these medications effective. CI A = (.67,.83) CI B =(.52,.70) Since the intervals overlap, it appears that there is no difference in the proportion of people who reported pain relieve between the two medicines. CI = (0.017, 0.261) Since zero is not in the interval, there is a difference in the proportion of people who reported pain relieve between the two medicines. SO – which is correct?

Hypothesis statements: H 0 : p 1 = p 2 H a : p 1 > p 2 H a : p 1 < p 2 H a : p 1 ≠ p 2 Be sure to define both p 1 & p 2 !

Since we assume that the population proportions are equal in the null hypothesis, the variances are equal. Therefore, we pool the variances!

Formula for Hypothesis test: p 1 = p 2 So... p 1 – p 2 =0

Example 4: A forest in Oregon has an infestation of spruce moths. In an effort to control the moth, one area has been regularly sprayed from airplanes. In this area, a random sample of 495 spruce trees showed that 81 had been killed by moths. A second nearby area receives no treatment. In this area, a random sample of 518 spruce trees showed that 92 had been killed by the moth. Do these data indicate that the proportion of spruce trees killed by the moth is different for these areas?

Assumptions: Have 2 independent SRS of spruce trees Both distributions are approximately normal since n 1 p 1 =81, n 1 (1-p 1 )=414, n 2 p 2 =92, n 2 (1-p 2 )=426 and all > 5 Population of spruce trees is at least 10,130. H 0 : p 1 =p 2 where p 1 is the true proportion of trees killed by moths H a : p 1 ≠p 2 in the treated area p 2 is the true proportion of trees killed by moths in the untreated area P-value =  = 0.05 Since p-value > , I fail to reject H 0. There is not sufficient evidence to suggest that the proportion of spruce trees killed by the moth is different for these areas