Part 5 Part 2

3 Guess the Mystery Proportion 1. You will select a random sample of 20 beads from the jar. Calculate the proportion of beads that are red (write this as a decimal). Proportion red:_______. This is your point estimate for the true proportion of red beads. 2. Identify the population, parameter, sample, and statistic. Population:______________________________ Parameter:____________________ Sample:________________________________ Statistic:____________________ 3. Now you are going to change your point estimate into an interval of values by adding and subtracting some value from your point estimate (the number you add and subtract is called your margin of error ). You can choose any amount to add and subtract, but remember, the smallest interval that captures the truth is the winner. What margin of error do you want to use? Why? 4. Use your point estimate and chosen margin of error to write an interval that you think contains the true proportion of red beads. 5. How confident do you feel that your interval captures the true proportion? Answer with a percentage. 6. One of the groups got (0.27, 0.33) as their interval. What was their point estimate? What was their margin of error? 7. One group claims that the true proportion of red beads 0.25. Does your interval support or deny this claim? Why? Mrs. Jones wants to implement a new reward system. When a student does something great, they randomly pick a bead from the bead jar. If the bead is red, they get a piece of candy. If not, they don’t win a prize. The questi on is, what are the chances that a student chooses a red bead? Each group will get a sample of beads and will create a confidence interval to estimate the true proportion of red beads. The group with the smallest interval that captures the true proportion wins a prize!

6 Gimme a Kiss! To take your sample, gather five Hershey’s kisses in your cup, shake them up, and drop them from about six inches above your desk. Count the number that land on their base. Repeat ten times to get a sample of size 50, recording your results in the table below. Let 𝑝̂ = the proportion of the Kisses that land flat. Toss Number Number that land on base Toss Number Number that land on base 1 6 2 7 3 8 4 9 5 10 Total 1. What is your point estimate for the true proportion that land flat? ________ 2. Identify the population, parameter, sample and statistic. Population: Parameter: Sample: Statistic: 3. Was the sample a random sample? Why is this important? 4. What is the formula for calculating the standard deviation of the sampling distribution of 𝑝 ̂ ? 5. What condition must be met to use this formula? Has it been met? 6. We don’t know the value of p (that’s the whole point of a confidence interval) so we will use 𝑝̂ instead. Calculate the standard deviation. 7. Would it be appropriate to use a normal distribution to model the sampling distribution of 𝑝̂ ? Justify your answer. 8. In a normal distribution, 95% of the data lies within ______ standard deviations of the mean. (Hint: Use InverseNorm to find.) This value is called the critical value. When you toss a Hershey Kiss, it sometimes lands flat and sometimes lands on its side. What proportion of tosses will land flat?

9 Check Your Understanding What do you want to be when you grow up? According to a nationwide survey of a random sample of 1000 kids under the age of 12, some kids want to be a ninja, a dragon keeper, a dancing unicorn, and even an octopus. Fifty-five of the 1000 kids want to be a doctor. We would like to use this study to find a 98% confidence interval for the true proportion of kids who want to be a doctor. a. Identify and define the parameter of interest. b. Check if the conditions for constructing a confidence interval for p are met. c. Find the critical value for a 98% confidence interval. Then calculate the interval. d. Interpret the interval in context.

12 What does “95% confident” mean? 1. Use the Confidence Intervals for Proportions applet. Set the population proportion to 0.5, the confidence level to 95% and the sample size to 75. 2. Click “Sample” to choose an SRS and display the resulting confidence interval. The confidence interval is displayed as a horizontal line segment with a dot representing the sample proportion in the middle of the interval. The true proportion (p) is the green vertical line. Did the first confidence interval capture the true proportion? Repeat this 10 times and sketch what you see to the right. How many of the intervals capture the true proportion? 3. “Reset” and then take a total of 100 confidence intervals (sample 25 four times). How many out of 100 captured the true proportion? Is this surprising? Why? Interpret the Confidence Level: In repeated samples of size _________, the _______ % confidence intervals created are expected to catch the true __________________ about __________ % of the time. The P.E. teacher of a large high school wants to estimate the proportion of free throws students at this school can do in one minute. He selects a random sample of 30 students from those who are there after school for sports practices. He records how many free throws each of the students in the sample can do in one minute. He determines that he is 90% confident that the interval from 0.50 to 0.65 captures the true proportion of free throws that students at this school can do in one minute. (a) Interpret the confidence level. In this Activity, you will use the applet at www.tinyurl.com/appletCI to learn what it means to say we are “95% confident” that our confidence interval captures the true proportion.

15 Is Mrs. Lee a good free throw shooter? 1. Identify the population, parameter, sample and statistic. Population: Parameter: Sample: Statistic: 2. There are two possible explanations for why Mrs. Lee only made 32/50 shots. 1) 2) To test Mrs. Lee’s claim, we will assume #1, she is an 80% free throw shooter , and examine the likelihood that she makes 32/50 shots through simulation. 3. Use an RNG to simulate 50 free throws shot by an 80% free throw shooter shooting 50 times. What is your sample proportion of shots made? 4. Repeat for another sample of 50 shots. Calculate the sample proportion. 5. Add your sample proportions to the dotplot on the board. Each person in your group should add two dots to the board. Sketch the dotplot below. proportion of FT made Mrs. Lee claims she is an 80% free throw shooter. To prove her skills she shoots 50 free throws and makes 32 shots. Is Mrs. Lee exaggerating about her free throw skills?

18 5.19 Notes day 2: Testing Hypothesis about Proportions #1- State: Hypotheses and parameter #2- Plan: Conditions, Name of test, Significance Level Conditions: 1) Random 2) Independence 3) Large Enough/~N Procedure: Significance Level (alpha): #3- Do: Mechanics – Mean, S.D., test statistic, sketch of model, and P-value. Test Statistic in General: Test Statistic for 1-Proportion Z-Test P-Value:

21 Check Your Understanding A 1996 report stated that 90% of all American homes have at least one smoke detector. A city's fire department has been running a public safety campaign, and wonders if this effort has increased the percentage of smoke detectors in homes. So, building inspectors visit 400 randomly selected homes and find that 376 of them have smoke detectors. Is this strong evidence that the local rate is higher than the national rate?

22 5.19 day 3 Notes: Testing Hypothesis about Proportions 1. a. Sometimes parents and grandparents like to recount how difficult life was when they were kids, such as having to walk 10+ miles to school (in the snow, uphill both ways). A random sample of 180 teenagers were selected and 40% had heard stories from their parents or grandparents about how difficult life was when they were kids. Do these data provide convincing evidence at the α = 0.05 significance level that the proportion of all teenagers who have heard stories from their parents or grandparents about how difficult life was when they were kids differs from 0.50? b. A 95% confidence interval for the proportion of all teenagers who have heard stories from their parents or grandparents about how difficult life was when they were kids is (0.328, 0.472). Explain how the confidence interval is consistent with, but gives more information than, the test.

23 2. The percent of male births has been claimed to be 52% in the US. There has been some changes in prenatal care over the past 10years, and we are wondering if the percent of male births has changed. We took a random sample of 550 births and found that 56.9% of them were male. Is this convincing evidence that the rate has changed? Assume Conditions are met. Check Your Understanding According to the National Institute for Occupational Safety and Health, job stress poses a major threat to the health of workers. A news report claims that 75% of restaurant employees feel that work stress has a negative impact on their personal lives. Managers of a large restaurant chain wonder whether this claim is valid for their employees. A random sample of 100 employees finds that 68 answer “Yes” when asked, “Does work stress have a negative impact on your personal life?” Do these data provide convin cing evidence at the α = 0.10 significance level that the proportion of all employees in this chain who would say “Yes” differs from 0.75?

24 Should Rockford switch to bottled water? The Wolverine Worldwide (a shoe company in Rockford) improperly disposed of chemicals (PFAS), which have leaked into the ground water. The state of Michigan says that if more than 7% of households in a city exceed the safe limit, the city needs to switch to bottled water. A concerned citizen takes a random sample of 100 households and finds that 12 have unsafe water. Do the data provide convincing evidence that Rockford should switch to bottled water? 1. State appropriate hypotheses for performing a significance test. Use α = 0.05. 2. a) After conducting a significance test, a P -value of 0.025 is found. Interpret this value. b) Reject H 0 or fail to reject H 0 ? Keep the current water or switch to bottled water? Explain. c) Let’s suppose this decision is wrong. What would be a consequence of this error? d) If the water is safe, what is the probability that this error will occur? 3. a) Now suppose the P-value was 0.217. Reject H 0 or fail to reject H 0 ? Keep the current water or switch to bottled water? Explain. (b) Let’s suppose this decision is wrong. What would be a consequence of this error? 4. Are the consequences in question #2 or question #3 more serious? Explain.

25 5.20 day 1: More About Tests and Intervals Errors Type I Error- Type II Error- Ho True Ho False Reject Ho Fail to Reject Ho Examples: Jury Trial Defendant is truly Innocent of Crime Defendant is truly Guilty of Crime Jury finds Defendant Guilty ERROR (Salem Witch Trials) Correct Result Jury finds Defendant Not Guilty Correct Result ERROR (OJ Simpson?) Medical Test - draw blood and perform test to see if patient has a certain disease Patient truly is healthy Patient truly has disease Test finds evidence of disease ERROR (false positive) Correct Result Test fails to find evidence of disease Correct Result ERROR (false negative)

26 Example: A manufacturer of hand-held calculators receives large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is selected for inspection. Information from the sample is then used to test H O : p = 0.05 versus H A : p > 0.05 where p is the true proportion of defective circuits in the shipment. If the null hypothesis is not rejected, the shipment is accepted, and the circuits are used in the production of calculators. If the null hypothesis is rejected, the entire shipment is returned to the supplier because of inferior quality (A shipment is defined to be of inferior quality if it contains more than 5% defective circuits.) a.) In this context, define Type I and Type II Errors. b.) From the calculator manufacturer’s point of view, which error is considered more serious and why? c.) From the circuit supplier’s point of view, which type of error is considered more serious and why?

27 Check Your Understanding Mr. Wilcox purchased a trick coin that is supposed to land heads up 75% of the time. One of his students volunteer to test this claim. The student flips the coin 50 times and finds that the coin lands heads up 35 times. The student then performs a test of the following hypotheses at the α = 0.10 significance level: 𝐻 0 : 𝑝 = 0.75 𝐻 𝑎 : 𝑝 < 0.75 where p = the true proportion of tosses of this coin that would land heads-up. 1. Describe a Type I error and a Type II error in this setting. 2. Which type of error may result in Mr. Wilcox returning the coin and writing a negative review of the product? 3. If the student were to use α = 0.05 instead of α = 0.10, would this make it more or less likely to reject the null hypothesis when the null hypothesis is true? Explain.

28 Extra Check Your Understanding The manager of a fast-food restaurant wants to reduce the proportion of drive thru customers who have to wait longer than 2 minutes to receive their food after placing an order. Based on store records, the proportion of customers who had to wait longer than 2 minutes was p = 0.63. To reduce this proportion, the manager assigns an additional employee to drive-thru orders. During the next month, the manager collects a random sample of 250 drive-thru times and finds that 𝑝̂ = 144 250 = 0.576 . The manager then performs a test of the following hypotheses at the α = 0.10 significance level: 𝐻 0 : 𝑝 = 0.63 𝐻 𝑎 : 𝑝 < 0.63 where p = the true proportion of drive-thru customers who have to wait longer than 2 minutes to receive their food. 1. Describe a Type I error and a Type II error in this setting. 2. Which type of error is more serious in this case? Justify your answer. 3. Based on your answer to Question 2 , do you agree with the company’s choice of α = 0.10? Why or why not? 4. The P - value of the manager’s test is 0.0385. Interpret the P -value.

29 Will Mrs. Lee Prove Herself?? 1. Write the appropriate hypotheses for the significance test. Be sure to define the parameter of interest. 2. Describe a Type II error in this setting. 3. Suppose that Mrs. Lee’s true free throw percentage is actually 66% (H a is true!). Enter all information into the applet at https://istats.shinyapps.io/power/ . Click to Display Type II error. It will show as a blue shaded region. (a) If Ha is true, what is the probability of a Type II error (bad decision)? _______ (b) If Ha is true, what would be the probability of a good decision here? _______ (c) Interpret the probability of this good decision in the context of the problem. 4. Click Display Power. We want to increase the power of our test. How could we adjust each of the following factors to increase our power? Use the applet to explore each. a. Sample size: b. Alpha level: c. Alternative p: Recall that Mrs. Lee claims she is an 80% free throw shooter. Mr. Mazzoni is skeptical and believes that she is a less than 80% free throw shooter. Mrs. Lee has decided to take another SRS of 50 free throws to prove herself.

30 5.20 Notes day 2: More About Tests and Intervals Probabilities of Errors and Power P(Type I Error) P(Type II Error) Power Interpret Power: Assuming the __________ (in context at the specific alternative value) is true, there is a ________ chances of finding convincing evidence to reject _______ (in context). Increasing Power: 3 things we can do to increase power are…. 1. Increasing the sample size A larger sample size gives more information about the true parameter. 2. Increase the alpha level Using a larger significance level makes it easier to reject H 0 when Ha is true. 3. Increasing the difference between the null and alternative hypothesis values It is easier to detect a bigger difference between the null and alternative parameter value. H 0 is actually true H 0 is actually false Reject H 0 Type I error (probability = ? ) Wrongly concluding that H 0 is false. Correct Conclusion (probability = 1 - ? ) This is equivalent to the statistical power of the test. Fail to Reject H 0 Correct Conclusion Type II error (probability = ? ) Wrongly concluding that H 0 is true. Example #1: You read that a statistical test at significance level α = 0.05 has power 0.78. What are the probabilities of Type I and Type II errors for this test? Example #2: You read that a statistical test at the α = 0.01 level has probability 0.14 of making a Type II error when a specific alternative is true. What is the power of the test against this alternative?

31 Example #3: A drug manufacturer claims that fewer than 10% of patients who take its new drug for treating Alzheimer’s disease will experience nausea. To test this claim, a significance test is carried out of H O : p = 0.10 H A : p < 0.10 You learn that the power of this test at the 5% significance level against the alternative p = 0.08 is 0.64. Explain in simple language what “power = 0.64” means in this setting. Check Your Understanding A statistics major and a finance major decide to get married. In order to investigate their per person expenses they select a random sample of guests they have invited to the wedding and records whether or not each person plans to attend the wedding. They decide to test H0: p = 0.75 versus Ha: p ≠ 0.75 where p = the true proportion of all guests that will attend the wedding. a. The power of the test to reject H 0 : p = 0.75 when p = 0.70 using α = 0.05 and n = 25 subjects is 0.10. Interpret this value. b. Find the probability of a Type I error and the probability of a Type II error for the test in part (a). c. Determine whether each of the following changes would increase or decrease the power of the test. Explain your answers. Use α = 0.01 instead of α = 0.05. Use n = 100 instead of n = 25.

32 5.20 Notes day 3: More About Tests and Intervals Confidence Intervals and Hypothesis Tests Confidence Intervals: Two-tailed Test One-tailed Test 1) Soon after the euro was introduced as currency in Europe, it was widely reported that someone had spun a euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. a) Estimate the true proportion of heads. Use a 95% confidence interval. Assume the conditions are met. b) Does your confidence interval provide evidence that the coin is unfair when spun? c) What is the level of significance of this test?

33 2) In Philadelphia it is believed that 12% of motorists run red lights. The city planners decide to put a few traffic cameras that will issue tickets to drivers that run red lights. After 2 months they check out a few corners where cameras had been installed. They found that out of 240 drivers that had the opportunity to run the red light 15 did so and were issued a ticket. a) Does this provide evidence of a decline in the percentage of drivers that run the red light? Use a level of significance of 5%. Assume conditions are met and just calculate the P-value b) Interpret the P-value. c) Are the results statistically significant? State the conclusion in context. d) What is the confidence level for an interval that matches the significance level for this test? e) Create the confidence interval. f) Is p = .12 in the interval? How does this confirm what was found in the significance test? g) Which type of error could have been made: Type I or Type II? h) What is a Type I error in this context? What is its probability?

34 Which grade is more likely to go to Prom? 1. What is the point estimate for… the proportion of Juniors planning on attending prom? 𝑝 1 ̂ = ________ the proportion of Seniors planning on attending prom? 𝑝 2 ̂ = ________ the difference in the proportion of Jrs and Srs planning on attending prom? 𝑝 1 ̂ − 𝑝 2 ̂ = ________ 2. Check the conditions needed in order to construct a confidence interval. Random: 10%: Large Counts: 3. Construct and interpret a 95% confidence interval for the difference in proportions of Juniors and Seniors who are planning on attending prom. 4. Does the interval provide convincing evidence that Juniors have a lower proportion planning on going to prom or is it plausible that there is no difference between the two classes? Explain. At many high schools, Prom is an annual dance that only Juniors and Seniors can purchase tickets for. The student council at a large high school is wondering if Juniors or Seniors are more likely to attend Prom. They take a random sample of 50 Juniors and find that 28 are planning on attending Prom. They select a random sample of 45 Seniors and 29 are planning on attending. Construct and interpret a 95% confidence interval for the difference in proportions of Juniors and Seniors who are planning on attending Prom.

35 5.21 day 1 Notes: Comparing Two Proportions Constructing a Confidence Interval for p 1 – p 2 State: Plan: 1) Random 2) Independence 3) Large Enough/~N 4) Independent Groups Procedure: Do: Conclude: We are ___________% confident that the true difference in proportions of _______________and ______________ is between _______________and ______________. If 𝑝̂ 1 − 𝑝̂ 2 and the interval is:

36 Check Your Understanding A recent random sample of 200 U.S. females revealed 110 use Twitter regularly. A separate random sample of 150 males revealed that 60 use Twitter regularly. Construct a 95% confidence interval for the true difference in proportions who use Twitter regularly (females – male).

37 5.21 Notes day 2: Comparing Two Proportions Conducting a Test of Significance for p 1 – p 2 State: Plan: Procedure: Do: Pooling: In the null hypothesis, we assume 𝐩 ? = 𝐩 ? , therefore we combine 𝑝̂ 1 and 𝑝̂ 2 to get a better estimate of the parameter. This is called the pooled (or combined) proportion. ˆ p 1 = ˆ p 2 = Pooled 𝑝̂ ( or 𝑝̂ 𝑐 ) = ______________ Standard Error (pooled): 𝜎 𝑝 ̂ 1 −𝑝 ̂ 2 = 𝜎 𝑝 ̂ 𝑐 = √ 𝑝 ̂ 𝑐 (1−𝑝 ̂ 𝑐 ) 𝑛 1 + 𝑝 ̂ 𝑐 (1−𝑝 ̂ 𝑐 ) 𝑛 2 = Test Statistic: *P-Value calculation is the same as 1-Proportion Conclude: Same first sentence: We reject/fail to reject … Second sentence: We have sufficient/insufficient evidence that the true proportion of _____________ is <, >,  the true proportion of _____________________. Or We have sufficient/insufficient evidence that the true difference in proportion of _______________ and ______________ is <, >, ≠ 0.

38 Example #1: According to phys.org, Black and Hispanic females are underrepresented in STEM programs compared to non-STEM programs. A certain university would like to see if this is true for their student population. They took a random sample of 300 STEM students and found that 12 were Black or Hispanic females. A separate random sample of 500 non-STEM students had 75 Black or Hispanic females. Do the data provide convincing evidence that Black and Hispanic females are underrepresented in STEM programs? Use a 5% significance level.

39 Check Your Understanding To study the long-term effects of preschool programs for poor children, researchers designed an experiment. They recruited 123 children who had never attended preschool from low-income families in Michigan. Researchers randomly assigned 62 of the children to attend preschool (paid for by the study budget) and the other 61 to serve as a control group who would not go to preschool. One response variable of interest was the need for social services as adults. Over a 10-year period, 38 children in the preschool group and 49 in the control group have needed social services. Do these data provide convincing evidence that preschool reduces the later need for social services for children like the ones in this study?

40 Extra Check Your Understanding Did you sleep with a comfort object as a child? Do you still? No need to stuff that teddy bear from your infant years in your closet the next time you have a guest over — the odds are good that he or she has a comfort object, too. A study conducted by Sleepopolis and OnePoll reported that 34% of adults still sleep with a stuffed animal, blanket, or other sentimental object, according to a survey of a random sample of 100 adults. A similar study, based on a random sample of 80 children reports that 75% of children sleep with a comfort object. Do these data provide convincing evidence that the proportion of adults who sleep with a comfort object is less than the proportion of children who sleep with a comfort object? Justify your answer.

41 What the AP FRQ Rubric will look for in a significance test: Section 1 1. Hypotheses 2. Identify parameters as proportions 3. Clearly define parameters in context 4. Identify test by name or formula Section 2 1. Random condition 2. Normal condition 3. 10% condition 4. Test statistic 5. P-value Section 3 1. Compare P-value to alpha 2. Correct conclusion consistent with the alternative hypothesis 3. Conclusion is in context Possible Section 4 Follow Up Questions: o Given your conclusion in part (a), which kind of mistake — a Type I error or a Type II error — could you have made? Explain what this mistake would mean in context. o Why do we check the (random, 10%, Large Counts) condition? o A simulation-based estimate of the P-value o Give students the confidence interval for the data in part (a) and ask them to explain why the confidence interval leads to the same conclusion as the significance test. o Ask students if the significance test reveals a causal relationship. If the data comes from an observational study, then we cannot infer causation. o The power of this test to detect the alternative (Ha) is _______. Interpret this value.