AP Test Practice

A student organization at a university is interested in estimating the proportion of students in favor of showing movies biweekly instead of monthly. How many students should be sampled to get a 90 percent confidence interval with a width of at most 0.08? (A) 27 (B) 64 (C) 106 (D) 256 (E) 423 E

A small kiosk at the Atlanta airport carries souvenirs in the price range of $3.99 to $29.99, with a mean price of $ The airport authorities decide to increase the rent charged for a kiosk by 5 percent. To make up for the increased rent, the kiosk owner decides to increase the prices of all items by 50 cents. As a result, which of the following will happen? (A) The mean price and the range of prices will increase by 50 cents. (B) The mean price will remain the same, but the range of prices will increase by 50 cents. (C) The mean price and the standard deviation of prices will increase by 50 cents. (D) The mean price will increase by 50 cents, but the standard deviation of prices will remain the same. (E) The mean price and the standard deviation will remain the same. D

The range and the standard deviation remain unaffected by the constant increment in all the measurements, but the mean will increase by 50 cents.

AP STATISTICS 10.3 Large-Sample Hypothesis Tests for a Population Proportion

10.3 Objectives:  Understand the definition of a test statistic and its role in hypothesis testing  Understand the definition of P-value and its role in hypothesis testing  Be able to use the hypothesis testing procedure to test a hypothesis about a population proportion

10.3 Objectives:  Have a sense that the hypothesis testing procedure for testing a hypothesis about a population proportion is based on the sampling distribution of the sample proportion, including relevant assumptions

Large-Sample Hypothesis Test for a Population Proportion The fundamental idea behind hypothesis testing is: We reject H 0 if the observed sample is very unlikely to occur if H 0 is true.

AP* Tips The AP Exam provides a formula sheet. The formula sheet gives the general form of the test statistic as follows:

3. When n is large, the sampling distribution of p is approximately normal. Recall the General Properties for Sampling Distributions of p As long as the sample size is less than 10% of the population These three properties imply that the standardized variable has an approximately standard normal distribution when n is large.

In June 2006, an Associated Press survey was conducted to investigate how people use the nutritional information provided on food packages. Interviews were conducted with 1003 randomly selected adult Americans, and each participant was asked a series of questions, including the Question 1: When purchasing packaged food, how often do you check the nutritional labeling on the package? Question 2: How often do you purchase food that is bad for you, even after you’ve checked the nutrition labels? following two: It was reported that 582 responded “frequently” to the question about checking labels and 441 responded “very often” or “somewhat often” to the question about purchasing bad foods even after checking the labels. Based on these data, is it reasonable to conclude that a majority of adult Americans frequently check nutritional labels when purchasing packaged foods?

Nutritional Labels Continued... H 0 : p =.5 H a : p >.5 p = true proportion of adult Americans who frequently check nutritional labels For this sample: We use p >.5 to test for a majority of adult Americans who frequently check nutritional labels. This observed sample proportion is greater than.5. Is it plausible a sample proportion of p =.58 occurred as a result of chance variation, or is it unusual to observe a sample proportion this large when p =.5? We will create a test statistic using: A test statistic indicates how many standard deviations the sample statistic (p) is from the population characteristic (p).

Since the P-value is so small, we reject H 0. There is convincing evidence to suggest that the majority of adult Americans frequently check the nutritional labels on packaged foods. Nutritional Labels Continued... H 0 : p =.5 H a : p >.5 p = true proportion of adult Americans who frequently check nutritional labels For this sample: Next we find the P-value for this test statistic. 0 In the standard normal curve, seeing a value of 5.08 or larger is unlikely. It’s probability is approximately 0. The P-value is the probability of obtaining a test statistic at least as inconsistent with H 0 as was observed, assuming H 0 is true. P-value ≈ 0

Computing P-values The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. H a : p > hypothesize value Calculated z z curve P-value = area in upper tail

Computing P-values The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. H a : p < hypothesize value Calculated z z curve P-value = area in lower tail

Computing P-values The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. H a : p ≠ hypothesize value Calculated z and –z z curve P-value = sum of area in two tails

Using P-values to make a decision: To decide whether or not to reject H 0, we compare the P-value to the significance level  If the P-value > , we “fail to reject” the null hypothesis. If the P-value < , we “reject” the null hypothesis.

Summary of the Large-Sample z Test for p Null hypothesis:H 0 : p = hypothesized value Test Statistic: Alternative Hypothesis:P-value: H a : p > hypothesized value Area to the right of calculated z H a : p < hypothesized value Area to the left of calculated z H a : p ≠ hypothesized value 2(Area to the right of z) of +z or 2(Area to the left of z) of -z

Summary of the Large-Sample z Test for p Continued... Assumptions: 1. p is a sample proportion from a random sample 2.The sample size n is large. (np > 10 and n(1 - p) > 10) 3.If sampling is without replacement, the sample size is no more than 10% of the population size

27 Recommended Steps in Hypothesis-Testing Analysis 1.Describe the population characteristic about which hypotheses are to be tested. 2.State the null hypothesis H 0. 3.State the alternative hypothesis H a. 4.Select the significance level a for the test. 5.Display the test statistic to be used, with substitution of the hypothesized value identified in Step 2 but without any computation at this point.

28 Recommended Steps in Hypothesis-Testing Analysis 6. Check to make sure that any assumptions required for the test are reasonable. 7. Compute all quantities appearing in the test statistic and then the value of the test statistic itself. 8. Determine the P-value associated with the observed value of the test statistic.

29 Hypothesis-Testing Analysis 9. State the conclusion (which is to reject H 0 if P-value ≤ α and not to reject H 0 otherwise). The conclusion should then be stated in the context of the problem, and the level of significance should be included.

A report states that nationwide, 61% of high school graduates go on to attend a two-year or four-year college the year after graduation. Suppose a random sample of 1500 high school graduates in 2009 from a particular state estimated the proportion of high school graduates that attend college the year after graduation to be 58%. Can we reasonably conclude that the proportion of this state’s high school graduates in 2009 who attended college the year after graduation is different from the national figure? Use  =.01. H 0 : p =.61 H a : p ≠.61 Where p is the proportion of all 2009 high school graduates in this state who attended college the year after graduation State the hypotheses.

College Attendance Continued... H 0 : p =.61 H a : p ≠.61 Where p is the proportion of all 2009 high school graduates in this state who attended college the year after graduation Assumptions: Given a random sample of 1500 high school graduates Since 1500(.61) > 10 and 1500(.39) > 10, sample size is large enough. Population size is much larger than the sample size.

College Attendance Continued... H 0 : p =.61 H a : p ≠.61 Where p is the proportion of all 2009 high school graduates in this state who attended college the year after graduation Test statistic: P-value = 2(.0087) =.0174 Since P-value > α, we fail to reject H 0. The evidence does not suggest that the proportion of 2009 high school graduates in this state who attended college the year after graduation differs from the national value. Use  =.01 The area to the left of is approximately.0087 What potential error could you have made? Type II

In December 2009, a county-wide water conservation campaign was conducted in a particular county. In 2010, a random sample of 500 homes was selected and water usage was recorded for each home in the sample. Suppose the sample results were that 220 households had reduced water consumption. The county supervisors wanted to know if their data supported the claim that fewer than half the households in the county reduced water consumption. H 0 : p =.5 H a : p <.5 where p is the proportion of all households in the county with reduced water usage State the hypotheses. Calculate p.

Water Usage Continued... H 0 : p =.5 H a : p <.5 2. Sample size n is large because np = 250 >10 and n(1-p) = 250 > It is reasonable that there are more than 5000 (10n) households in the county. 1. p is from a random sample of households Verify assumptions where p is the proportion of all households in the county with reduced water usage

Water Usage Continued... H 0 : p =.5 H a : p <.5 Calculate the test statistic and P-value where p is the proportion of all households in the county with reduced water usage Look this value up in the table of z curve areas P-value =.0037 Since P-value < , we reject H 0. There is convincing evidence that the proportion of households with reduced water usage is less than half. Use  =.01 What potential error could you have made? Type I

Water Usage Continued... H 0 : p =.5 H a : p <.5 Compute a 98% confidence interval: where p is the proportion of all households in the county with reduced water usage Since P-value < , we reject H 0. Used  =.01 Let’s create a confidence interval with this data. What is the appropriate confidence level to use? Since we are testing H a : p <.5,  would also be in the lower tail. Confidence intervals are two- tailed, so we need to put.01 in the upper tail (since the curve is symmetrical)..5 With.01 in each tail, that puts.98 in the middle – this is the appropriate confidence level.98 Notice that the hypothesized value of.5 is NOT in the 98% confidence interval and that we “rejected” H 0 !

College Attendance Revisited... H 0 : p =.61 H a : p ≠.61 Where p is the proportion of all 2009 high school graduates in this state who attended college the year after graduation Since P-value > , we fail to reject H 0. Use  =.01 Let’s compute a confidence interval for this problem. This is a two-tailed test so  gets split evenly into both tails, leaving 99% in the middle..99 Notice that the hypothesized value of.61 IS in the 98% confidence interval and that we “failed to reject” H 0 !

AP* Tips Understanding what a P-value represents is critical to being able to draw a conclusion in a hypothesis test.

AP* Tips When carrying out a hypothesis test on the AP exam, be sure to include all the steps. Omitting any of the steps will reduce your score on a hypothesis test exam question

AP* Tips A common mistake on the AP exam is not stating and checking required conditions. You should know the conditions for each test and remember to check them.

10.3 Objectives: Understand the definition of a test statistic and its role in hypothesis testing Understand the definition of P-value and its role in hypothesis testing Be able to use the hypothesis testing procedure to test a hypothesis about a population proportion

10.3 Objectives: Have a sense that the hypothesis testing procedure for testing a hypothesis about a population proportion is based on the sampling distribution of the sample proportion, including relevant assumptions

Homework: P599: 10.25, 10.26, 10.27, 10.37, 10.38, 10.69, 10.70

HW: Read 10.3: Large-Sample Hypothesis Tests for a Population Proportion Kaplan For Tonight: P599: 10.25, 10.26, 10.27, 10.37, 10.38, 10.69, 10.70

FRQs – What past FRQs relate to this topic? 1998A: #5a 2003B: #3b 2005A: #4 2006B: #6a