2006 Ap Statistics Response Questions Answers

2006 Ap Statistics Response Questions Answers Unveiling the Secrets 2006 AP Statistics Response Questions and Answers The AP Statistics exam notorious for its challenging freeresponse questions is a crucial hurdle for students aspiring to gain college credit While the exam changes annually understanding past questions provides valuable insight into the exams structure content and the skills tested This article dives into the 2006 AP Statistics freeresponse questions offering detailed solutions and analysis Exam Overview The 2006 AP Statistics exam consisted of two sections Multiple Choice 40 of the score and Free Response 60 of the score The Free Response section comprised six questions each worth 9 points Question 1 Sampling and Bias Question A large company is introducing a new product The company wants to know what percentage of the population would be interested in buying the product They conduct a survey of 200 randomly selected people from the population Of the 200 people surveyed 80 said they would be interested in buying the product a What is the population of interest in this study b What is the sample in this study c What is the population parameter of interest d What is the statistic that the company calculated from the sample e Is this a good way to estimate the percentage of the population who would be interested in buying the product Explain your answer Solution a The population of interest is all people in the population b The sample is the 200 people surveyed c The population parameter of interest is the percentage of the entire population who would 2 be interested in buying the product d The statistic calculated is the sample proportion which is 80200 04 or 40 e This is a good way to estimate the population percentage if the sample is truly random and representative of the population However there is always a possibility of sampling bias where the sample doesnt accurately reflect the population Key Concepts Population Sample Parameter Statistic Random Sampling Sampling Bias Question 2 Probability and Simulation Question A company that produces breakfast cereal puts a small prize in each box of cereal There are 5 different prizes and each prize is equally likely to be included in any box of cereal a What is the probability that a customer who buys one box of cereal will get a particular prize b What is the probability that a customer who buys two boxes of cereal will get the same prize in both boxes c A customer wants to collect all 5 different prizes Describe a simulation that could be used to estimate the average number of boxes of cereal the customer would need to buy to collect all 5 prizes Solution a The probability of getting a particular prize is 15 or 20 b The probability of getting the same prize in two boxes is 1515 125 c Heres a simulation 1 Generate a random number between 1 and 5 to represent the prize in each box 2 Repeat the process for each box of cereal until all 5 numbers have been generated 3 Record the number of boxes needed to collect all 5 prizes 4 Repeat the simulation many times at least 100 and calculate the average number of boxes needed Key Concepts Probability Independent Events Simulation Random Number Generation Question 3 Confidence Intervals Question A researcher is interested in estimating the average amount of time students spend studying for a particular exam The researcher randomly selects a sample of 40 3 students and finds that the average study time for the sample is 85 hours with a standard deviation of 23 hours a Construct a 95 confidence interval for the population mean study time b Interpret the confidence interval in the context of the problem c The researcher believes that the average study time for all students is actually 10 hours Based on the confidence interval what can the researcher conclude about this belief Solution a Using a tdistribution since the population standard deviation is unknown the 95 confidence interval is 85 2023 23 sqrt40 803 897 hours b We are 95 confident that the true average study time for all students is between 803 and 897 hours c Since 10 hours falls outside the 95 confidence interval the researcher can reject the belief that the average study time is 10 hours at the 5 significance level Key Concepts Confidence Interval tdistribution Interpretation of Confidence Interval Hypothesis Testing Question 4 Hypothesis Testing Question A pharmaceutical company is testing a new drug to treat a certain illness They randomly assign 100 patients to either the treatment group receiving the new drug or the control group receiving a placebo The company wants to know if the new drug is effective in reducing symptoms of the illness a State the null and alternative hypotheses for this test b Describe how the company could use a twosample ttest to test the hypothesis c What are the possible conclusions that the company could draw from the results of the hypothesis test Solution a Null Hypothesis H0 The new drug has no effect on reducing symptoms of the illness Alternative Hypothesis Ha The new drug is effective in reducing symptoms of the illness b The company can compare the average symptom reduction in the treatment group to the average symptom reduction in the control group using a twosample ttest 4 c Based on the pvalue the company can either Reject the null hypothesis The drug is effective in reducing symptoms Fail to reject the null hypothesis There is not enough evidence to conclude that the drug is effective Key Concepts Hypothesis Testing Null Hypothesis Alternative Hypothesis TwoSample t test Pvalue Question 5 ChiSquare Test Question A researcher is interested in studying the relationship between gender and opinion on a particular political issue The researcher randomly selects a sample of 100 people and asks them their gender and their opinion on the issue The results are summarized in the following table Gender Opinion Male Agree 30 Male Disagree 20 Female Agree 25 Female Disagree 25 a Calculate the expected counts for each cell in the table under the assumption of independence between gender and opinion b Perform a chisquare test to determine if there is a statistically significant relationship between gender and opinion on the issue c State the conclusion of the chisquare test in the context of the problem Solution a The expected counts are calculated based on the row and column totals For example the expected count for Male and Agree is 45 55 100 2475 b The chisquare statistic is calculated by summing the squared differences between observed and expected counts divided by expected counts In this case the chisquare statistic is approximately 274 c The pvalue for this chisquare test is approximately 0098 Since the pvalue is greater than 005 we fail to reject the null hypothesis There is not enough evidence to conclude that there is a statistically significant relationship between gender and opinion on the issue 5 Key Concepts ChiSquare Test Independence Expected Counts ChiSquare Statistic P value Conclusion Question 6 Regression Question A researcher is interested in studying the relationship between the number of hours students study for an exam and their exam score The researcher collects data on 10 students and obtains the following scatterplot Include a scatterplot depicting a positive linear relationship a Describe the relationship between the number of hours studied and the exam score b Calculate the correlation coefficient for the data c Fit a linear regression model to the data and interpret the slope of the regression line d Predict the exam score for a student who studies for 5 hours Solution a The scatterplot shows a positive linear relationship between hours studied and exam score As the number of hours studied increases the exam score tends to increase b The correlation coefficient for this data is approximately 092 indicating a strong positive linear relationship c The regression line equation is Exam Score a b Hours Studied The slope b represents the average increase in exam score for each additional hour studied The interpretation of the slope depends on the specific regression equation calculated d Substitute 5 hours into the regression equation to predict the exam score for a student who studies for 5 hours Key Concepts Scatterplot Linear Relationship Correlation Coefficient Linear Regression Slope Interpretation Prediction Conclusion By dissecting the 2006 AP Statistics freeresponse questions we gain a deeper understanding of the exams format and the skills emphasized This analysis highlights the importance of mastering topics like sampling probability confidence intervals hypothesis testing chi square tests and regression Remember practice is key to success on the AP Statistics exam Understanding past questions and their solutions allows you to identify your strengths and weaknesses refine your problemsolving skills and ultimately improve your chances of 6 scoring well on the exam