2017 ap statistics free response solutions

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17-12-2024

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2017 AP Statistics Free Response Questions: Solutions and Explanations

The 2017 AP Statistics exam presented students with challenging free-response questions. This article provides solutions and detailed explanations for each problem, helping you understand the concepts and improve your problem-solving skills. Remember that understanding the underlying statistical principles is crucial, not just memorizing the solutions.

Question 1: Randomized Experiment

This question involved a randomized experiment comparing two different methods for teaching statistics.

(a) Identify the treatments and experimental units.

• Treatments: Method A and Method B (the two teaching methods).
• Experimental Units: The students enrolled in the statistics course.

(b) Describe how the researchers could randomly assign the students to the two treatments.

Several methods are acceptable, but the key is randomization. For example:

• Assign each student a unique number. Use a random number generator to assign numbers to Method A and the remaining to Method B.
• Write each student's name on a slip of paper, mix them thoroughly, and then draw half to assign to Method A.

(c) Explain why it is important to randomly assign the students to the treatments.

Random assignment helps ensure that the groups are as similar as possible before the experiment begins. This minimizes bias and allows researchers to more confidently attribute any observed differences in outcomes to the treatments themselves, rather than pre-existing differences between the groups.

(d) Describe a potential confounding variable and explain how it could affect the results.

A confounding variable is a factor that influences both the treatment and the response variable, making it difficult to isolate the effect of the treatment. Possible examples include:

• Prior statistical knowledge: Students with stronger prior knowledge might perform better regardless of the teaching method.
• Study habits: Students with better study habits might do better regardless of the method.

Question 2: Inference for Two Proportions

This question involved analyzing the results of a survey about smartphone usage.

(a) Calculate the sample proportions.

This involves calculating the proportion of smartphone users in each group (e.g., male and female). The formula is: sample proportion = (number of successes) / (sample size).

(b) Construct a 95% confidence interval for the difference in population proportions.

Use the two-proportion z-interval formula. This requires calculating the pooled proportion, the standard error, and then applying the formula. Remember to interpret the interval in context – it gives a range of plausible values for the difference in the proportions.

(c) State the hypotheses for a two-sided test to compare the population proportions.

• Null hypothesis (H₀): p₁ - p₂ = 0 (There is no difference in population proportions)
• Alternative hypothesis (Hₐ): p₁ - p₂ ≠ 0 (There is a difference in population proportions)

(d) Perform the hypothesis test and state your conclusion in the context of the problem.

Use the two-proportion z-test. Calculate the test statistic and the p-value. Compare the p-value to the significance level (alpha, often 0.05). If the p-value is less than alpha, reject the null hypothesis and conclude that there is a significant difference. Otherwise, fail to reject the null hypothesis. Remember to state your conclusion in the context of the problem.

Question 3: Inference for Regression

This question examined the relationship between two variables using regression analysis.

(a) Interpret the slope and y-intercept in context.

The slope represents the change in the response variable for a one-unit increase in the explanatory variable. The y-intercept is the predicted value of the response variable when the explanatory variable is zero. Always interpret these values in the context of the problem.

(b) Calculate and interpret the coefficient of determination (R²).

R² represents the proportion of the variation in the response variable that is explained by the linear relationship with the explanatory variable. A higher R² indicates a stronger linear relationship.

(c) Construct and interpret a 95% confidence interval for the slope.

Use the formula for a confidence interval for the slope of a regression line. This involves the estimated slope, the standard error of the slope, and the t-critical value. The interval gives a range of plausible values for the true slope of the population regression line.

(d) Perform a hypothesis test for the slope.

Test whether the slope is significantly different from zero. The null hypothesis is that the slope is zero (no linear relationship). The alternative hypothesis is that the slope is not zero (there is a linear relationship). Calculate the test statistic and p-value and draw your conclusion.

Remember: This is a simplified overview. Each question in the 2017 AP Statistics free response section required detailed calculations and explanations. Consult the official AP Statistics scoring guidelines and your textbook for more detailed solutions and to understand the nuances of each problem. Practice is key to mastering AP Statistics. Work through additional practice problems and review the concepts to improve your understanding and test-taking skills.