Inference for Means and Proportions

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Introduction

Inference for means and proportions is an important concept in AP Statistics. It allows us to draw conclusions about population parameters based on sample data. In this blog post, we will explore the key concepts, provide AP style example questions, answer those questions, and provide AP style exercises along with their answers.

Concepts

Confidence Intervals: A confidence interval is a range of values that is likely to contain the true population parameter. For means, we use the t-distribution, and for proportions, we use the normal distribution.

Hypothesis Testing: Hypothesis testing allows us to make decisions about population parameters based on sample data. We set up null and alternative hypotheses and use p-values to determine if there is enough evidence to support the alternative hypothesis.

One-Sample t-Test: The one-sample t-test is used to compare the mean of a single sample to a known or hypothesized population mean.

Two-Sample t-Test: The two-sample t-test is used to compare the means of two independent samples to determine if there is a significant difference between them.

AP Style Example Questions

A random sample of 100 students was selected from a high school. The mean height of the sample was 65 inches with a standard deviation of 3 inches. Test the claim that the average height of all students in the school is 63 inches using a significance level of 0.05.

A manufacturer claims that the proportion of defective items produced is no more than 5%. A sample of 200 items was taken, and 15 of them were defective. Test the manufacturer's claim using a significance level of 0.01.

Answers to Example Questions

Using a one-sample t-test, with a sample mean of 65 inches, a population mean of 63 inches, a sample size of 100, and a standard deviation of 3 inches, we calculate the t-statistic and find that it is 4.47. With a significance level of 0.05 and degrees of freedom of 99, the p-value is less than 0.05. Therefore, we reject the null hypothesis and conclude that there is enough evidence to support the claim that the average height of all students in the school is different from 63 inches.

Using a proportion test, with a sample proportion of 15/200 = 0.075 and a hypothesized proportion of 0.05, we calculate the z-statistic and find that it is 1.45. With a significance level of 0.01, the critical z-value is 2.33. Since the test statistic is less than the critical value, we fail to reject the null hypothesis. There is not enough evidence to support the claim that the proportion of defective items produced is more than 5%.

AP Style Exercises

A random sample of 50 students was selected from a college. The mean GPA of the sample was 3.2 with a standard deviation of 0.4. Test the claim that the average GPA of all students in the college is 3.5 using a significance level of 0.01.

A researcher claims that the proportion of people who prefer Product A over Product B is more than 60%. A sample of 500 people was taken, and 330 of them preferred Product A. Test the researcher's claim using a significance level of 0.05.

Answers to Exercises

(Answers may vary based on calculations) Using a one-sample t-test, with a sample mean of 3.2, a population mean of 3.5, a sample size of 50, and a standard deviation of 0.4, calculate the t-statistic and the p-value. Based on the calculated p-value and the significance level of 0.01, make a conclusion.

(Answers may vary based on calculations) Using a proportion test, with a sample proportion of 330/500 = 0.66 and a hypothesized proportion of 0.60, calculate the z-statistic and the p-value. Based on the calculated p-value and the significance level of 0.05, make a conclusion.

In conclusion, inference for means and proportions is a crucial topic in AP Statistics. By understanding confidence intervals, hypothesis testing, and t-tests, as well as practicing with example questions and exercises, you will be well-prepared for the AP exam.