Math Medic Lesson 3.2 AP Statistics Answer Key

This comprehensive answer key provides detailed solutions for all the exercises and problems included in Math Medic Lesson 3.2 of the AP Statistics course.

Confidence Intervals for Proportions

Problem 1:

A survey of 200 adults found that 120 of them have a college degree. Construct a 95% confidence interval for the proportion of adults with college degrees.

Solution:

1. Calculate the sample proportion: p = 120/200 = 0.6
2. Calculate the standard error: SE = sqrt((0.6 * 0.4) / 200) = 0.04
3. Calculate the margin of error: MOE = 1.96 * 0.04 = 0.079
4. Construct the confidence interval: (p – MOE, p + MOE) = (0.6 – 0.079, 0.6 + 0.079) = (0.521, 0.679)

Problem 2:

A company claims that 80% of its customers are satisfied with their products. If a random sample of 100 customers is taken and only 70% of them express satisfaction, test the claim at a significance level of 0.05.

Solution:

1. State the hypotheses:
   – Null hypothesis: p = 0.8
   – Alternative hypothesis: p != 0.8
2. Calculate the sample proportion: p = 70/100 = 0.7
3. Calculate the standard error: SE = sqrt((0.8 * 0.2) / 100) = 0.04
4. Calculate the test statistic: z = (p – 0.8) / SE = -2.5
5. Determine the p-value: p-value = 2 * P(z < -2.5) = 0.012
6. Since the p-value < 0.05, we reject the null hypothesis. There is sufficient evidence to conclude that the proportion of satisfied customers is not 80%.

Hypothesis Testing for Proportions

Problem 3:

A political candidate claims that at least 55% of the voters in her district support her campaign. A random sample of 300 voters is surveyed, and 175 of them indicate support for the candidate. Test the candidate’s claim at a significance level of 0.01.

Solution:

1. State the hypotheses:
   – Null hypothesis: p <= 0.55
   – Alternative hypothesis: p > 0.55
2. Calculate the sample proportion: p = 175/300 = 0.583
3. Calculate the standard error: SE = sqrt((0.55 * 0.45) / 300) = 0.032
4. Calculate the test statistic: z = (p – 0.55) / SE = 1
5. Determine the p-value: p-value = P(z > 1) = 0.1587
6. Since the p-value > 0.01, we fail to reject the null hypothesis. There is not sufficient evidence to support the candidate’s claim.

Problem 4:

A manufacturer claims that the defect rate of its products is less than 5%. A sample of 500 products is tested, and 25 of them are found to be defective. Test the manufacturer’s claim at a significance level of 0.05.

Solution:

1. State the hypotheses:
   – Null hypothesis: p >= 0.05
   – Alternative hypothesis: p < 0.05
2. Calculate the sample proportion: p = 25/500 = 0.05
3. Calculate the standard error: SE = sqrt((0.05 * 0.95) / 500) = 0.014
4. Calculate the test statistic: z = (p – 0.05) / SE = 0
5. Determine the p-value: p-value = P(z < 0) = 0.5
6. Since the p-value > 0.05, we fail to reject the null hypothesis. There is not sufficient evidence to support the alternative hypothesis that the defect rate is less than 5%.

Tips and Tricks

• Remember that the standard error is calculated as SE = sqrt((p * q) / n), where p is the sample proportion, q is 1 – p, and n is the sample size.
• When constructing confidence intervals, use the formula (p +/- MOE), where MOE is the margin of error.
• In hypothesis testing, determine the direction of the alternative hypothesis (less than, greater than, or not equal to) based on the claim or research question.
• Check the sample size to ensure that it is large enough to apply the normal approximation for z-scores (n * p * q >= 10).

Common Mistakes to Avoid

• Incorrectly calculating the sample proportion or standard error. Verify your calculations carefully.
• Confusing the alternative and null hypotheses. Make sure to state the hypotheses correctly based on the claim or research question.
• Using the wrong p-value. The p-value should be based on the direction of the alternative hypothesis and the test statistic.
• Drawing conclusions beyond the scope of the data. The results of hypothesis testing apply only to the population represented by the sample.

Why Confidence Intervals and Hypothesis Testing Matter

Confidence intervals provide a range of plausible values for a population parameter, allowing for uncertainty in the sample. Hypothesis testing helps researchers make informed decisions about whether to accept or reject a claim based on sample evidence. These statistical methods are essential for drawing valid inferences from data in a variety of fields, including psychology, medicine, and business.

Benefits of Confidence Intervals and Hypothesis Testing

• Enhance the accuracy and reliability of population estimates.
• Provide a basis for making informed decisions about claims or hypotheses.
• Allow for the quantification of uncertainty in statistical inferences.
• Facilitate the comparison of different populations or groups.
• Form the foundation for more advanced statistical analyses, such as regression and analysis of variance.

New Applications

Predictive Analytics: Confidence intervals can be used to predict future values or outcomes, such as the probability of a customer purchasing a product.

Quality Control: Hypothesis testing can be applied to monitor production processes, ensuring that products meet specified standards.

Medical Diagnosis: Confidence intervals can help clinicians estimate the likelihood of a patient having a particular disease or condition.

Data-Driven Marketing: Hypothesis testing can be used to evaluate marketing campaigns, determining the effectiveness of different advertising strategies.

Table 1: Effect of Sample Size on Confidence Interval Width

Table 2: Hypothesis Testing for Proportions

Table 3: Examples of Hypothesis Statements