Step-by-Step Guidance for Hypothesis Testing with Proportions

Q1. Write the null and alternative hypotheses for each situation:

Background

Topic: Hypothesis Testing for Proportions

This question is about formulating null and alternative hypotheses for different real-world scenarios involving proportions. Understanding how to set up hypotheses is a foundational step in hypothesis testing.

Key Terms:

• Null Hypothesis (H0): The default assumption (e.g., no change, no effect).

• Alternative Hypothesis (Ha): The claim you are testing for (e.g., a difference, an effect).

• Proportion (p): The probability or percentage of success in the population.

Step-by-Step Guidance

1. Read each scenario carefully and identify what is being tested (e.g., fairness of a coin, effectiveness of a product, change in a percentage).

2. Determine the value of the population proportion under the null hypothesis ($p_0$).

3. Decide if the alternative hypothesis is one-sided (greater than or less than) or two-sided (not equal to).

4. Write $H_0$ as $p = p_0$ and $H_a$ as $p \neq p_0$, $p > p_0$, or $p < p_0$ as appropriate.

Try writing the hypotheses for each scenario before checking the provided answers!

Q2. Drug manufacturer claims fewer than 10% of patients experience nausea. In a sample of 250, 23 experienced nausea.

Background

Topic: One-Proportion Z-Test

This question tests your ability to set up and perform a hypothesis test for a single population proportion, including calculating the sample proportion and standard error.

Key Terms and Formulas:

• Sample Proportion ($\hat{p}$): $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes and $n$ is the sample size.

• Standard Error (SE): $SE = \sqrt{\frac{p_0(1-p_0)}{n}}$

• Test Statistic (z): $z = \frac{\hat{p} - p_0}{SE}$

Step-by-Step Guidance

1. State the null and alternative hypotheses: $H_0: p = 0.10$, $H_a: p < 0.10$.

2. Calculate the sample proportion: $\hat{p} = \frac{23}{250}$.

3. Compute the standard error using $p_0 = 0.10$ and $n = 250$:

4. $SE = \sqrt{\frac{0.10 \times (1 - 0.10)}{250}}$

5. Set up the formula for the z-score, but stop before plugging in the final values:

6. $z = \frac{\hat{p} - 0.10}{SE}$

Try calculating the sample proportion and standard error before moving on!

Q3. Sleep apnea in men: Is the percentage greater than 5.8%? Sample of 100 men, 10 have sleep apnea. $\alpha = 0.05$

Background

Topic: One-Proportion Z-Test (Right-Tailed)

This question asks you to test whether the proportion of men with sleep apnea is greater than a known value, using a sample proportion and significance level.

Key Terms and Formulas:

• Null Hypothesis ($H_0$): $p = 0.058$

• Alternative Hypothesis ($H_a$): $p > 0.058$

• Sample Proportion ($\hat{p}$): $\hat{p} = \frac{10}{100}$

• Standard Error (SE): $SE = \sqrt{\frac{0.058 \times (1 - 0.058)}{100}}$

• Test Statistic (z): $z = \frac{\hat{p} - 0.058}{SE}$

Step-by-Step Guidance

1. Write the hypotheses: $H_0: p = 0.058$, $H_a: p > 0.058$.

2. Calculate the sample proportion: $\hat{p} = \frac{10}{100}$.

3. Compute the standard error using $p_0 = 0.058$ and $n = 100$:

4. $SE = \sqrt{\frac{0.058 \times (1 - 0.058)}{100}}$

5. Set up the formula for the z-score, but do not compute the final value:

6. $z = \frac{\hat{p} - 0.058}{SE}$

Try setting up the calculations and see if you can find the z-score!

Q4. National Academy of Science: Is the percentage of US-authored math research articles no longer 40%? Sample of 130, 62 US authors. $\alpha = 0.10$

Background

Topic: Two-Tailed Proportion Test

This question involves testing whether a population proportion has changed from a historical value, using a two-tailed test.

Key Terms and Formulas:

• Null Hypothesis ($H_0$): $p = 0.40$

• Alternative Hypothesis ($H_a$): $p \neq 0.40$

• Sample Proportion ($\hat{p}$): $\hat{p} = \frac{62}{130}$

• Standard Error (SE): $SE = \sqrt{\frac{0.40 \times (1 - 0.40)}{130}}$

• Test Statistic (z): $z = \frac{\hat{p} - 0.40}{SE}$

Step-by-Step Guidance

1. State the hypotheses: $H_0: p = 0.40$, $H_a: p \neq 0.40$.

2. Calculate the sample proportion: $\hat{p} = \frac{62}{130}$.

3. Compute the standard error using $p_0 = 0.40$ and $n = 130$:

4. $SE = \sqrt{\frac{0.40 \times (1 - 0.40)}{130}}$

5. Set up the formula for the z-score, but do not compute the final value:

6. $z = \frac{\hat{p} - 0.40}{SE}$

Try working through the setup and see if you can calculate the z-score!

Q5. HIV infection rate among IV drug users: Is the percentage less than 2%? Sample of 415, 8 HIV positive. $\alpha = 0.01$

Background

Topic: One-Proportion Z-Test (Left-Tailed)

This question tests whether the proportion of HIV-positive IV drug users is now less than a previously reported value, using a left-tailed test.

Key Terms and Formulas:

• Null Hypothesis ($H_0$): $p = 0.02$

• Alternative Hypothesis ($H_a$): $p < 0.02$

• Sample Proportion ($\hat{p}$): $\hat{p} = \frac{8}{415}$

• Standard Error (SE): $SE = \sqrt{\frac{0.02 \times (1 - 0.02)}{415}}$

• Test Statistic (z): $z = \frac{\hat{p} - 0.02}{SE}$

Step-by-Step Guidance

1. Write the hypotheses: $H_0: p = 0.02$, $H_a: p < 0.02$.

2. Calculate the sample proportion: $\hat{p} = \frac{8}{415}$.

3. Compute the standard error using $p_0 = 0.02$ and $n = 415$:

4. $SE = \sqrt{\frac{0.02 \times (1 - 0.02)}{415}}$

5. Set up the formula for the z-score, but do not compute the final value:

6. $z = \frac{\hat{p} - 0.02}{SE}$

Try setting up the calculations and see if you can find the z-score!