6.6 Concluding a Test for a Population Proportion

• The null hypothesis often states that there is no change or difference in the population proportion.

  True
• What does the independence condition require for hypothesis testing?

  Observations must be independent
• The denominator in the test statistic formula represents the standard error
• What does the alternative hypothesis claim about the population proportion?

  It is different from H₀
• Match the hypothesis type with its statement and example:

  Null Hypothesis (H₀) ↔️ Asserts the population proportion equals a specified value; \(H₀: p = 0.5\)

  Alternative Hypothesis (H₁) ↔️ Claims the population proportion differs from the null value; \(H₁: p \neq 0.5\)
• The test statistic for testing a population proportion is denoted by the letter z.
• Match the type of hypothesis test with its p-value calculation:

  Two-Tailed Test ↔️ \(2 \times P(Z \geq |z|)\)

  One-Tailed Test (p > p₀) ↔️ \(P(Z \geq z)\)

  One-Tailed Test (p < p₀) ↔️ \(P(Z \leq z)\)
• If the p-value is less than \(\alpha\), we reject the null hypothesis.
• Why is a smaller $\alpha$ preferred in medical research?


  Minimize rejecting treatment effectiveness
• If the p-value is 0.03 and $\alpha$ is 0.05, we would reject
• In a survey of 200 individuals, the null hypothesis states the preference is 50%. The large sample size condition is met because $np =$$200 \times 0.5 =$$100 \geq$ 10
• What does the null hypothesis state about the population proportion?

  It equals a specified value
• The normal distribution condition is ensured by the large sample size condition.


  True
• Steps to calculate the p-value for a hypothesis test

  1️⃣ Calculate the test statistic

  2️⃣ Use the standard normal distribution

  3️⃣ Determine the type of test (one-tailed or two-tailed)

  4️⃣ Calculate the $p$-value
• A smaller p-value indicates stronger evidence to reject the null hypothesis.
• Independence is a condition that must be checked before conducting a hypothesis test for a population proportion.


  True
• If \(\hat{p} = 0.55\), \(p_0 = 0.5\), and \(n = 100\), the calculated test statistic \(z = 1\).

  True
• The significance level $\alpha$ represents the probability of rejecting a true null hypothesis.


  True
• Common values for $\alpha$ are 0.05 (5%) and 0.01
• Independence is a condition that must be checked before conducting a hypothesis test.


  True
• What does the denominator of the test statistic formula represent?

  Standard error
• What does the denominator in the test statistic formula represent?

  Standard error
• The p-value is determined using the standard normal distribution.
• What is the common value for the significance level ($\alpha$)?


  0.05
• What action do we take if the p-value is greater than or equal to the significance level ($\alpha$)?


  Fail to reject H₀
• The alternative hypothesis can be one-sided or two-sided
• The large sample size condition requires that $n \geq 10$ and $np \geq 10$ and n(1-p) \geq 10
• What is the value of the test statistic if \(\hat{p} = 0.55\), \(p_0 = 0.5\), and \(n = 100\)?

  1
• Steps to calculate the p-value for a hypothesis test on a population proportion

  1️⃣ Calculate the test statistic \(z\)

  2️⃣ Use the standard normal distribution

  3️⃣ Determine the p-value based on the type of test
• What is the null hypothesis in the example provided for a marketing team testing a conversion rate?

  \(H₀: p = 0.15\)
• What does the denominator in the test statistic formula represent?

  Standard error
• What is the p-value if \(z = 1.5\) in a one-tailed test where \(p > p_0\)?

  0.0668
• What does $\alpha$ represent in hypothesis testing?


  Type I error probability
• Steps in comparing the p-value to \alpha</latex> in hypothesis testing

  1️⃣ Calculate the p-value

  2️⃣ Choose the significance level $\alpha$

  3️⃣ Compare the p-value to $\alpha$

  4️⃣ Reject or fail to reject the null hypothesis
• What does the null hypothesis (H₀) assume about the population proportion?

  It equals a specified value
• Match the term with its description in the test statistic formula:

  \(\hat{p}\) ↔️ Sample proportion

  \(p_0\) ↔️ Proportion in the null hypothesis

  \(n\) ↔️ Sample size
• In the formula for the test statistic, \hat{p} represents the sample proportion.
• What is the p-value in hypothesis testing?

  Probability of test statistic
• The significance level ($\alpha$) represents the maximum acceptable risk of committing a Type I error.
• If the p-value is less than \alpha, we reject the null hypothesis.