6.4 Setting Up a Test for a Population Proportion

• What does the alternative hypothesis ($H_{a}$) state about the population proportion?


  There is a difference
• Steps in setting up a hypothesis test for a population proportion

  1️⃣ Identify the population parameter of interest

  2️⃣ State the null hypothesis

  3️⃣ State the alternative hypothesis

  4️⃣ Check conditions for a z-test

  5️⃣ Choose the level of significance

  6️⃣ Determine the critical z-values or p-value

  7️⃣ Calculate the test statistic

  8️⃣ Make a decision and interpret the conclusion
• The population mean is the average value of a quantity
• The null hypothesis $H_{0}: p =$$0.5$ assumes 50% of the population has a certain characteristic


  True
• The null hypothesis proposes no effect or difference
• The conditions for a z-test include $np \ge 10$ and n(1 - p) \ge 10</latex>


  True
• The population proportion is relevant when dealing with categorical data


  True
• What does the population standard deviation measure?

  Dispersion around the mean
• What is the population proportion in the example where H_{0}: p = 0.5</latex>?

  0.5
• The alternative hypothesis H_{1}: p \neq 0.5</latex> indicates a two-tailed test


  True
• The alternative hypothesis indicates that there is a significant effect or difference
• Unlike the population mean, the population proportion deals with the percentage of a group sharing a common characteristic
• The alternative hypothesis proposes that there is a real effect or difference in the data.


  True
• The condition $np \ge 10$ ensures that the sample is large enough to approximate a normal distribution.


  True
• A random sample ensures that each individual has an equal chance of being included in the sample.
• If $n =$$100$ and $p =$$0.2$, what is the value of $n(1 - p)$?


  80
• Independent observations mean that the outcome of one observation depends on another.

  False
• What is the most frequently used level of significance in statistical analysis?

  $\alpha =$$0.05$
• The level of significance $\alpha =$$0.05$ is preferred when a very low risk of Type I error is needed.


  False
• Match the parameter type with its definition:

  Population Proportion ($p$) ↔️ Percentage with a characteristic

  Population Mean ($\mu$) ↔️ Average value of a quantity

  Population Standard Deviation ($\sigma$) ↔️ Dispersion of values around the mean
• The alternative hypothesis proposes that there is a significant difference
• If the null hypothesis is $H_{0}: p =$$0.5$, the alternative hypothesis for a one-tailed test with $p > 0.5$ is $H_{a}: p > < blank_{s}tart > 0.5 < blank_{e}nd >$<distractors>0 ||| 1 ||| 0.2</distractors><cloze_end>

  
  

  <cloze_start>One condition for a valid z-test is that the sample must be random
• A lower level of significance reduces the chance of rejecting a true null hypothesis.


  True
• Match the z-test approach with its characteristic:

  Critical Z-Values Approach ↔️ Compares test statistic with critical values

  P-Value Approach ↔️ Calculates the probability of extreme results
• In a one-tailed test, the null hypothesis is rejected if the test statistic falls beyond the critical z-value.


  True
• In the p-value approach, the null hypothesis is rejected if the p-value is greater than $\alpha$.


  False
• A calculated z-statistic is used to determine whether to reject the null hypothesis based on the chosen level of significance.


  True
• In the p-value approach, the null hypothesis is rejected if the p-value is less than or equal to $\alpha$.


  True
• The population mean measures the average value of a quantity, while the population proportion measures the percentage of a group sharing a common characteristic.


  True
• What does the alternative hypothesis ($H_{a}$) propose compared to the null hypothesis ($H_{0}$)?


  Effect or difference
• What is the key difference between the null and alternative hypotheses in hypothesis testing?

  Effect vs no effect
• Why is a random sample necessary for a valid z-test?

  Representative of population
• What does independence of observations mean in the context of hypothesis testing?

  Outcome unrelated to others
• What does the level of significance ($\alpha$) represent in hypothesis testing?


  Type I error probability
• What is the p-value in hypothesis testing?

  Probability of test statistic
• In the $z$-formula, $\hat{p}$ represents the sample proportion
• What is the population parameter of interest in this study material?

  Population proportion ($p$)
• What are two approaches to determine the critical value in a hypothesis test for proportions?

  Critical z-values or p-value
• Match the parameter type with its definition:

  Population Proportion ↔️ Percentage of individuals with a characteristic

  Population Mean ↔️ Average value of a quantity

  Population Standard Deviation ↔️ Dispersion of values around the mean
• What does the null hypothesis state about the population proportion?

  No significant difference