2.5.2 Hypothesis testing for the binomial distribution

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What does the null hypothesis assume about populations or variables being studied?
No effect or difference
In hypothesis testing for a fair coin, the null hypothesis is expressed as p = 0.5.
The alternative hypothesis can be either directional (one-tailed) or non-directional (two-tailed).
Match the type of alternative hypothesis with its description:
One-tailed ↔️ States the direction of the effect
Two-tailed ↔️ States the effect is different without specifying direction
When is a directional (one-tailed) hypothesis used?
Effect has a specific direction
A non-directional hypothesis states that the probability of getting heads is different from 0.5.
The alternative hypothesis asserts that there is no effect or difference between populations or variables being studied.
False
When is a non-directional (two-tailed) hypothesis used?
Effect direction is unknown
Match the alternative hypothesis with its example:
One-tailed ↔️ The proportion of defective items is greater than 0.05
Two-tailed ↔️ The average score is different from 75
The alternative hypothesis asserts that there is an effect or a difference between populations or variables being studied.
What are the two types of alternative hypotheses?
One-tailed and two-tailed
What does a directional hypothesis state about the effect being tested?
Direction of the effect
A two-tailed hypothesis is used when the direction of the effect is not known or expected
The alternative hypothesis (H1) contradicts the null hypothesis (H0).
The significance level (α) represents the probability of a Type I error
A smaller α reduces the risk of a Type I error but increases the risk of a Type II error.
What is the purpose of the test statistic in hypothesis testing?
Evaluate the null hypothesis
In the z-score formula, $\hat{p}$ represents the sample proportion
What does $p_{0}$ represent in the z-score formula? 
Population proportion under H₀
Steps to calculate the test statistic
1️⃣ Identify the sample proportion ( $\hat{p}$ )
2️⃣ Identify the population proportion ( $p_{0}$ ) under H₀
3️⃣ Determine the sample size ( $n$ )
4️⃣ Apply the z-score formula
How does the test statistic differ from a confidence interval in hypothesis testing?
Evaluates a sample value
The p-value is the probability of observing data as extreme as, or more extreme than, the sample data if H0 is true.
In a one-tailed test, the p-value is the area in the tail corresponding to the direction specified by the alternative hypothesis
What does a p-value of 0.0228 indicate at a significance level of 0.05?
Reject the null hypothesis
What does the null hypothesis assume about the populations or variables being studied?
No effect or difference
How is the p-value calculated in a two-tailed test?
Twice the tail area
In a one-tailed test, the p-value is the area in the tail corresponding to the z-score.
In a two-tailed test, the p-value is twice the area in the tail corresponding to the absolute value of the z-score
What is the null hypothesis when testing if a coin is biased toward heads?
p = 0.5</latex>
If the p-value is less than the significance level α, we reject the null hypothesis.
What conclusion is drawn if the p-value is 0.0228 and α = 0.05?
Reject the null hypothesis
The null hypothesis assumes there is no effect or no difference
Provide an example of a null hypothesis assuming the average test score is 75.
$H_{0}: \mu =$ $75$
Match the type of alternative hypothesis with its example:
One-tailed ↔️ $H_{1}: p > 0.05$
Two-tailed ↔️ $H_{1}: \mu \neq 75$
A one-tailed hypothesis is used when the direction of the effect is known.
A two-tailed hypothesis is used when the direction of the effect is not known
What is the alternative hypothesis when testing if a coin is fair in a two-tailed test?
$H_{1}: p \neq 0.5$
A Type I error occurs when we reject a true null hypothesis.
Common values for the significance level α are 5% (0.05) and 1% (0.01)
What happens to the risk of a Type II error when α is reduced?
It increases