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You are testing the hypothesis \(p=0.7\) and have decided to reject this hypothesis if after 15 trials you observe 14 or more successes. a. If the null hypothesis is true and you observe 13 successes, which of the following will you do? (1) Correctly fail to reject \(H_{o} .\) (2) Correctly reject \(H_{o} .(3)\) Commit a type I error. (4) Commit a type II error. b. Find the significance level of your test. c. If the true probability of success is \(1 / 2\) and you observe 13 successes, which of the following will you do? (1) Correctly fail to reject \(H_{o} .\) (2) Correctly reject \(H_{o} .(3)\) Commit a type I error. (4) Commit a type II error. d. Calculate the \(p\) -value for your hypothesis test after 13 successes are observed.

Step by step solution

01

Understanding the Null Hypothesis

The null hypothesis, denoted \(H_{0}\), is that the probability \(p = 0.7\). We reject this hypothesis if we observe 14 or more successes out of 15 trials.

02

Answering question (a)

If the null hypothesis is true and we observe 13 successes, we fail to reject the null hypothesis since our rule was to reject it in favor of the alternative hypothesis only if we observe 14 or more successes. So, the answer is (1) Correctly fail to reject \(H_{0}\).

03

Significance Level Calculation

The significance level of a test is the value below which we reject the null hypothesis. The significance level is calculated using the binomial distribution function. The cumulative probability of observing 13 or fewer successes if \(p = 0.7\) can be obtained, and this value is subtracted from 1 to obtain the significance level of the test.

04

Answering question (c)

If the true probability of success is \(1/2\) and we observe 13 successes, the observed data does not provide sufficient evidence to reject the null hypothesis. If we fail to reject the null hypothesis when it is false in reality, we are committing a type II error. So, the answer is (4) Commit a type II error.

05

P-value Calculation

The p-value is the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is correct. So, following the same steps as in the calculation of the significance level, we calculate the p-value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error

In hypothesis testing, an error can result from either incorrectly rejecting or failing to reject a null hypothesis. When we make a Type I error, we reject a true null hypothesis. It is like saying there is an effect or a difference when none actually exists. Imagine accusing someone of a crime they did not commit; that's akin to a Type I error.

Type I errors can occur in any hypothesis testing, so we should manage them carefully:

• Type I error is traditionally denoted by the symbol \(\alpha\).
• It represents a false positive result.
• The probability of making a Type I error is equal to the significance level of the test, which we'll dive into shortly.

By keeping the significance level low, researchers minimize the risk of Type I errors, ensuring that their conclusions are reliable.

Type II Error

A Type II error occurs when we fail to reject a null hypothesis that is actually false. It's like missing an effect or relationship that truly exists. Consider it as a situation where you're letting a guilty person go free; this is similar to committing a Type II error.

Understanding the Type II error is crucial in hypothesis testing:

• Type II error is represented by the symbol \(\beta\).
• It is known as a false negative result.
• The power of a test, which is 1 minus \(\beta\), represents the probability of correctly rejecting a false null hypothesis.

Researchers aim to optimize the power of their tests to minimize the chance of a Type II error, ensuring that genuine findings are not overlooked.

Significance Level

The significance level is a critical threshold in hypothesis testing that helps to determine when to reject the null hypothesis. It is a predefined value that sets the maximum acceptable probability of making a Type I error.

Here's what you need to know about significance level:

• Typically denoted by \(\alpha\), it is often set at \(0.05\) or \(0.01\), which corresponds to a 5% or 1% risk of committing a Type I error.
• Setting a significance level helps decide whether the evidence is strong enough to reject the null hypothesis.
• If the significance level is set too low, one risks missing a true effect because it is too difficult to achieve. Conversely, if it's too high, false effects may appear significant.

Choosing an appropriate significance level involves considering the risk of Type I and Type II errors and the context of the research.

p-value

The p-value is a vital concept in hypothesis testing. It quantifies the evidence against the null hypothesis by measuring the probability of observing results at least as extreme as the actual results, assuming the null hypothesis is true.

Here's why the p-value is important:

• A small p-value indicates strong evidence against the null hypothesis, suggesting that it may be false.
• If the p-value is less than or equal to the significance level \(\alpha\), the null hypothesis is rejected.
• Conversely, a large p-value means there isn't enough evidence to reject the null hypothesis.

When interpreting p-values, it's crucial to remember that they do not measure the size of an effect or the importance of a result. Instead, they help understand whether the observed data is consistent with the null hypothesis or not.