Refer to the following. In a random sample of 525 teenagers,...

Refer to the following. In a random sample of 525 teenagers, eye colors are cross-classified with favorite colors from among green, red, blue, and purple. Below is a two-way table of the responses. For these data, $\chi^{2}=18.591$ with a $P$ -value of 0.290 . Assuming a significance level of $0.05,$ which of the following is true? (A) A Type-I error may have been committed, but a TypeII error was not committed. (B) A Type-II error may have been committed, but a Type-I error was not committed. (C) Both Type-I and Type-II errors may have been committed. (D) Neither a Type-I nor a Type-II error could have been committed. (E) There is not sufficient information to make any statements about the possibility of a Type-I or a Type-II error being committed.

Refer to the following. In a random sample of 525 teenagers, eye colors are cross-classified with favorite colors from among green, red, blue, and purple. Below is a two-way table of the responses.
For these data, $\chi^{2}=18.591$ with a $P$ -value of 0.290 . Assuming a significance level of $0.05,$ which of the following is true?
(A) A Type-I error may have been committed, but a TypeII error was not committed.
(B) A Type-II error may have been committed, but a Type-I error was not committed.
(C) Both Type-I and Type-II errors may have been committed.
(D) Neither a Type-I nor a Type-II error could have been committed.
(E) There is not sufficient information to make any statements about the possibility of a Type-I or a Type-II error being committed.

Key Concepts

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is in fact true. It is also known as a 'false positive' and is directly related to the significance level established for the test.

P-value

The p-value is a probability measure that indicates the evidence against the null hypothesis provided by the sample data. It represents the likelihood of obtaining results as extreme as, or more extreme than, those observed, assuming that the null hypothesis is true, and is compared against the significance level to decide on rejecting or not rejecting the null hypothesis.

Type II Error

A Type II error takes place when the null hypothesis is not rejected despite being false. This 'false negative' error occurs when the test fails to detect a true effect or difference, and its likelihood is influenced by factors such as the sample size and statistical power.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves setting up null and alternative hypotheses and then determining, based on the sample data and a chosen significance level, whether to reject the null hypothesis or not.

Significance Level

The significance level is a predetermined threshold (often 0.05) that defines the probability of rejecting the null hypothesis when it is actually true, which is known as a Type I error. It serves as a criterion for whether the observed data is sufficiently unlikely under the assumption that the null hypothesis holds.

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