91Ó°ÊÓ

For each of the following situations, state whether a Type I, a Type II, or neither error has been made. Explain briefly. a. A bank wants to know if the enrollment on their website is above \(30 \%\) based on a small sample of customers. They test \(\mathrm{H}_{0}: p=0.3\) vs. \(\mathrm{H}_{\mathrm{A}}: p>0.3\) and reject the null hypothesis. Later they find out that actually \(28 \%\) of all customers enrolled. b. A student tests 100 students to determine whether other students on her campus prefer Coke or Pepsi and finds no evidence that preference for Coke is not 0.5 . Later, a marketing company tests all students on campus and finds no difference. c. A human resource analyst wants to know if the applicants this year score, on average, higher on their placement exam than the 52.5 points the candidates averaged last year. She samples 50 recent tests and finds the average to be 54.1 points. She fails to reject the null hypothesis that the mean is 52.5 points. At the end of the year, they find that the candidates this year had a mean of 55.3 points. d. A pharmaceutical company tests whether a drug lifts the headache relief rate from the \(25 \%\) achieved by the placebo. They fail to reject the null hypothesis because the P-value is \(0.465 .\) Further testing shows that the drug actually relieves headaches in \(38 \%\) of people.

Step by step solution

01

Identify Type of Error for Situation A

For situation a, the bank mistakenly rejects the null hypothesis (\(H_{0}: p=0.3\)) based on their sample even though in reality the true proportion is less (only 28% of all customers have actually enrolled). This is a Type I error, a false rejection of the null hypothesis.

02

Identify Type of Error for Situation B

In situation b, the student's test concludes accurately that there is no preference for Coke (i.e., probability of preference is 0.5), and later surveys also confirm this. There is no error in this case, neither a Type I nor a Type II.

03

Identify Type of Error for Situation C

In situation c, the HR analyst fails to reject the null hypothesis (\(H_{0}: \mu = 52.5\)), despite the true average score being higher (55.3 points). This is a Type II error. She fails to reject the null hypothesis when it was false.

04

Identify Type of Error for Situation D

For situation d, the pharmaceutical company fails to reject the null hypothesis (that the headache relief rate equals to the one achieved by the placebo), although further testing reveals that the drug actually has a higher relief rate (38%). This, again, is a Type II error, failing to reject the null hypothesis when it was false.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method that allows you to make inferences or educated guesses about a population parameter based on sample data. This methodology involves setting up two opposing hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_A\)). The null hypothesis usually states that there is no effect or no difference, while the alternative hypothesis suggests the opposite.To conduct a hypothesis test, you:

• Define the null and alternative hypotheses.
• Select a significance level, typically 0.05.
• Use sample data to calculate a test statistic.
• Compare this statistic to a critical value or use a p-value to decide whether to reject the null hypothesis.

Rejecting the null hypothesis suggests that there is statistical evidence supporting the alternative hypothesis, whereas failing to reject it implies that there is not enough evidence to move away from the null hypothesis.

Statistical Errors

Statistical errors occur when incorrect conclusions are drawn about the population based on the sample data. These errors are two types: Type I error and Type II error.

Type I Error

A Type I error occurs when the null hypothesis is true, but is wrongly rejected. This is like a false positive result. For example, deciding that a new medicine is effective when it is not.

Type II Error

A Type II error happens when the null hypothesis is false but is incorrectly accepted. This is similar to a false negative result. Suppose a test fails to detect a disease, even though the disease is present. Both types of errors are crucial because they influence how we interpret results and make decisions based on statistical tests. Minimizing these errors requires careful experiment design and analysis.

Null Hypothesis

The null hypothesis (\(H_0\)) is a statement of no effect, no difference, or no change in a given context. It serves as a starting point for hypothesis testing. For example, if you want to test whether a new drug is more effective than an old one, the null hypothesis might state that both drugs have equal effectiveness.The null hypothesis is typically formulated as:

• There is no difference in means, i.e., \( \mu_1 = \mu_2 \).
• The observed effect is due to chance.
• The population parameter equals a specified value, such as \( p = 0.3 \).

Testing the null hypothesis allows researchers to evaluate the evidence against it. If the evidence is strong enough, the null hypothesis is rejected, which often means accepting the alternative hypothesis.

Statistical Significance

Statistical significance is a measure of whether observed results are meaningful or occurred merely by chance. This concept is crucial in hypothesis testing as it helps determine whether to reject the null hypothesis. Usually, a result is considered statistically significant if the p-value is less than the predetermined alpha level (e.g., 0.05). A p-value indicates the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. Statistical significance does not imply practical significance; it merely suggests that the observed effect is unlikely due to random variation. It aids researchers in making inferences about populations based on sample data, ensuring that findings are robust and reliable. In summary, understanding statistical significance allows for informed decisions about whether to accept or reject hypotheses in scientific research.