Math Problem Statement

A significance test is based on the following idea:

Make an assumption about the population (The null hypothesis - an assumption that the parameter equals a value). Then test to see if, under the assumption that the null hypothesis is true how likely is the sample that you got? If the sample you got is likely (within the range of normal defined by or with probability greater then ) then it is possible that both the sample and the null hypothesis is true. If the sample is not likely then both it and the null hypothesis are probably not both true. Since the sample was collected (under the assumption it was collected well) then the null hypothesis is probably not true and we reject the null hypothesis.

There are 3 types of significance tests.

Left-sided test: Unusual defined by the test statistic being less than the critical value which cuts off area in the left tail (note these will be negative values, less then means bigger negative) or a p-value less then , where p-value is the probability of being less than the test statistic under the assumption that the null hypothesis gives the value of the parameter.

Right-sided test: Unusual defined by the test statistic being more than the critical value which cuts off area in the right tail or a p-value less then , where p-value is the probability of being more than the test statistic under the assumption that the null hypothesis gives the value of the parameter.

Two-sided test: Unusual defined by the test statistic being more than the critical value which cuts off total area in the TWO tails or a p-value less then , where p-value is the probability of being less than the negative test statistic OR more than the positive test statistic (that is area in two tails) under the assumption that the null hypothesis gives the value of the parameter.

If the significance level is = 5% and p-value = 0.07, what is the conclusion?

Group of answer choices

fail to reject the alternate hypothesis

reject the alternate hypothesis

fail to reject the null hypothesis

reject the null hypothesis

Solution

To determine the correct answer, let's break down the information given:

  1. Significance Level (α): 5% or 0.05.
  2. P-Value: 0.07.

In a significance test:

  • If the p-value is less than or equal to the significance level (α), we reject the null hypothesis. This means the sample provides enough evidence to conclude that the null hypothesis is unlikely to be true.
  • If the p-value is greater than the significance level (α), we fail to reject the null hypothesis. This indicates that there is insufficient evidence against the null hypothesis, so we do not reject it.

Since the p-value (0.07) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Answer:

The correct answer is:

  • Fail to reject the null hypothesis.

Would you like more details on this conclusion, or do you have any questions?

Here are 5 related questions that expand on this concept:

  1. What does it mean if the p-value was exactly equal to the significance level?
  2. How do we interpret a two-sided test differently from one-sided tests when it comes to critical values?
  3. What are common significance levels, and why is 5% often used?
  4. What are Type I and Type II errors in hypothesis testing, and how do they relate to significance levels?
  5. How is the p-value calculated for a given test statistic?

Tip: Always compare the p-value to the significance level to decide whether to reject or fail to reject the null hypothesis.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Statistical Significance
P-Value Analysis

Formulas

p-value comparison to significance level

Theorems

Null Hypothesis
Type I Error

Suitable Grade Level

Grades 11-12

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