Math Problem Statement

Identify the P-value.

The claim is that for 12 AM body temperatures, the mean is μ > 98.6°F. The sample size is 5 and the test statistic is t = 1.567. Use technology to find the P-value for the hypothesis test described.

Solution

The image describes a hypothesis test where:

  • The null hypothesis H0H_0 is that the mean body temperature μ98.6F\mu \leq 98.6^\circ F.
  • The alternative hypothesis H1H_1 is that μ>98.6F\mu > 98.6^\circ F (a one-tailed test).
  • The sample size is n=5n = 5.
  • The test statistic is t=1.567t = 1.567.
  • The significance level is α=0.05\alpha = 0.05.

To find the P-value for this one-tailed t-test, we use the degrees of freedom df=n1=51=4\text{df} = n - 1 = 5 - 1 = 4.

Next, we can calculate the P-value using the t-distribution with 4 degrees of freedom, by looking up the t-score of 1.567 or using a calculator.

Let me compute the P-value for you.The P-value for the given hypothesis test is approximately 0.0961.

Since the P-value (0.0961) is greater than the significance level α=0.05\alpha = 0.05, we fail to reject the null hypothesis. This means that the data does not provide strong enough evidence to support the claim that the mean body temperature is greater than 98.6F98.6^\circ F.

Do you have any questions, or would you like further details?

Here are five related questions for you:

  1. What is the importance of the P-value in hypothesis testing?
  2. How do we interpret the result if the P-value is less than the significance level?
  3. What are the consequences of rejecting the null hypothesis when it is true (Type I error)?
  4. Why is a one-tailed test used in this scenario?
  5. How does the sample size affect the degrees of freedom and the test outcome?

Tip: Always ensure to check the assumptions of the t-test, such as normality, especially with small sample sizes.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
T-distribution
Significance Level

Formulas

t = (x̄ - μ) / (s / √n)
Degrees of freedom = n - 1

Theorems

T-distribution Theorem

Suitable Grade Level

College-level Statistics