Question

A study by a researcher in a lab is to address the question of whether the mean body temperature of humans is 37℃. The body temperatures of six graduate students in the lab were measured {35.5, 36, 37.5, 35, 36.5, 37.8}. Note that the sample mean is x=36.383. Assume that the standard deviation of the body temperatures of humans is 0.5℃.

Test at the 5% significance level whether the average body temperature of humans is different from 37℃.

Answer 1

To test if the average human body temperature differs from 37°C, we calculate a t-test statistic with the sample data and compare it to the critical t-values at a 5% significance level. If the t-statistic falls outside the critical range, we reject the null hypothesis, suggesting a significant difference from 37°C.

Step-by-step explanation:

The question asks us to perform a hypothesis test to determine whether the average body temperature of humans is different from 37°C. The null hypothesis ($0) claims there is no difference, while the alternative hypothesis ($1) suggests a difference exists. Given the sample mean (x = 36.383°C), the known population standard deviation (σ = 0.5°C), and the sample size of 6, a t-test is appropriate due to the small sample size.

0: μ = 37°C (the population mean is 37°C)
1: μ ≠ 37°C (the population mean is not 37°C)

First, calculate the test statistic using the formula:

t = (x - μ) / (σ/√)
t = (36.383 - 37) / (0.5/√<6>)
t ≈ -3.72

Next, determine the t-critical value for 5% significance level (two-tailed test) and degree of freedom (df = n - 1 = 5). If the calculated t-value falls outside the range of the t-critical values, we reject the null hypothesis.

Since the calculated t-value is likely to be beyond the critical t-values, the evidence suggests rejecting the null hypothesis, indicating that the average body temperature is significantly different from 37°C. However, the exact t-critical value must be checked against a t-distribution table or software.