# Why does a p-value of 2.4 indicate a very small probability that rejecting H0 is incorrect?

I am working with body temperatures in R, conducting a t-test.

It is widely believed that the average body temperature for healthy adults is 98.6 degrees Fahrenheit.

The body temperatures of $n = 130$ healthy adults were measured:

$\bar{x} = 98.249$ with standard deviation $s = 0.7332$

So, the null hypothesis is $\mu = \mu_0 = 98.6$

Two-sided test: $H_a : µ \neq µ_0$, with $α = 0.05$

Critical limit (c) given by $t_{n−1,1−α/2} = t_{129,0.975} = 1.979$ This is upper limit, lower limit is then $−1.979$

(98.249-98.6) / (0.7332/sqrt(130))
[1] -5.458287


The test statistic $t$ falls in the rejection region $(−5.46 < −1.98)$, so we reject $H_0$ at $α = 0.05$

However, when we run the t-test in R, the p-value is $2.411e-07$:

t.test (bodytemp $temp, mu=98.6, alternative="two.sided") One Sample t-test data: bodytemp$temp

t = -5.4548, df = 129, p-value = 2.411e-07

alternative hypothesis: true mean is not equal to 98.6


95 percent confidence interval:

98.12200 98.37646 ## sample estimates:


mean of x

98.24923


Therefore, if the p-value is greater than 0.5 (given by α), we would accept H0, but we know this is not correct given the test statistic t falling in the rejection region, and the mean of x being 98.24.

SO, shouldn't the p-value be less than 0.5? Why is the p-value 2.4?

• Please fix the significance level you talk about at the end of your question. It is NOT 0.5 – Glen_b -Reinstate Monica Jun 25 '16 at 8:09

R gives you a p-value of $2.411\times 10^{-7}$ , thus much smaller than .05. The convention of writing this as 2.411e-7 is used in many contexts in programming and in no way specific to R.