Question

It has long been stated that the mean temperature of humans is 98.6 degrees F. ​However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6 degrees F. They measured the temperatures of 44 healthy adults 1 to 4 times daily for 3​ days, obtaining 200 measurements. The sample data resulted in a sample mean of 98.3 degrees F and a sample standard deviation of 1 degrees F. Use the​ P-value approach to conduct a hypothesis test to judge whether the mean temperature of humans is less than 98.6 degrees F at the alpha equals 0.01 level of significance. State the hypotheses.

Upper H 0​: ▼ mu p sigma ▼ not equals less than equals greater than 98.6 degrees F

Upper H 1​: ▼ sigma mu p ▼ not equals equals less than greater than 98.6 degrees F

Find the test statistic. t 0 equals nothing ​(Round to two decimal places as​ needed.) The​ P-value is nothing. ​(Round to three decimal places as​ needed.) What can be​ concluded?

A. Do not reject Upper H 0 since the​ P-value is not less than the significance level.

B. Do not reject Upper H 0 since the​ P-value is less than the significance level.

C. Reject Upper H 0 since the​ P-value is not less than the significance level.

D. Reject Upper H 0 since the​ P-value is less than the significance level.

Answer 1

Solution:

Here, we have to use one sample t test for the population mean.

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The mean temperature of humans is 98.6 degrees F.

Alternative hypothesis: H₁: The mean temperature of humans is less than 98.6 degrees F.

H₀: µ = 98.6

H₁: µ < 98.6

This is a lower tailed or left tailed (one tailed) test.

We are given a level of significance = α = 0.01

The test statistic formula is given as below:

t = (Xbar - µ)/[S/sqrt(n)]

We are given

Xbar = 98.3

S = 1

n = 200

df = n – 1 = 200 – 1 = 199

α = 0.01

Critical t value = -2.3452

(by using t-table)

t = (Xbar - µ)/[S/sqrt(n)]

t = (98.3 – 98.6)/[1/sqrt(200)]

t = -0.3/ 0.0707

t = -4.2426

Test statistic = t₀ = -4.24 (Answer)

P-value = 0.000 (Answer)

(by using t-table)

P-value < α = 0.01

So, we reject the null hypothesis

D. Reject H₀ since the​ P-value is less than the significance level. (Answer)

There is sufficient evidence to conclude that the mean temperature of humans is less than 98.6 degrees F.