Question

It has long been stated that the mean temperature of humans is 98.6degreesF. ​However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6degreesF. 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 98.3degreesF and a sample standard deviation of 1 degrees F.

a. 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 α= 0.01 level of significance.
b. State the hypotheses.

Answer 1

We conclude that the mean temperature of humans is more than or equal to 98.6°F.

Explanation:

We are given that it has long been stated that the mean temperature of humans is 98.6°F. ​However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6°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°F and a sample standard deviation of 1°F.

Let
`\mu` = true mean temperature of humans.

SO, Null Hypothesis,
`H_0` :
`\mu \geq` 98.6°F {means that the mean temperature of humans is more than or equal to 98.6°F}

Alternate Hypothesis,
`H_A` :
`\mu` < 98.6°F {means that the mean temperature of humans is less than 98.6°F}

The test statistics that will be used here is One-sample t test statistics as we don't know about the population standard deviation;

T.S. =
`(\bar X -\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean temperature of 44 adults = 98.3°F

s = sample standard deviation = 1°F

n = sample of healthy adults = 44

So, test statistics =
`(98.3-98.6)/((1)/(√(43) ) )` ~
`t_4_3`

= -1.967

Hence, the value of test statistics is -1.967.

Now, P-value of the test statistics is given by;

P-value = P(
`t_4_3` < -1.967) = 0.029 or 2.9%

• If the P-value of test statistics is more than the level of significance, then we will not reject our null hypothesis as it will not fall in the rejection region.

• If the P-value of test statistics is less than the level of significance, then we will reject our null hypothesis as it will fall in the rejection region.

Now, here the P-value is 0.029 which is clearly higher than the level of significance of 0.01, so we will not reject our null hypothesis as it will not fall in the rejection region.

Therefore, we conclude that the mean temperature of humans is more than or equal to 98.6°F.