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 56 healthy adults 1 to 4 times daily for 3 days, obtaining 250 measurements. The sample data resulted in a sample mean of 98.2 degrees F and a sample standard deviation of 0.9 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 = 0.01 level of significance.

1. State the hypotheses.

A. \( H_0 \): \(\mu = 98.6^\circ F\) (The mean temperature of humans is 98.6 degrees F.)

B. \( H_1 \): \(\mu < 98.6^\circ F\) (The mean temperature of humans is less than 98.6 degrees F.)

2. Find the test statistic.

a. \( t_0 = ? \)

b. The P-value is: ____.

3. What can be concluded?

A. Reject \( H_0 \) since the P-value is less than the significance level.

B. Reject \( H_0 \) since the P-value is not less than the significance level.

C. Do not reject \( H_0 \) since the P-value is less than the significance level.

D. Do not reject \( H_0 \) since the P-value is not less than the significance level.

We are given that 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.

The sample data resulted in a sample mean of 98.2 degrees F and a sample standard deviation of 0.9 degrees F.

Let [tex]\mu[/tex] = mean temperature of humans.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu\geq[/tex] 98.6°F {means that the mean temperature of humans is more than or equal to 98.6°F}

Alternate Hypothesis, [tex]H_A[/tex] : p < 98.6°F {means that the mean temperature of humans is less than 98.6°F}

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

T.S. = [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_n_-_1[/tex]

where, [tex]\bar X[/tex] = sample mean temperature = 98.2°F

[tex]\sigma[/tex] = sample standard deviation = 0.9°F

n = sample of healthy adults = 56

So, test statistics = [tex]\frac{98.2-98.6}{\frac{0.9}{\sqrt{56} } }[/tex] ~ [tex]t_5_5[/tex]

= -3.326

The value of t test statistics is -3.326.

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

P-value = P( [tex]t_5_5[/tex] < -3.326) = 0.00077 or 0.08%

Since, P-value of the test statistics is less than the level of significance as 0.08% < 1%, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the mean temperature of humans is less than 98.6°F.

BillPaxton BillPaxton · Answer 2 · 2025-02-14T20:34:42-05:00

Final answer:

To determine if the mean human temperature is less than 98.6°F, we conducted a left-tailed t-test using the provided sample data. We calculated the t-test statistic and then used it to find the p-value. We then used the p-value to make our final decision on whether to reject or fail to reject our null hypothesis.

Explanation:

In this scenario, we are performing a hypothesis test about the average human temperature. Let's formulate our hypothesis first:

Null hypothesis (H0): The average human temperature equals 98.6 F. Mathematically, H0: μ = 98.6 F.
Alternative hypothesis (H1): The average human temperature is less than 98.6 F. Mathematically, H1: μ < 98.6 F.

We will conduct a left-tailed t-test because we are testing whether the average human temperature is less than a stated value.

Given the data: the sample size n = 250, the sample mean (x_bar) = 98.2 F, and the standard deviation (s) = 0.9 F.

To calculate the t-test statistic, use the formula: t0 = (x_bar - μ) / (s/√n)

For getting the p-value, you would use a statistical table or software with the above t statistic and degree of freedom (which is n-1 in this case).

In the end, if your p-value is less than the significance level (α = 0.01 in this case), we reject the null hypothesis, if not, we fail to reject the null hypothesis.

If the P-value is less than α, we would conclude that the research may be correct, and the average human temperature is indeed lower than 98.6 F (37.0 °C).

Learn more about Hypothesis Testing here:

https://brainly.com/question/34171008

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