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Question 2 of 20

It has long been stated that the mean temperature of humans is [tex]$98.6^{\circ} F$[/tex]. However, two researchers currently involved in the subject thought that the mean temperature is different. 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 [tex]$98.2^{\circ} F$[/tex] and a standard deviation of [tex]$0.9^{\circ} F$[/tex]. Use the [tex]$P$[/tex]-value approach to conduct a hypothesis test to judge whether the mean temperature of humans is less than [tex]$98.6^{\circ} F$[/tex] at the [tex]$\alpha = 0.01$[/tex] level of significance.

State the hypotheses.
[tex]$H _0$[/tex] : [tex]$\mu = 98.6^{\circ} F$[/tex]
[tex]$H _1$[/tex] : [tex]$\mu \ \textless \ 98.6^{\circ} F$[/tex]

Find the test statistic.
[tex]$t_0=$[/tex] [tex]$\square$[/tex]
(Round to two decimal places as needed.)

The [tex]$P$[/tex]-value is [tex]$\square$[/tex].
(Round to three decimal places as needed.)

What can be concluded?
A. Reject [tex]$H _0$[/tex] since the [tex]$P$[/tex]-value is less than the significance level.
B. Do not reject [tex]$H_0$[/tex] since the [tex]$P$[/tex]-value is not less than the significance level.
C. Do not reject [tex]$H_0$[/tex] since the [tex]$P$[/tex]-value is less than the significance level.
D. Reject [tex]$H_0$[/tex] since the [tex]$P$[/tex]-value is not less than the significance level.


Sagot :

Step-by-step solution:

1. State the hypotheses:
- The null hypothesis, [tex]\(H_0\)[/tex]: [tex]\(\mu = 98.6^{\circ} F\)[/tex]
- The alternative hypothesis, [tex]\(H_1\)[/tex]: [tex]\(\mu < 98.6^{\circ} F\)[/tex]

So, we have:
[tex]\[ H_0 : \mu = 98.6^{\circ} F \][/tex]
[tex]\[ H_1 : \mu < 98.6^{\circ} F \][/tex]

2. Find the test statistic:
- Given:
- Sample size ([tex]\(n\)[/tex]) = 200
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 98.2
- Population mean ([tex]\(\mu\)[/tex]) = 98.6
- Sample standard deviation ([tex]\(s\)[/tex]) = 0.9
- Significance level ([tex]\(\alpha\)[/tex]) = 0.01

The t-score (test statistic) is calculated using the formula:
[tex]\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \][/tex]
Plugging in the values:
[tex]\[ t = \frac{98.2 - 98.6}{0.9 / \sqrt{200}} \][/tex]
The calculated t-score is:
[tex]\[ t_0 = -6.29 \][/tex]
(Rounded to two decimal places as needed.)

3. Determine the P-value:
- The calculated t-score is -6.29.
- Degrees of freedom = [tex]\( n - 1 = 200 - 1 = 199 \)[/tex]

Using a t-distribution table or software, the P-value corresponding to [tex]\( t_0 = -6.29 \)[/tex] for 199 degrees of freedom is:
[tex]\[ \text{P-value} = 0.000 \][/tex]
(Rounded to three decimal places as needed.)

4. Conclusion:
Given the information:
- The significance level ([tex]\(\alpha\)[/tex]) = 0.01
- The P-value = 0.000

Since the P-value (0.000) is less than the significance level (0.01), we reject the null hypothesis.

Therefore, the correct conclusion is:
[tex]\[ \text{A. Reject \(H_0\) since the P-value is less than the significance level.} \][/tex]