Answer :
Certainly! Let's work through this hypothesis testing problem step by step.
### Step 1: Understand the Hypotheses
First, let's establish our null and alternative hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean body temperature of healthy adults is 98.6°F.
[tex]\(H_0: \mu = 98.6\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The mean body temperature of healthy adults is less than 98.6°F.
[tex]\(H_a: \mu < 98.6\)[/tex]
We are conducting a left-tailed test since we believe the true mean is less than 98.6°F.
### Step 2: Gather the Sample Data
You have a sample of body temperatures from 13 adults:
[tex]\[ 97, 99.9, 98.6, 100.6, 96.7, 95.5, 97.3, 96.4, 100.3, 99.3, 94.9, 98.4, 97.3 \][/tex]
### Step 3: Calculate the Sample Mean and Standard Deviation
The sample mean is approximately 97.86°F, and the sample standard deviation is roughly 1.82°F.
### Step 4: Determine the Sample Size
The sample size, [tex]\(n\)[/tex], is 13.
### Step 5: Calculate the t-Score
The t-score is a measure that helps us understand how far our sample mean is from the hypothesized population mean in terms of standard errors. For this data, the t-score is approximately -1.46.
### Step 6: Find the Critical t-Value
Using a significance level ([tex]\(\alpha\)[/tex]) of 0.10 and 12 degrees of freedom (since [tex]\(n-1 = 12\)[/tex]), the critical t-value for a one-tailed test is about -1.36.
### Step 7: Make the Decision
To make a decision, compare the t-score to the critical t-value:
- Since the calculated t-score of -1.46 is less than the critical t-value of -1.36, we reject the null hypothesis.
### Conclusion
There is enough statistical evidence at the 0.10 level of significance to conclude that the mean body temperature of healthy adults is lower than 98.6°F.
### Step 1: Understand the Hypotheses
First, let's establish our null and alternative hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean body temperature of healthy adults is 98.6°F.
[tex]\(H_0: \mu = 98.6\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The mean body temperature of healthy adults is less than 98.6°F.
[tex]\(H_a: \mu < 98.6\)[/tex]
We are conducting a left-tailed test since we believe the true mean is less than 98.6°F.
### Step 2: Gather the Sample Data
You have a sample of body temperatures from 13 adults:
[tex]\[ 97, 99.9, 98.6, 100.6, 96.7, 95.5, 97.3, 96.4, 100.3, 99.3, 94.9, 98.4, 97.3 \][/tex]
### Step 3: Calculate the Sample Mean and Standard Deviation
The sample mean is approximately 97.86°F, and the sample standard deviation is roughly 1.82°F.
### Step 4: Determine the Sample Size
The sample size, [tex]\(n\)[/tex], is 13.
### Step 5: Calculate the t-Score
The t-score is a measure that helps us understand how far our sample mean is from the hypothesized population mean in terms of standard errors. For this data, the t-score is approximately -1.46.
### Step 6: Find the Critical t-Value
Using a significance level ([tex]\(\alpha\)[/tex]) of 0.10 and 12 degrees of freedom (since [tex]\(n-1 = 12\)[/tex]), the critical t-value for a one-tailed test is about -1.36.
### Step 7: Make the Decision
To make a decision, compare the t-score to the critical t-value:
- Since the calculated t-score of -1.46 is less than the critical t-value of -1.36, we reject the null hypothesis.
### Conclusion
There is enough statistical evidence at the 0.10 level of significance to conclude that the mean body temperature of healthy adults is lower than 98.6°F.
