To determine whether we can reject the null hypothesis [tex]\( H_0: \mu = 98.6 \)[/tex] versus the alternative hypothesis [tex]\( H_1: \mu \neq 98.6 \)[/tex], we need to see if the test value [tex]\( 98.6 \)[/tex] lies within the given 99% confidence interval.
### Step-by-Step Approach:
1. State the Hypotheses:
- Null hypothesis ([tex]\( H_0 \)[/tex]): [tex]\(\mu = 98.6\)[/tex]
- Alternative hypothesis ([tex]\( H_1 \)[/tex]): [tex]\(\mu \neq 98.6\)[/tex]
2. Given Data:
- The 99% confidence interval for the mean body temperature based upon a random sample of 100 adults is [tex]\( (97.5, 99.2) \)[/tex].
3. Understanding Confidence Intervals:
- A 99% confidence interval means we are 99% confident that the true population mean [tex]\(\mu\)[/tex] lies within the interval [tex]\( (97.5, 99.2) \)[/tex].
- If the test value 98.6 falls within this interval, we do not have sufficient evidence to reject the null hypothesis.
4. Evaluate the Test Value:
- Check if [tex]\( 98.6 \)[/tex] lies between [tex]\( 97.5 \)[/tex] and [tex]\( 99.2 \)[/tex].
5. Conclusion:
- Since [tex]\( 98.6 \)[/tex] is within the interval [tex]\( (97.5, 99.2) \)[/tex], the interval contains the null hypothesis value.
### Interpretation:
- At Significance Levels [tex]\( \alpha = 0.01 \)[/tex] and [tex]\( \alpha = 0.005 \)[/tex]:
- Since [tex]\( 98.6 \)[/tex] is within the 99% confidence interval, we fail to reject the null hypothesis at both significance levels [tex]\( \alpha = 0.01 \)[/tex] and [tex]\( \alpha = 0.005 \)[/tex].
- This is because the confidence interval suggests that the true mean could reasonably be 98.6, hence no significant evidence against [tex]\( H_0 \)[/tex].
Thus, the correct conclusions are:
- No, the null hypothesis cannot be rejected at the significance level [tex]\(\alpha = 0.01\)[/tex], because 98.6 is contained in the 99% confidence interval.
- No, the null hypothesis cannot be rejected at the significance level [tex]\(\alpha = 0.005\)[/tex], because 98.6 is contained in the 99% confidence interval.
