Answer :
### 1. Using a T-Test to Find the p-value and Hypothesis Testing
Given Information:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 97.9
- Sample standard deviation ([tex]\(s\)[/tex]) = 1.6
- Sample size ([tex]\(n\)[/tex]) = 30
- Population mean ([tex]\(\mu\)[/tex]) = Generally assumed to be 98.6°F for human body temperature
- We have three significance levels ([tex]\(\alpha\)[/tex]): 0.10, 0.05, and 0.01
Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 98.6\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu \neq 98.6\)[/tex] (two-tailed test)
p-value Computed: 0.0116
Decisions:
- At [tex]\(\alpha = 0.10\)[/tex]: Since 0.0116 < 0.10, reject [tex]\(H_0\)[/tex].
- At [tex]\(\alpha = 0.05\)[/tex]: Since 0.0116 < 0.05, reject [tex]\(H_0\)[/tex].
- At [tex]\(\alpha = 0.01\)[/tex]: Since 0.0116 > 0.01, do not reject [tex]\(H_0\)[/tex].
### 2. Using TInterval to Find Confidence Intervals
Confidence Intervals Computed:
- 90% Confidence Interval: (97.40, 98.40)
- 95% Confidence Interval: (97.30, 98.50)
- 99% Confidence Interval: (97.09, 98.71)
Decisions:
- The population mean of 98.6 is not within the 90% and 95% confidence intervals, so we reject [tex]\(H_0\)[/tex] for these cases.
- The 98.6 is within the 99% confidence interval, so we do not reject [tex]\(H_0\)[/tex] for the 99% confidence interval.
### 3. Connection Between T-Test and Confidence Intervals
The connection between the two methods (T-Test and Confidence Intervals) is that both are tools used to determine whether to reject the null hypothesis. If the significance level [tex]\(\alpha\)[/tex] is small enough to cause rejection of [tex]\(H_0\)[/tex] in a T-test, then the corresponding confidence interval will not contain the hypothesized mean (98.6). Conversely, if the hypothesized mean is contained within the calculated confidence interval, one would typically not reject [tex]\(H_0\)[/tex]. This relationship highlights how confidence intervals provide a range of plausible values for the population mean, supporting the decisions from hypothesis testing at specific significance levels.
Given Information:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 97.9
- Sample standard deviation ([tex]\(s\)[/tex]) = 1.6
- Sample size ([tex]\(n\)[/tex]) = 30
- Population mean ([tex]\(\mu\)[/tex]) = Generally assumed to be 98.6°F for human body temperature
- We have three significance levels ([tex]\(\alpha\)[/tex]): 0.10, 0.05, and 0.01
Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 98.6\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu \neq 98.6\)[/tex] (two-tailed test)
p-value Computed: 0.0116
Decisions:
- At [tex]\(\alpha = 0.10\)[/tex]: Since 0.0116 < 0.10, reject [tex]\(H_0\)[/tex].
- At [tex]\(\alpha = 0.05\)[/tex]: Since 0.0116 < 0.05, reject [tex]\(H_0\)[/tex].
- At [tex]\(\alpha = 0.01\)[/tex]: Since 0.0116 > 0.01, do not reject [tex]\(H_0\)[/tex].
### 2. Using TInterval to Find Confidence Intervals
Confidence Intervals Computed:
- 90% Confidence Interval: (97.40, 98.40)
- 95% Confidence Interval: (97.30, 98.50)
- 99% Confidence Interval: (97.09, 98.71)
Decisions:
- The population mean of 98.6 is not within the 90% and 95% confidence intervals, so we reject [tex]\(H_0\)[/tex] for these cases.
- The 98.6 is within the 99% confidence interval, so we do not reject [tex]\(H_0\)[/tex] for the 99% confidence interval.
### 3. Connection Between T-Test and Confidence Intervals
The connection between the two methods (T-Test and Confidence Intervals) is that both are tools used to determine whether to reject the null hypothesis. If the significance level [tex]\(\alpha\)[/tex] is small enough to cause rejection of [tex]\(H_0\)[/tex] in a T-test, then the corresponding confidence interval will not contain the hypothesized mean (98.6). Conversely, if the hypothesized mean is contained within the calculated confidence interval, one would typically not reject [tex]\(H_0\)[/tex]. This relationship highlights how confidence intervals provide a range of plausible values for the population mean, supporting the decisions from hypothesis testing at specific significance levels.
