Answer :
We are given the following hypotheses to test at a significance level of $\alpha=0.01$:
$$
\begin{aligned}
H_0 &: \mu = 84.8, \\
H_a &: \mu > 84.8.
\end{aligned}
$$
and the sample data:
$$
93.4,\; 83.7,\; 95.8,\; 54.4,\; 97.4.
$$
Below are the steps to solve this one-sample right-tailed $t$-test:
1. **Determine the sample size and compute the sample statistics.**
The sample size is $n = 5$.
The sample mean is calculated (using all data points) as $\overline{x}$.
The sample standard deviation, $s$, is computed using $n - 1$ in the denominator.
2. **Calculate the standard error of the mean.**
The standard error (SE) is given by:
$$
\text{SE} = \frac{s}{\sqrt{n}}.
$$
3. **Compute the test statistic.**
The test statistic for a one-sample $t$-test is:
$$
t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}},
$$
where $\mu_0 = 84.8$.
For this data, the computed value is:
$$
t \approx 0.018 \quad \text{(rounded to three decimal places)}.
$$
4. **Determine the $p$-value.**
Since the alternative hypothesis is $\mu > 84.8$, we evaluate the $p$-value as the probability of obtaining a $t$-statistic greater than the observed value using the $t$-distribution with $n-1 = 4$ degrees of freedom:
$$
p\text{-value} = P(T > t_{\text{obs}})
$$
For $t_{\text{obs}} \approx 0.018$, the $p$-value is found to be approximately:
$$
p\text{-value} \approx 0.4934 \quad \text{(rounded to four decimal places)}.
$$
5. **Compare the $p$-value with the significance level.**
Here, $\alpha = 0.01$. Since
$$
0.4934 > 0.01,
$$
the $p$-value is greater than the significance level.
6. **Make a decision regarding the null hypothesis.**
Since the $p$-value exceeds $\alpha$, we do not have sufficient evidence to reject the null hypothesis. In other words, we:
$$
\textbf{fail to reject } H_0.
$$
7. **State the final conclusion.**
Based on the test, there is insufficient evidence to conclude that the population mean $\mu$ is greater than $84.8$.
To summarize:
- a. Test statistic: $t \approx 0.018$
- b. $p$-value: $\approx 0.4934$
- c. The $p$-value is greater than $\alpha$.
- d. The decision is to fail to reject the null hypothesis.
- e. Final Conclusion: There is insufficient evidence to support that the population mean is greater than $84.8$.
$$
\begin{aligned}
H_0 &: \mu = 84.8, \\
H_a &: \mu > 84.8.
\end{aligned}
$$
and the sample data:
$$
93.4,\; 83.7,\; 95.8,\; 54.4,\; 97.4.
$$
Below are the steps to solve this one-sample right-tailed $t$-test:
1. **Determine the sample size and compute the sample statistics.**
The sample size is $n = 5$.
The sample mean is calculated (using all data points) as $\overline{x}$.
The sample standard deviation, $s$, is computed using $n - 1$ in the denominator.
2. **Calculate the standard error of the mean.**
The standard error (SE) is given by:
$$
\text{SE} = \frac{s}{\sqrt{n}}.
$$
3. **Compute the test statistic.**
The test statistic for a one-sample $t$-test is:
$$
t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}},
$$
where $\mu_0 = 84.8$.
For this data, the computed value is:
$$
t \approx 0.018 \quad \text{(rounded to three decimal places)}.
$$
4. **Determine the $p$-value.**
Since the alternative hypothesis is $\mu > 84.8$, we evaluate the $p$-value as the probability of obtaining a $t$-statistic greater than the observed value using the $t$-distribution with $n-1 = 4$ degrees of freedom:
$$
p\text{-value} = P(T > t_{\text{obs}})
$$
For $t_{\text{obs}} \approx 0.018$, the $p$-value is found to be approximately:
$$
p\text{-value} \approx 0.4934 \quad \text{(rounded to four decimal places)}.
$$
5. **Compare the $p$-value with the significance level.**
Here, $\alpha = 0.01$. Since
$$
0.4934 > 0.01,
$$
the $p$-value is greater than the significance level.
6. **Make a decision regarding the null hypothesis.**
Since the $p$-value exceeds $\alpha$, we do not have sufficient evidence to reject the null hypothesis. In other words, we:
$$
\textbf{fail to reject } H_0.
$$
7. **State the final conclusion.**
Based on the test, there is insufficient evidence to conclude that the population mean $\mu$ is greater than $84.8$.
To summarize:
- a. Test statistic: $t \approx 0.018$
- b. $p$-value: $\approx 0.4934$
- c. The $p$-value is greater than $\alpha$.
- d. The decision is to fail to reject the null hypothesis.
- e. Final Conclusion: There is insufficient evidence to support that the population mean is greater than $84.8$.
