To solve the problem presented, I’ll guide you through step-by-step:

Problem Overview:

• Claim (Hₐ): Population mean $\mu > 78.1$
• Null Hypothesis (H₀): Population mean $\mu = 78.1$
• Significance level: $\alpha = 0.01$
• Data points: 83.7, 65.2, 111, 80, 96.7, 75.3
• Population standard deviation is unknown.

We will calculate:

1. Test statistic ($t$-statistic).
2. P-value to assess whether the null hypothesis is rejected.
3. Decision and conclusion.

Step 1: Compute the Sample Statistics

The sample data is: $83.7, 65.2, 111, 80, 96.7, 75.3$.

1. Sample mean ($\bar{x}$): $\bar{x} = \frac{\text{Sum of data points}}{\text{Number of data points}}$

2. Sample standard deviation ($s$): $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Step 2: Calculate the Test Statistic

$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ where:

• $\mu_0 = 78.1$ (hypothesized mean),
• $n = 6$ (sample size),
• $s$ is the sample standard deviation,
• $\bar{x}$ is the sample mean.

Step 3: Determine the P-value

Using the $t$-distribution table or software, find the P-value corresponding to the calculated $t$-statistic with $n-1$ degrees of freedom (df = 5).

Step 4: Compare the P-value to $\alpha$

Decide whether to reject $H_0$ based on:

• If $\text{P-value} \leq \alpha$, reject $H_0$.
• If $\text{P-value} > \alpha$, fail to reject $H_0$.

Step 5: Conclude

Interpret the result based on the decision to reject or fail to reject $H_0$.

If you'd like, I can calculate these values for you step-by-step or directly display the computations. Let me know!