To determine if the population mean differs from 9, we can perform a hypothesis test. Specifically, we will use a one-sample z-test since we know the population standard deviation. Here's a step-by-step breakdown of the process:
Step 1: Set Up Hypotheses
- Null Hypothesis (H₀): μ = 9 (the population mean is equal to 9)
- Alternative Hypothesis (H₁): μ ≠ 9 (the population mean is not equal to 9)
Step 2: Calculate the Test Statistic
We'll use the z-test statistic formula:
\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]
Where:
- \(\bar{x}\) = sample mean = 7.26
- \(\mu_0\) = hypothesized population mean = 9
- \(\sigma\) = population standard deviation = 3.72
- \(n\) = sample size = 28
Plugging in the values:
\[ z = \frac{7.26 - 9}{3.72 / \sqrt{28}} \]
First, calculate the standard error (SE):
\[ SE = \frac{3.72}{\sqrt{28}} \approx \frac{3.72}{5.2915} \approx 0.702 \]
Now calculate the z-value:
\[ z = \frac{7.26 - 9}{0.702} \approx \frac{-1.74}{0.702} \approx -2.48 \]
Step 3: Determine the Critical Value and p-value
Given that we are performing a two-tailed test, we need to find the critical z-values for a significance level (α) of 0.05. The critical z-values are approximately ±1.96.
Now let's check the p-value corresponding to our calculated z-value of -2.48. Using z-tables or software, we find:
\[ \text{p-value} \approx 0.013 (two-tailed) \]
Step 4: Compare and Conclude
Since the p-value (approximately 0.013) is less than the significance level of α = 0.05, we reject the null hypothesis.
Conclusion
There is sufficient evidence to conclude that the population mean differs from 9.