To determine whether the mean laying rate for Sarah's hens is statistically higher than the mean rate for all Golden Comets, we need to calculate a test statistic. We'll use the Z-test formula for the population mean with the known population standard deviation. Here is a detailed, step-by-step solution to the problem:
Step 1: Identify the given values.
- Population mean (μ): 5.2 eggs/day
- Sample mean ([tex]\( \bar{x} \)[/tex]): 5.4 eggs/day
- Population standard deviation (σ): 1.1 eggs/day
- Sample size (n): 46 hens
Step 2: Write down the Z-test formula for the population mean:
[tex]\[ Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \][/tex]
Step 3: Substitute the given values into the formula.
[tex]\[ Z = \frac{5.4 - 5.2}{\frac{1.1}{\sqrt{46}}} \][/tex]
Step 4: Simplify the denominator by calculating [tex]\( \frac{1.1}{\sqrt{46}} \)[/tex].
[tex]\[ \sqrt{46} \approx 6.782 \][/tex]
[tex]\[ \frac{1.1}{6.782} \approx 0.1622 \][/tex]
Step 5: Plug the simplified value back into the formula.
[tex]\[ Z = \frac{0.2}{0.1622} \][/tex]
Step 6: Perform the division.
[tex]\[ Z \approx 1.233150906022777 \][/tex]
So, the computed value for the test statistic is approximately 1.23.
Answer:
C) 1.23
