Final answer:
To find the probability that the mean length is between 44 cm and 46 cm in a sample of 100 individual grouse, calculate the standard error, convert the sample mean bounds to z-scores, and use a standard normal distribution table or calculator to find the area under the curve between the z-scores.
Step-by-step explanation:
To find the probability that the mean length of the sample is between 44 cm and 46 cm, we can use the Central Limit Theorem. Firstly, we calculate the standard error, which is the standard deviation of the population divided by the square root of the sample size (√n). In this case, the standard error is 4 / √100 = 4 / 10 = 0.4 cm. Next, we convert the sample mean bounds (44 cm and 46 cm) to z-scores using the formula z = (x - μ) / σ, where x is the sample mean, μ is the population mean, and σ is the standard error. The z-score for 44 cm is (44 - 45) / 0.4 = -2.5, and the z-score for 46 cm is (46 - 45) / 0.4 = 2.5. Finally, we use a standard normal distribution table or calculator to find the area under the curve between the z-scores. The probability that the mean length is between 44 cm and 46 cm is the difference between the two areas, which is approximately 0.9876 - 0.0124 = 0.9752, or 97.52%.