Final answer:
The distribution of the average length of 31 independent dog walks, each with a mean of 2.1 miles and a standard deviation of 0.2 miles, will be normally distributed with a mean of 2.1 miles and a standard error of approximately 0.0359 miles.
Step-by-step explanation:
You take your dog on a walk every morning. The lengths of your walks have a mean of 2.1 miles and a standard deviation of 0.2 miles. Assuming independence between different walks and considering a month with 31 days, the distribution of the average length of the 31 walks will follow a normal distribution. This is due to the Central Limit Theorem which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough.
The mean (μ) of the sampling distribution of the sample mean will be the same as the population mean, which is 2.1 miles. The standard deviation (σ) of the sampling distribution of the sample mean, also known as the standard error (SE), will be the population standard deviation divided by the square root of the sample size (n). Therefore, SE = 0.2 / √31. Now we can find the standard error:
- SE = 0.2 / √31
- SE ≈ 0.2 / 5.567764363
- SE ≈ 0.0359 miles
The distribution of the 31 walks' average length will be approximately N(2.1, 0.0359).