Answer:
a. The shape of the sampling distribution of sample means will be approximately normal. This is because of the central limit theorem, which states that for a large enough sample size (n > 30), the distribution of sample means will be approximately normal regardless of the shape of the original population distribution.
b. The mean of the sampling distribution of sample means will be the same as the mean of the population, which is 25. The variance of the sampling distribution of sample means will be equal to the population variance divided by the sample size, which is 7^2/35 = 1.4.
c. To find the probability that a single random sample of size 35 drawn from this population will yield a mean between 22 and 29, we need to standardize the distribution using the formula:
z = (x - μ) / (σ / sqrt(n))
where x is the sample mean we are interested in (in this case, 22 and 29), μ is the population mean (25), σ is the population standard deviation (7), and n is the sample size (35).
For x = 22,
z = (22 - 25) / (7 / sqrt(35)) = -1.91
For x = 29,
z = (29 - 25) / (7 / sqrt(35)) = 1.91
Using a standard normal distribution table or a calculator, we can find that the probability of getting a z-score between -1.91 and 1.91 is approximately 0.859. Therefore, the probability that a single random sample of size 35 drawn from this population will yield a mean between 22 and 29 is 0.859 or 85.9%.