Question

An important aspect of a federal economic plan was that consumers would save a substantial portion of the money that they received from an income tax reduction. Suppose that early estimates of the portion of total tax saved, based on a random sampling of 35 economists, had mean 26% and standard deviation 12%.

Required:
What is the approximate probability that a sample mean estimate, based on a random sample of n = 35 economists, will lie within 1% of the mean of the population of the estimates of all economists?

Answer 1

To find the approximate probability that a sample mean estimate will lie within 1% of the population mean, we need to use the Central Limit Theorem and the Z-score formula. The approximate probability is 0.0144, or 1.44%.

Step-by-step explanation:

To solve this problem, we need to use the Central Limit Theorem. The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. In this case, the sample mean estimate has a mean of 26% and a standard deviation of 12%. To find the approximate probability that a sample mean estimate will lie within 1% of the population mean, we need to standardize the values using the Z-score formula.

Z-score = (x - µ) / (σ/sqrt(n))

Z(-0.01) = -0.0072

Z(0.01) = 0.0072

The probability that the sample mean estimate will lie within 1% of the population mean is approximately equal to the area between the Z-scores, which can be calculated as:

Probability = Z(0.01) - Z(-0.01) = 0.0072 - (-0.0072) = 0.0144

Therefore, the approximate probability that a sample mean estimate will lie within 1% of the population mean is 0.0144, or 1.44%.