If you have no preliminary estimate for p, how many business phone calls should you include in a random sample to be 98% sure that the point estimate will be within a distance of 0.034 from p

To determine the necessary sample size, we need the desired level of confidence, margin of error, and the estimated value of p. However, since you mentioned that you have no preliminary estimate for p, we can use the worst-case scenario where p = 0.5. This assumption will provide the maximum sample size required.

Let's break down the process step-by-step to find the necessary sample size:

1. Determine the z-score corresponding to the desired level of confidence. For 98% confidence, the z-score is approximately 2.33 (obtained from standard normal distribution tables).

2. Calculate the margin of error (E) using the formula:
E = z * √(p(1-p) / n), where n is the sample size.

3. Since we want the point estimate to be within a distance of 0.034 from p, the margin of error (E) is 0.034:
0.034 = 2.33 * √(0.5*(1-0.5) / n)

4. Square both sides of the equation to eliminate the square root:
(0.034 / 2.33)^2 = 0.5*(1-0.5) / n

5. Solve for n:
n = 0.5*(1-0.5) / (0.034 / 2.33)^2

6. Simplify the expression:
n = 0.25 / (0.034 / 2.33)^2

7. Calculate the required sample size:
n ≈ 1948 (rounded up to the nearest whole number)

Therefore, you should include at least 1948 business phone calls in your random sample to be 98% sure that the point estimate will be within a distance of 0.034 from p when you have no preliminary estimate for p.