Final answer:
To find the sample size n that ensures P(4900 < X < 5100) ≥ 0.99, we need to use the z-score formula. The z-score represents the number of standard deviations away from the mean a particular value is.
Step-by-step explanation:
To find the sample size n that ensures P(4900 < X < 5100) ≥ 0.99, we need to use the z-score formula. The z-score represents the number of standard deviations away from the mean a particular value is. We can calculate the z-score using the formula: z = (x - mean) / standard deviation. In this case, the mean is 5000 psi and the standard deviation is 400 psi. We want to find the z-score for the lower bound 4900 psi: z1 = (4900 - 5000) / 400 = -0.25. We want to find the z-score for the upper bound 5100 psi: z2 = (5100 - 5000) / 400 = 0.25. To find the sample size n, we need to find the corresponding z-score from the z-table that gives us a cumulative probability of 0.99. From the z-table, the z-score is approximately 2.33. Now we can use the formula for the z-score: n = (z-score * standard deviation / margin of error)^2. Since we want to find the sample size that gives us a probability of 0.99, we can subtract the lower cumulative probability of 0.01 from 1 to get the margin of error: margin of error = 1 - 0.99 = 0.01. Substituting the values in the formula, we have: n = (2.33 * 400 / 200)^2 = 5.29^2 ≈ 27.97. Therefore, the sample size n should be at least 28 to ensure P(4900 < X < 5100) ≥ 0.99.