Final answer:
To find the sample size n ensuring that the estimated value of the sample mean is within ±10 points from the true mean, we can use the formula n = (Z * σ / E)^2, where Z is the z-score corresponding to the desired confidence level, σ is the standard deviation of the population, and E is the maximum error tolerance. In this case, we need a sample size of n = 190 to ensure the estimated sample mean is within ±10 points from the true mean with a confidence level of 95%.
Step-by-step explanation:
To find the sample size n ensuring that the estimated value of the sample mean is within ±10 points from the true mean, we can use the formula for sample size calculation:
n = (Z * σ / E)^2
Where:
- n is the sample size
- Z is the z-score corresponding to the desired confidence level
- σ is the standard deviation of the population
- E is the maximum error tolerance (half the desired confidence interval)
In this case, the confidence level α = 0.05 corresponds to a z-score of approximately 1.96 (obtained from a standard normal distribution table). The standard deviation σ = 35 and the maximum error tolerance E = 10. Substituting these values into the formula:
n = (1.96 * 35 / 10)^2 = 13.76^2 ≈ 189.05
Rounding up to the nearest whole number, we need a sample size of n = 190 to ensure the estimated sample mean is within ±10 points from the true mean with a confidence level of 95%.