Final answer:
To calculate the probability that the sample proportion will be within 0.03 of the population proportion, use the normal distribution for proportions. The probability is 0.6894. To calculate the probability that the sample proportion will be within 0.05 of the population proportion, follow the same steps. The probability is 0.6936.
Step-by-step explanation:
To calculate the probability that the sample proportion will be within 0.03 of the population proportion, we need to use the normal distribution for proportions.
First, calculate the standard deviation of the sample proportion: = √((0.30 * 0.70) / 100) = 0.0488
Next, calculate the z-scores for the upper and lower limits: z = (0.03 - 0) / 0.0488 = 0.613 and z = (-0.03 - 0) / 0.0488 = -0.613
Lastly, use a z-table or a calculator to find the probability associated with these z-scores: P(-0.613 < z < 0.613) = 0.6894
Therefore, the probability that the sample proportion will be within 0.03 of the population proportion is 0.6894.
To calculate the probability that the sample proportion will be within 0.05 of the population proportion, follow the same steps as above. The z-scores are z = (0.05 - 0) / 0.0488 = 1.024 and z = (-0.05 - 0) / 0.0488 = -1.024. Using a z-table or calculator, find the probability: P(-1.024 < z < 1.024) = 0.6936
Therefore, the probability that the sample proportion will be within 0.05 of the population proportion is 0.6936.