Final answer:
To find the probability that between 43% and 51% of the sampled adults work during summer vacation, calculate the z-scores for both proportions and find the corresponding probabilities using the z-score table. The probability of over 65% of the sampled adults working during summer vacation is 1.
Step-by-step explanation:
To determine the probability that between 43% and 51% of the sampled adults work during summer vacation, we need to calculate the z-scores for both proportions and use the z-score table to find the corresponding probabilities.
First, we calculate the z-score for 43%:
z = (43 - 47) / sqrt(0.47 * (1 - 0.47) / 400) = -2
Next, we calculate the z-score for 51%:
z = (51 - 47) / sqrt(0.47 * (1 - 0.47) / 400) = 2
Using the z-score table, we find the corresponding probabilities for z = -2 and z = 2:
Probability for z = -2: 0.0228
Probability for z = 2: 0.9772
To find the probability between 43% and 51%, we subtract the probability for z = 2 from the probability for z = -2:
Probability = 0.9772 - 0.0228 = 0.9544
Therefore, the probability that between 43% and 51% of the sampled adults work during summer vacation is 0.9544 (rounded to three decimal places).
To determine the probability that over 65% of the sampled adults work during summer vacation, we need to calculate the z-score for 65%:
z = (65 - 47) / sqrt(0.47 * (1 - 0.47) / 400) = 4.977
Using the z-score table, we find the corresponding probability for z = 4.977:
Probability = 1 (since it is beyond the limit of the z-score table)
Therefore, the probability that over 65% of the sampled adults work during summer vacation is 1 (rounded to three decimal places).