Final answer:
To find the probability that the number of expectant mothers that gave birth by caesarean section falls within two standard deviations of the mean, we need to calculate the mean and standard deviation of the number of births by caesarean section.
Step-by-step explanation:
To find the probability that the number of expectant mothers that gave birth by caesarean section falls within two standard deviations of the mean, we need to calculate the mean and standard deviation of the number of births by caesarean section.
From the given information, we know that 32.8% of births occur by caesarean section.
Therefore, the mean of the number of expectant mothers that gave birth by caesarean section is 0.328 * 65 = 21.32.
The standard deviation can be calculated using the formula: sqrt(npq), where n is the number of trials, p is the probability of success, and q is the probability of failure.
In this case, n = 65, p = 0.328, and q = 1 - 0.328 = 0.672.
Plugging in the values, we get sqrt(65 * 0.328 * 0.672) = 4.764.
Now, we can calculate the z-score for two standard deviations above and below the mean using the formula: z = (x - mean) / standard deviation.
For two standard deviations below the mean: z = (0 - 21.32) / 4.764 = -4.476
For two standard deviations above the mean: z = (43 - 21.32) / 4.764 = 4.544
Using a z-table, we can find the probability associated with these z-scores. The probability that the number of expectant mothers that gave birth by caesarean section falls within two standard deviations of the mean is the difference between the probability of two standard deviations above the mean and two standard deviations below the mean.
Let's calculate the values using the z-table.