Question

The ideal number of children. "What do you think is the ideal number of children for a family to have?" A Gallup Poll asked this question of 1020 randomly chosen adults. Roughly 41% thought that a total of three or more children was ideal. 29 Suppose that p=0.41 is exactly true for the population of all adults. Gallup announced a margin of error of 4 percentage points for this poll.

What is the probability that the sample proportion p∧ for an SRS of size n=1020 falls between 0.37 and 0.45 ? You see that it is likely, but not certain, that polls like this give results that are correct within their margin of error.

Answer 1

To find the probability that the sample proportion falls between 0.37 and 0.45, we use the normal distribution and the z-score formula. We calculate the standard deviation of the sample proportion using the given population proportion and sample size, and then find the z-scores for the lower and upper limits. Using a standard normal distribution table or calculator, we find the probability to be approximately 0.9907.

Step-by-step explanation:

To find the probability that the sample proportion for an SRS of size n=1020 falls between 0.37 and 0.45, we need to use the normal distribution. Since the sample proportion follows a normal distribution when the sample size is large, we can use the z-score formula to standardize the sample proportion. The formula for z-score is:

z = (p - p) / σ

where p is the sample proportion, p is the population proportion, and σ is the standard deviation of the sample proportion. In this case, p is 0.41 and σ can be approximated by:

σ = √((p(1-p))/n)

Substituting the values, we have:

σ = √((0.41(1-0.41))/1020)

Calculating gives us σ ≈ 0.0153

Now we can calculate the z-scores for the lower and upper limits:

z1 = (0.37 - 0.41) / 0.0153 ≈ -2.614

z2 = (0.45 - 0.41) / 0.0153 ≈ 2.614

Using a standard normal distribution table or calculator, we can find the probability that a standard normal variable falls between -2.614 and 2.614 to be approximately 0.9907.