Problem 5 Use the relatively small number ... [FREE SOLUTION] (2024)

Expert verified

Imagine you asked a group of women whether they purchase books online. Their responses might vary. If you surveyed a new group each time, you’d also get different outcomes. This variation is what we call the sampling distribution. It showcases the range of possible results we might see if we took multiple random samples from the same population.

In our exercise, instead of surveying many groups, we used bootstrap samples (re-sampling with replacement of the original group). The proportions from these samples create an approximate sampling distribution. For example, one bootstrap sample gave us a proportion of 0%, another 25%, and another 50%. This range represents different results we might get if we surveyed different groups of women multiple times. This method helps us understand how variable our estimates can be and explains why no single survey gives a complete picture.

When we talk about a proportion estimate, we're referring to the fraction or percentage of a whole. Here, we're estimating the proportion of women who purchase books online.

From the original responses, we get: 'no', 'yes', 'no', 'no'—numerically, that's 0, 1, 0, 0. If we calculate the proportion of 'yes' responses, which represent book purchases, we'd get \(\frac{1}{4} = 0.25\) or 25%.

However, in our exercise, we go beyond the initial group and use bootstrap samples to estimate. With bootstrap sampling, we re-sample the original responses multiple times to mimic surveying more groups. We then calculate the proportion of 'yes' (purchase) in each sample. For instance, one bootstrap sample might be \[1, 0, 1, 0\] or 50%, and another might be \[0, 0, 0, 0\] or 0%. These varied proportions help us better understand the possible range of our proportion estimate.

A confidence interval gives us a range in which we expect the true proportion to lie, with a certain level of confidence (like 90%, 95%, etc.). It reflects how sure we are about where the 'true' proportion falls, based on our sample data.

Let's say we want a 90% confidence interval for our survey question. We first gather several bootstrap samples and calculate the proportion of 'yes' for each. We then order these proportions. To form a 90% confidence interval, we remove the lowest 5% and the highest 5% of our ordered proportions. For our exercise, we had 10 samples, and removing the top and bottom values leaves us with the 90% middle range. This range is our 90% confidence interval.

In our specific case, after sorting the proportions, we removed the extreme values. Our remaining range, from 0 (0%) to 0.5 (50%), is our 90% confidence interval. This suggests that we're 90% confident the true proportion of women who purchase books online lies somewhere between 0 and 50%.