91Ó°ÊÓ

Understanding the success/failure condition is key to determining the shape of the sampling distribution for a sample proportion. This condition is part of a set of requirements needed to approximate a binomial distribution with a normal distribution, which is often more practical to work with.

The success/failure condition necessitates that the expected number of successes (np) and the expected number of failures (n(1-p)) in the sample must both be greater than or equal to 5. This rule ensures that the sample size is sufficiently large so that the sampling distribution can be considered symmetric and bell-shaped, resembling the normal distribution that we often rely on for further statistical inferences.

In the given exercise, with a sample size of 200 and a true proportion of smartphone users being 36%, we calculate both np (72) and n(1-p) (128), which are indeed greater than 5. Hence, the condition is satisfied, indicating that the sampling distribution of the sample proportion will likely be normal, allowing us to use z-scores and other normal distribution properties for analysis.

The sample proportion, denoted as \( \hat{p} \), represents the fraction of successes in a sample and plays a pivotal role when assessing statistics related to proportions within a population. It's a statistic that estimates a population parameter – the population proportion p.

When a random sample is taken, as in the investment company’s case, the sample proportion is used to estimate the true population proportion of customers using smartphones to access the website. Given a true population proportion of 36%, we can infer, just for this sample, that about 36 out of every 100 customers are expected to access the site via smartphone.

However, due to natural variability, we can expect different samples to yield different sample proportions. This is where the concept of a sampling distribution comes into play – it allows us to understand and quantify the variations among these sample proportions.

The standard deviation of the sampling distribution, often denoted as \( \sigma_{\hat{p}} \), measures the variability of the sample proportions from the true population proportion. It's an essential concept because it quantifies how much the sample proportion is likely to fluctuate from sample to sample.

The formula to calculate this standard deviation is \( \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} \), where p is the population proportion and n is the sample size. A smaller standard deviation indicates that the sample proportions are tightly clustered around the population proportion, whereas a larger standard deviation suggests more spread out sample proportions.

In the exercise, we are given the population proportion (0.36) and the sample size (200), allowing us to compute the standard deviation \( \sigma_{\hat{p}} \) as approximately 0.03416. This value is crucial for constructing confidence intervals or conducting hypothesis tests regarding the population proportion.