91Ó°ÊÓ

When estimating a proportion of a population who might respond positively to an offer or display a certain behavior, the sample proportion is crucial. It is denoted by \( \hat{p} \). This value represents the fraction of the sample showing the desired characteristic. In our scenario, the bank initially targets 100 individuals. But since none responded affirmatively to the credit card offer, their sample proportion is 0.Understanding the sample proportion helps illustrate what percentage of the surveyed group acts a certain way. If one or two people had accepted out of the 100, the sample proportion would have been \( \frac{1}{100} \) or \( \frac{2}{100} \), illustrating that even small numbers can change the perception of acceptability in different contexts.

The Standard Error (SE) is fundamental in constructing a confidence interval. It measures how much variability there is in a sample proportion and indicates how much our \(\hat{p}\) might differ from the real population proportion \(p\). It gets smaller as the sample size increases.To calculate SE for a proportion, use the following formula: \\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]Here, \(\hat{p}\) is 0 for our situation because no one accepted the offer, and \(n\) is 100.With this formula:

• \( \hat{p} = 0 \)
• \( 1-\hat{p} = 1 \)
• \( n = 100 \)

Therefore, our standard error is \( \sqrt{\frac{0 \times 1}{100}} = 0 \).Having the SE as zero meant the variability in reading acceptance was minimal in this sample.

The Critical Value aids in determining the range of values that are plausible estimates of the population parameter at our selected confidence level. It's linked to the confidence level, acting as a "cut-off" point for how extreme a result can be before it’s considered unlikely.In our case, with a \(95\%\) confidence level, we use the critical value from the standard normal distribution (Z-distribution), which commonly is \(1.96\). This represents the number of standard errors we move away from \(\hat{p}\) to develop a range that likely covers the true population parameter.Getting this critical value right ensures that our confidence interval provides a reasonable assurance regarding where the true proportion lies — crucial in determining whether our bank's assumptions about acceptance rates hold under statistical scrutiny.

This is a statistical concept that offers us the faith or certainty with which we can state our interval estimate contains the true population parameter. In our problem, a \(95\%\) confidence level implies that if we could repeat our mail-out over numerous samples, approximately 95 out of 100 of those samples' intervals would contain the true proportion of advertisement acceptors.Choosing a \(95\%\) confidence level is standard as it strikes a balance between being too lenient and overly stringent. However, selecting a higher confidence level increases the interval's breadth, lowering our precision but improving security against missing the true parameter. On the flip side, a lower confidence level tightens our interval — meaning our estimation is more narrow, but also the risk increases that our true parameter lies outside this interval.