(Hypothetical.) A bank wants to estimate the amount of...

(Hypothetical.) A bank wants to estimate the amount of change people carry. They take a simple random sample of 100 people, and find that on the average, people in the sample carry 73$\Phi$ in change. They figure the standard error is 4$\phi$ , because $$ \sqrt{100} \times \sqrt{0.73 \times 0.27} \approx 4, \quad 4 / 100=.04 $$ Are they right? Answer yes or no, and explain.

(Hypothetical.) A bank wants to estimate the amount of change people carry.
They take a simple random sample of 100 people, and find that on the average,
people in the sample carry 73$\Phi$ in change. They figure the standard error is 4$\phi$ ,
because
$$
\sqrt{100} \times \sqrt{0.73 \times 0.27} \approx 4, \quad 4 / 100=.04
$$
Are they right? Answer yes or no, and explain.

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Key Concepts

Central Limit Theorem

The central limit theorem states that the distribution of the sample mean will be approximately normal for a large enough sample size, regardless of the population’s distribution. This theorem justifies using the standard error to assess sampling variation for the mean in many practical applications.

Sampling Distribution

The sampling distribution is the theoretical distribution of a statistic over many samples drawn from the same population. It is crucial in constructing confidence intervals and performing hypothesis testing by understanding the variability across different samples.

Standard Error

The standard error measures how much the statistic (such as the sample mean) would vary from sample to sample. It is typically calculated as the sample standard deviation divided by the square root of the sample size when estimating a mean. Using the wrong formula can lead to misestimating the uncertainty of an estimate.

Proper Use of Statistical Formulas

It is essential to apply the correct formulas based on the type of data and statistic being estimated. In this scenario, the bank incorrectly applied a formula appropriate for a proportion (using p(1-p)) to calculate the standard error for a mean value, which is not valid when the data are continuous monetary amounts.

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