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College students with checking accounts typically write relatively few checks in any given month, whereas nonstudent residents typically write many more checks during a month. Suppose that \(50 \%\) of a bank's accounts are held by students and that \(50 \%\) are held by nonstudent residents. Let \(x\) denote the number of checks written in a given month by a randomly selected bank customer. a. Give a sketch of what the probability distribution of \(x\) might look like. b. Suppose that the mean value of \(x\) is \(22.0\) and that the standard deviation is \(16.5 .\) If a random sample of \(n=100\) customers is to be selected and \(\bar{x}\) denotes the sample average number of checks written during a particular month, where is the sampling distribution of \(\bar{x}\) centered, and what is the standard deviation of the \(\bar{x}\) distribution? Sketch a rough picture of the sampling distribution. c. Referring to Part (b), what is the approximate probability that \(\bar{x}\) is at most 20 ? at least 25 ?

Step by step solution

01

Sketch the Probability Distribution of \(x\)

Given the varying checking habits between students and nonstudents, the distribution is likely to be bimodal with two peaks - one for students who typically write fewer checks and one for nonstudents who typically write more checks.

02

Determine the Center and Standard Deviation of the Sampling Distribution

The center (mean) of the sampling distribution of \(\bar{x}\) is the same as the mean of the population, which is 22.0. The standard deviation of the sampling distribution (known as the standard error) is the standard deviation of the population divided by the square root of the sample size, or \(16.5 / \sqrt{100}\), which equals 1.65.

03

Sketch the Sampling Distribution of \(\bar{x}\)

The sampling distribution should be approximately normally distributed, centered at 22.0, with a spread determined by the standard error of 1.65.

04

Calculate the Probabilities

To calculate the probability that \(\bar{x}\) is at most 20, first find the z-score using the formula \((20 - 22) / 1.65\). Look up this z-score in a standard normal (z) table to find the probability. The same process applies for finding the probability that \(\bar{x}\) is at least 25, but note that this is a right-tail probability, so you may need to subtract the table value from 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution

A probability distribution showcases how likely different outcomes are in an experiment or process. It assigns probabilities to each possible value of a random variable. If you're imagining it visually, it might be represented as a graph, a table, or a formula. For example, in the exercise about the number of checks a bank customer writes, we would expect a bimodal distribution because there are two distinct groups—students and nonstudents—with different behaviors.

When plotting a probability distribution, you might notice patterns. In a bimodal distribution, there would be two peaks on the graph, each peak representing the concentration of values around a common point (mean) for each group’s number of checks written.

Standard Deviation

Standard deviation is a measure that tells us how much variation there is from the average (mean), or expected value. A low standard deviation indicates that the data points are close to the mean, while a high standard deviation indicates that data points are spread out over a larger range of values. In the bank example, a standard deviation of 16.5 for the number of checks suggests that there's a wide variation in the number of checks bank customers write in a month.

Understanding standard deviation is pivotal since it gives us a sense of how 'spread out' a set of data is, which is critical when you're trying to predict or understand the likelihood of different outcomes.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is bell-shaped and characterized by its mean and standard deviation. In practice, many random variables have distributions that are close to normal.

The normal distribution is important because of the Central Limit Theorem, which implies that the sampling distribution of the sample mean will be approximately normally distributed, assuming that the sample size is large enough and regardless of the population distribution shape.

Z-score

A z-score measures exactly how many standard deviations an element is from the mean. It's a way of standardizing scores on the same scale to easily compare results from different data sets. To find a z-score, you use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \(X\) is the value, \(\mu\) is the mean of the population, and \(\sigma\) is the standard deviation. In the context of our bank account example, calculating the z-score helps indicate how typical or atypical a value is when compared to the average number of checks written.

Central Limit Theorem

The Central Limit Theorem (CLT) is a statistical theory that states that given a sufficiently large sample size, the sample means will be approximately normally distributed, regardless of the population's distribution. Hence, even if the original population distribution is not normal (like a bimodal distribution in our bank customer example), the distribution of the sample means will tend to be normal if the sample size is large enough. The CLT is fundamental because it allows statisticians to make inferences about population parameters using sample statistics.

For instance, in the exercise, the mean of the sampling distribution of \(\bar{x}\) is centered at the population mean, and the standard deviation (standard error) is reduced by the sample size. With the CLT, we can approximate the probability of the sample mean being within a certain range, which facilitates hypothesis testing and confidence interval building.