The amount of time that a drive-through bank teller spends...

The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean $\mu=3.2$ minutes and a standard deviation $\sigma=1.6$ minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller's counter is (a) at most 2.7 minutes: (b) more than 3.5 minutes; (c) at least 3.2 minutes but less than 3.4 minutes.

The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean $\mu=3.2$ minutes and a standard deviation $\sigma=1.6$ minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller's counter is
(a) at most 2.7 minutes:
(b) more than 3.5 minutes;
(c) at least 3.2 minutes but less than 3.4 minutes.

Key Concepts

Central Limit Theorem

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will approximate a normal distribution, regardless of the original data’s distribution, as long as the sample size is sufficiently large. This concept is crucial for making inferences about the population mean based on a sample mean.

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of all possible sample means from samples of a given size drawn from a population. It is used to calculate probabilities associated with the sample mean and is central to inferential statistics.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size, and it quantifies the variability of the sample mean around the population mean.

Z-score

A Z-score represents the number of standard deviations a particular value (in this case, a sample mean) is away from the population mean. It is used to convert a value into a standard normal variable so that probabilities can be determined from the standard normal distribution.

Normal Distribution

The Normal Distribution is a symmetric, bell-shaped distribution that is commonly used in statistics. When the sampling distribution of the sample mean approximates a normal distribution (often by the CLT), it allows for straightforward probability calculations using standard normal tables or software.

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