Problem 28 - Basic Statistics for Business and Economics with Student

Question

At the downtown office of First National Bank there are five tellers. Last week the tellers made the following number of errors each: \(2,3,5,3,\) and \(5 .\) a. How many different samples of 2 tellers are possible? b. List all possible samples of size 2 and compute the mean of each. c. Compute the mean of the sample means and compare it to the population mean.

Step-by-Step Solution

Answer

a. 10 samples; b. Means: 2.5, 3.5, 2.5, 3.5, 4, 4, 3, 4, 5; c. Sample mean: 3.2, Population mean: 3.6, similar but not equal.

Step 1: Identify Possible Samples

First, we need to determine how many different samples of 2 tellers can be selected from the 5 tellers. Since the order doesn't matter, we use the combination formula \( \binom{n}{k} \) where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. Here, \( n = 5 \) and \( k = 2 \).

Step 2: Calculate Combinations

We compute the number of combinations using the formula for combinations: \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \). Thus, there are 10 different samples of 2 tellers possible.

Step 3: List All Possible Samples

List all the combinations of samples of 2 from the tellers' errors: \((2,3), (2,5), (2,3), (2,5), (3,5), (3,5), (3,3), (3,5), (5,5)\).

Step 4: Compute Sample Means

Find the mean error for each of the samples:1. \( (2,3) \) has a mean of \( \frac{2+3}{2} = 2.5 \).2. \( (2,5) \) has a mean of \( \frac{2+5}{2} = 3.5 \).3. \( (2,3) \) has a mean of \( \frac{2+3}{2} = 2.5 \).4. \( (2,5) \) has a mean of \( \frac{2+5}{2} = 3.5 \).5. \( (3,5) \) has a mean of \( \frac{3+5}{2} = 4 \).6. \( (3,5) \) has a mean of \( \frac{3+5}{2} = 4 \).7. \( (3,3) \) has a mean of \( \frac{3+3}{2} = 3 \).8. \( (3,5) \) has a mean of \( \frac{3+5}{2} = 4 \).9. \( (5,5) \) has a mean of \( \frac{5+5}{2} = 5 \).

Step 5: Mean of Sample Means

Compute the mean of all sample means calculated: \( \frac{2.5 + 3.5 + 2.5 + 3.5 + 4 + 4 + 3 + 4 + 5}{10} = \frac{32}{10} = 3.2 \).

Step 6: Compute Population Mean

Calculate the population mean of the entire set of tellers' errors: \( \frac{2 + 3 + 5 + 3 + 5}{5} = \frac{18}{5} = 3.6 \).

Step 7: Compare Means

Compare the mean of the sample means \( (3.2) \) to the population mean \( (3.6) \). The mean of sample means is typically an unbiased estimator of the population mean, though it may differ slightly due to sample size and selection variations.

Key Concepts

Sample Mean

A sample mean is a way to find the average from a selected group in a larger set of data. It helps us understand what is happening in our smaller group without looking at every single part of the whole data set. Imagine you have a bag of mixed fruit and you want to know how sweet it is overall. Instead of tasting every single fruit, you pick a few and taste those.

In statistics, calculating the sample mean involves adding up all the numbers in your sample and then dividing by how many numbers you added. It's a simple division problem. For instance, if the errors made by two tellers in a bank are 2 and 3, the sample mean would be \[\frac{2+3}{2} = 2.5\]which gives an average error of 2.5 per teller for that sample.

• Helps in making estimates about the whole group.
• Easy to calculate and understand.
• Useful in comparing different samples.

Population Mean

A population mean serves as a central value for an entire data set. It gives us a bigger picture, representing the average across all data points. When you want to know something about the entire population, not just a sample, you use the population mean.

Think of this as checking the sweetness of all the fruits in your big bag. You taste each one, note the sweetness level, and divide by the number of fruits to find the true average sweetness. In our bank example, the errors made by all five tellers are 2, 3, 5, 3, and 5. Adding them up gives \(18\), and there are five tellers, so:\[\frac{18}{5} = 3.6\]This is the population mean for the errors.

• Provides a complete and true average for the entire group.
• Important for comparing theoretical models to real-world data.
• Usually remains constant until new data is added.

Combination Formula

The combination formula is used to determine how many different ways you can select 'k' items from a group of 'n' items, without focusing on the order. It's a mathematical way to model scenarios where the arrangement doesn't matter.

The formula itself is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(n!\) stands for factorial, the product of all positive integers up to \(n\).

In the bank example, with 5 tellers and wanting samples of 2, so\[\binom{5}{2} = \frac{5!}{2!(3)!} = \frac{5 \times 4}{2 \times 1} = 10\]This calculation tells us there are 10 possible pairs of tellers,

• Useful for calculating probabilities in statistics.
• Helps understand all possible selections.
• Ensures accuracy in analysis of combinations.