Problem 45 - Basic Statistics for Business and Economics with Student

Question

Nike's annual report says that the average American buys 6.5 pairs of sports shoes per year. Suppose the population standard deviation is 2.1 and that a sample of 81 customers will be examined next year. a. What is the standard error of the mean in this experiment? b. What is the probability that the sample mean is between 6 and 7 pairs of sports shoes? c. What is the probability that the difference between the sample mean and the population mean is less than 0.25 pairs? d. What is the likelihood the sample mean is greater than 7 pairs?

Step-by-Step Solution

Answer

a) 0.2333, b) 0.9674, c) 0.8577, d) 0.0164.

Step 1: Calculate the Standard Error of the Mean

The standard error of the mean (SEM) is calculated using the formula: \[ SEM = \frac{\sigma}{\sqrt{n}} \]where \( \sigma = 2.1 \) is the population standard deviation and \( n = 81 \) is the sample size. Plugging in these values, we get:\[ SEM = \frac{2.1}{\sqrt{81}} = \frac{2.1}{9} = 0.2333 \]

Step 2: Probability of Sample Mean Between 6 and 7

First, find the z-scores for 6 and 7. The z-score is calculated as:\[ z = \frac{X - \mu}{SEM} \]For 6:\[ z_6 = \frac{6 - 6.5}{0.2333} \approx -2.14 \]For 7:\[ z_7 = \frac{7 - 6.5}{0.2333} \approx 2.14 \]Using the standard normal distribution table:\[ P(-2.14 < Z < 2.14) \approx 0.9836 - 0.0162 = 0.9674 \]

Step 3: Probability Difference less than 0.25

The probability that the sample mean is within 0.25 of the population mean is equivalent to finding the probability that the z-score is within:\[ z = \frac{0.25}{0.2333} \approx 1.0719 \]Hence:\[ P(-1.0719 < Z < 1.0719) \approx 0.8577 \]

Step 4: Probability Sample Mean Greater Than 7

Calculate the z-score for 7 as:\[ z = \frac{7 - 6.5}{0.2333} \approx 2.14 \]Using the standard normal distribution table, the probability of z being greater than 2.14 is:\[ P(Z > 2.14) = 1 - P(Z < 2.14) \approx 1 - 0.9836 = 0.0164 \]

Key Concepts

Probability Distribution

Probability distribution is a key concept in statistics that describes how the values of a random variable, like the number of sports shoes purchased, are distributed. In this context, we're looking at a normal distribution, which is the famous bell-shaped curve. This distribution is defined by two parameters: the mean and the standard deviation.

• The mean tells us the average value. For Nike's case, it's 6.5 pairs of sports shoes per year.
• The standard deviation, which is 2.1 pairs here, gives us an idea about the variability or spread of the data.

The importance of probability distribution cannot be overstated, as it helps us determine the likelihood of specific outcomes. For instance, understanding how often customers buy more or less than 6.5 pairs. This distribution allows us to predict probabilities using z-scores and p-values, which are crucial for statistical inferencing.

Z-Score Calculation

Calculating a z-score is like finding the location of a data point in the context of its distribution. The z-score tells you how many standard deviations away a particular data point is from the mean. The formula is quite simple:

• Z = \( \frac{X - \mu}{SEM} \)

Where:

• \( X \) is the value you are examining.
• \( \mu \) is the mean.
• SEM is the standard error of the mean, in our case, 0.2333.

In the exercise, we computed the z-score for several values like 6 and 7. This was done to determine how unusual or typical these values were in our sample. By converting raw data points into z-scores, we can utilize standard normal distribution tables to find probabilities related to these values. This method helps us to understand the probability of certain sample means being observed if we were to take many samples from the population repeatedly.

Sample Mean

The sample mean is a simple but incredibly important concept in statistics. It refers to the average value of a sample, which is a subset of a larger population. In this exercise, we considered a sample of 81 customers to predict shoe-buying behavior.
The significance of the sample mean is highlighted by the fact that it serves as an estimate of the population mean, which is the average value calculated from the entire population, not just the sample. The formula for the sample mean is straightforward:

• \( ar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \)

Where \( ar{x} \) is the sample mean and \( x_i \) are the individual observations.

• The larger your sample size, the more reliable your sample mean becomes as an approximation of the population mean.
• In statistical practice, comparing the sample mean to the population mean can reveal insights about trends and anomalies within the data.

This step of analysis tells us not only about our current sample but also informs predictions about future samples.