Problem 38 - Basic Statistics for Business and Economics

Question

According to a government study among adults in the 25 - to 34 -year age group, the mean amount spent per year on reading and entertainment is \(\$ 1,994\) (www. infoplease.com/ipa/A0908759.html. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of \(\$ 450 .\) a. What percent of the adults spend more than \(\$ 2,500\) per year on reading and entertainment? b. What percent spend between \(\$ 2,500\) and \(\$ 3,000\) per year on reading and entertainment? c. What percent spend less than \(\$ 1,000\) per year on reading and entertainment?

Step-by-Step Solution

Answer

13.03% spend more than $2500; 11.74% spend between $2500 and $3000; 1.36% spend less than $1000.

Step 1: Understanding the Problem

The problem provides that the distribution of amounts spent by adults in the 25-34 age group on reading and entertainment follows a normal distribution with a mean of $1994 and a standard deviation of $450. We are asked to find percentages of adults spending more than $2500, between $2500 and $3000, and less than $1000.

Step 2: Calculating Z-scores

The Z-score is given by the formula \( Z = \frac{X - \mu}{\sigma} \). Calculate the Z-scores for the given amounts: - For \(2500: \( Z = \frac{2500 - 1994}{450} \approx 1.124 \) - For \)3000: \( Z = \frac{3000 - 1994}{450} \approx 2.236 \) - For $1000: \( Z = \frac{1000 - 1994}{450} \approx -2.209 \)

Step 3: Finding Percentages Using Z-tables

Utilize Z-tables or standard normal distribution calculators to find probabilities:- For \( Z = 1.124 \): Probability (spending less than \(2500) ≈ 0.8697.- For \( Z = 2.236 \): Probability (spending less than \)3000) ≈ 0.9871.- For \( Z = -2.209 \): Probability (spending less than $1000) ≈ 0.0136.

Step 4: Calculating Required Percentages

Convert probabilities into percentages:- To find the percentage spending more than \(2500: \( 1 - 0.8697 \approx 0.1303 \) or 13.03%.- To find the percentage spending between \)2500 and \(3000: \( 0.9871 - 0.8697 \approx 0.1174 \) or 11.74%.- To find the percentage spending less than \)1000: Approximately 1.36%.

Key Concepts

Z-score

The Z-score is a way to understand how far away a particular data point is from the mean in a normal distribution. It's like a measuring tape for how "unusual" or "typical" a specific data point is. The formula for calculating a Z-score is:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:

• \( X \) is the value we are examining.
• \( \mu \) represents the mean of the data.
• \( \sigma \) is the standard deviation.

The Z-score tells you how many standard deviations the value \( X \) is from the mean. For example, in the exercise, the Z-score for spending \(2500 is computed as approximately 1.124. This means that \)2500 is 1.124 standard deviations above the mean spending amount of $1994.
When you have a Z-score, you can use it with a Z-table. This handy table helps determine what percentage of the data lies below a certain Z-score in a standard normal distribution. So, from there, you can calculate how unusual it is for someone to spend more or less than a set amount.

Standard Deviation

The standard deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of data values. Imagine if every person spent exactly the same amount on reading and entertainment. The standard deviation would be zero because there's no variation.
In this exercise, the standard deviation is given as $450. This value helps to determine how spread out people's spending habits are around the average of $1994.
A small standard deviation means that the data points are generally close to the mean, showing consistency. A larger one indicates that they're more spread out, signifying more variability in spending habits. Standard deviation is crucial in calculating Z-scores, as it acts as a unit of measurement to gauge how far a particular data point deviates from the average.

Probability

Probability is the measure of the likelihood of an event occurring. In the context of normal distributions, it helps us determine the chances of a specific outcome or range of outcomes happening.
In our exercise, we use probability to find out how likely it is for someone to spend more than $2500, between $2500 and $3000, or less than $1000 on reading and entertainment.
Here's how you might look at this:

• The probability of spending more than a certain amount (e.g., $2500) is calculated by finding the area under the normal distribution curve to the right of that amount. If Z-tables are used, you find this by subtracting the Z-score probability from 1.
• For ranges, like spending between $2500 and $3000, you find the area between those two Z-scores.
• For spending less than a specific amount, such as $1000, the probability is the area under the curve to the left of that Z-score.

Understanding probability with the help of Z-scores lets us gauge spending habits across a population, providing insights into how typical or rare certain behaviors are within the data set.