Problem 21 - Basic Statistics for Business and Economics

Question

WNAE, an all-news AM station, finds that the distribution of the lengths of time listeners are tuned to the station follows the normal distribution. The mean of the distribution is 15.0 minutes and the standard deviation is 3.5 minutes. What is the probability that a particular listener will tune in: a. More than 20 minutes? b. For 20 minutes or less? c. Between 10 and 12 minutes?

Step-by-Step Solution

Answer

a. 0.0764 b. 0.9236 c. 0.1185

Step 1: Understand the Problem

We need to calculate probabilities involving a normal distribution, where the mean (\( \mu \) ) is 15.0 minutes and the standard deviation (\( \sigma \) ) is 3.5 minutes. The questions ask us to find probabilities for specific time intervals.

Step 2: Convert to Standard Normal Distribution (Z-Score)

To solve probability questions related to a normal distribution, we convert the raw scores to Z-scores using the formula: \[ Z = \frac{(X - \mu)}{\sigma} \] where \(X\) is the time in minutes. This step will convert the questions to standard normal distribution probabilities.

Step 3: Calculate Z-score for 20 minutes

For part (a), calculate \(Z\) for 20 minutes: \[ Z = \frac{20 - 15}{3.5} = \frac{5}{3.5} \approx 1.43 \]

Step 4: Probability for More than 20 Minutes

Using the Z-table or standard normal distribution calculator, find the probability \(P(Z > 1.43)\). This is the probability of a listener tuning in for more than 20 minutes. Using a Z-table, this probability is approximately 0.0764.

Step 5: Probability for 20 Minutes or Less

For part (b), find \(P(X \leq 20)\), which is the same as \(P(Z \leq 1.43)\). From the Z-table, this probability is 1 - 0.0764 = 0.9236.

Step 6: Calculate Z-scores for 10 and 12 Minutes

For part (c), calculate Z-scores for 10 and 12 minutes: \[ Z_{10} = \frac{10 - 15}{3.5} = \frac{-5}{3.5} \approx -1.43 \] \[ Z_{12} = \frac{12 - 15}{3.5} = \frac{-3}{3.5} \approx -0.86 \]

Step 7: Probability Between 10 and 12 Minutes

Find the probability between these two Z-scores: \(P(-1.43 < Z < -0.86) = P(Z < -0.86) - P(Z < -1.43)\). From the Z-table, \(P(Z < -0.86) \approx 0.1949\) and \(P(Z < -1.43) \approx 0.0764\). Thus, \(P(-1.43 < Z < -0.86) \approx 0.1949 - 0.0764 = 0.1185\).

Key Concepts

Z-score Calculation

Understanding how to work with normal distributions starts with grasping the concept of a Z-score. A Z-score tells us how many standard deviations a particular value is from the mean. For example, if you're looking at how long a listener tunes into a radio station, you might wonder how unusual a 20-minute listening period is, considering the average time.

To calculate a Z-score, you start by subtracting the mean (\(\mu\)) from your value (\(X\)), then divide by the standard deviation (\(\sigma\)). It's like standardizing your value:

• Identify your observation: the duration in minutes.
• Subtract the mean time from your observation.
• Divide this result by the standard deviation.

So, to find the Z-score for a listener tuning in for 20 minutes when the mean is 15 and the standard deviation is 3.5, you calculate: \[Z = \frac{20 - 15}{3.5} = \frac{5}{3.5} \approx 1.43\]This calculation informs us that 20 minutes is approximately 1.43 standard deviations above the average listening time.

Probability Analysis

After calculating the Z-score, we move to analyzing the probability related to this score. The purpose is to understand the likelihood of an event, such as a listener choosing to tune in for more than 20 minutes.

In probability analysis:

• The Z-score helps us find the probability using a Z-table or a calculator.
• This table or calculator gives the area to the left of the Z-score under the standard normal curve.

For example, a Z-score of 1.43 gives us a probability of about 0.0764 when looking for the probability of listening for more than 20 minutes. This probability refers to the tiny "tail" section of the curve beyond 1.43.
When dealing with the probability of listening 20 minutes or less, we look for the area to the left of 1.43, i.e., the cumulative probability, which is 1 - 0.0764 = 0.9236. Thus, there's a 92.36% chance a listener will tune in for 20 minutes or less.

Standard Deviation

Consider standard deviation as the heartbeat of a normal distribution. It measures how spread out values are around the mean. A small standard deviation means data points are close to the mean, while a larger one indicates more variation.

In our case, listeners' tuning times have a standard deviation of 3.5 minutes. This number, when coupled with the mean, paints a picture of the likely range of listening times.

• If standard deviation were smaller - say 1 minute - most listeners would tune in around 15 minutes.
• A larger standard deviation implies listeners tune in over a more varied number of minutes - perhaps some tune in for significantly shorter or longer periods than the average.

Standard deviation is crucial because it affects the shape and spread of the normal distribution. It's the factor that scales the difference when we calculate the Z-score, showing how typical or atypical a particular tuning level is compared to the average.