Problem 39 - Basic Statistics for Business & Economics

Question

An auditor for Health Maintenance Services of Georgia reports \(40 \%\) of policyholders 55 years or older submit a claim during the year. Fifteen policyholders are randomly selected for company records. a. How many of the policyholders would you expect to have filed a claim within the last year? b. What is the probability that 10 of the selected policyholders submitted a claim last year? c. What is the probability that 10 or more of the selected policyholders submitted a claim last year? d. What is the probability that more than 10 of the selected policyholders submitted a claim last year?

Step-by-Step Solution

Answer

a) Expect 6. b) Probability is 0.0148. c) Probability is 0.0331. d) Probability is 0.0183.

Step 1: Expected Number of Policyholders Filing a Claim

We want to calculate the expected value of policyholders who have filed a claim. The expected value in a binomial distribution is calculated as \[ E(X) = n imes p \]where \( n = 15 \) is the number of policyholders, and \( p = 0.40 \) is the probability of a policyholder submitting a claim.Substitute the values:\[ E(X) = 15 \times 0.40 = 6 \]So, we expect 6 policyholders to have filed a claim.

Step 2: Probability of Exactly 10 Policyholders Filing a Claim

This is calculated using the binomial probability formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Where \( k = 10 \), \( n = 15 \), and \( p = 0.40 \). Substitute the values:\[P(X = 10) = \binom{15}{10} (0.40)^{10} (0.60)^{5}\]Calculate:\[P(X = 10) \approx 3003 \times 0.0001048576 \times 0.07776 = 0.0148\]

Step 3: Probability of 10 or More Policyholders Filing a Claim

To find this probability, calculate \( P(X \geq 10)\) by summing the probabilities from 10 to 15 policyholders:\[P(X \geq 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)\]Using a binomial table or calculator, find the probabilities for each, sum them, which yields:\[P(X \geq 10) \approx 0.0331\]

Step 4: Probability of More Than 10 Policyholders Filing a Claim

This is obtained by calculating the sum of probabilities for 11 to 15 policyholders:\[P(X > 10) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)\]Calculate these using a binomial table or technology, which results in:\[P(X > 10) \approx 0.0183\]

Key Concepts

Probability

Probability is a fundamental concept in statistics that expresses the likelihood of an event occurring. In our case, we are dealing with the probability that policyholders submit a claim. This exercise involves determining probabilities within a binomial distribution. The binomial distribution applies because each policyholder either submits a claim or doesn't, which are mutually exclusive outcomes.

In statistical terms:

• The probability that a singular event occurs, such as a policyholder making a claim, is denoted as \(p\).
• The binomial formula allows us to calculate the probability of exactly \(k\) successes (claims) out of \(n\) trials (policyholders tested), expressed as: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
• In our specific example, the probability \(p\) is 0.40 and \(n\) is 15.

Understanding probability helps us predict the outcomes of random events, aiding in risk assessment for tasks like auditing insurance claims.

Expected Value

The expected value in statistics is akin to the mean or average outcome we'd anticipate if we could repeat the experiment an infinite number of times. It's a critical concept because it provides a summary measure of the likelihood-weighted average outcome.

For a binomial distribution, the expected value (E) can be calculated as:

• The formula for the expected value of a binomially distributed random variable is \( E(X) = n \times p \).
• In this exercise, with \(n = 15\) and \(p = 0.40\), the expected number of policyholders to submit a claim is \(E(X) = 15 \times 0.40 = 6\).
• This means we would expect, on average, 6 out of the 15 policyholders to submit a claim yearly.

By understanding expected value, companies can better anticipate their financial planning related to policyholder claims.

Random Selection

Random selection represents a cornerstone of statistical sampling, ensuring unbiased results by selecting samples where each member has an equal chance of selection. In this exercise, we randomly selected 15 policyholders from a larger pool.

Random selection is crucial in:

• Providing each policyholder an equal opportunity of being included in the sample, thereby avoiding bias.
• Ensuring that the sample is representative of the entire population, in our case, the community of policyholders.
• Facilitating valid generalizations from the sample results, such as predicting probabilities and expectations regarding future claims.

A well-defined process of random selection forms the basis for reliable and accurate statistical analysis.

Statistical Analysis

Statistical analysis involves collecting, examining, and interpreting data to form conclusions. The binomial distribution example in this exercise is a perfect illustration of how statistical analysis aids decision-making.

Key parts of statistical analysis include:

• Using binomial probability formulas to calculate the chances of specific numbers of policyholders submitting claims, as seen in the steps above.
• Interpreting these probabilities to forecast potential outcomes, which informs business strategies and resource allocation.
• Assessing probabilities for multiple scenarios: exactly 10 claims (0.0148), 10 or more claims (0.0331), and more than 10 claims (0.0183).

Mastering statistical analysis empowers companies to forecast and mitigate risks, optimize resources, and ensure informed policy decisions.