Question

The recession has caused many people to use their credit cards far less. In fact, in the United States, \(60 \%\) of consumers say they are committed to living with fewer credit cards. \({ }^{15} \mathrm{~A}\) sample of \(n=400\) consumers with credit cards are randomly selected. a. What is the average number of consumers in the sample who said they are committed to living with fewer credit cards? b. What is the standard deviation of the number in the sample who said they are committed to living with fewer credit cards? c. Within what range would you expect to find the number in the sample who said they are committed to living with fewer credit cards? d. If only 200 of the sample of consumers said they were committed to living with fewer credit cards, would you consider this unusual? Explain. What conclusion might you draw from this sample information?

Short answer

Answer: Yes, a sample with only 200 committed consumers would be considered unusual, as it falls outside the calculated range of approximately 220 to 260.

Step-by-step solution

Calculate the mean of the binomial distribution

Using the given proportion p=0.60 and the sample size n=400, we can calculate the mean (μ) of the binomial distribution as follows: μ = n * p μ = 400 * 0.60 = 240 The average number of consumers in the sample who are committed to living with fewer credit cards is 240.

Calculate the standard deviation of the binomial distribution

Using the given proportion p=0.60 and its complement q=1-p=0.40, as well as the sample size n=400, we can calculate the standard deviation (σ) of the binomial distribution as follows: σ = sqrt(n * p * q) σ = sqrt(400 * 0.60 * 0.40) = sqrt(96) ≈ 9.80 The standard deviation of the number of consumers in the sample who are committed to living with fewer credit cards is approximately 9.80.

Calculate the range within which we expect to find committed consumers

To find the range within which we expect to find the number of committed consumers in the sample, we will use the mean and standard deviation calculated in the previous steps. A common range used is within two standard deviations from the mean. Lower Bound = μ - 2σ ≈ 240 - 2 * 9.80 ≈ 220.40 Upper Bound = μ + 2σ ≈ 240 + 2 * 9.80 ≈ 259.60 We would expect to find between approximately 220 and 260 consumers in the sample who are committed to living with fewer credit cards.

Evaluate if a sample of 200 committed consumers is unusual

Given that we calculated a range within which we expect to find the number of committed consumers in the sample, we can now determine if a sample with only 200 committed consumers would be unusual. Since 200 is outside the range calculated in Step 3 (220 to 260), this result would be considered unusual.

Conclusion

In conclusion, the average number of consumers in the sample who said they are committed to living with fewer credit cards is 240, with a standard deviation of approximately 9.80. We would usually expect to find between 220 and 260 committed consumers in such a sample. If only 200 consumers in the sample were committed to living with fewer credit cards, this would be considered unusual, and it might indicate that the actual proportion of committed consumers is lower than the 60% stated in the problem or that the sample is not representative of the entire population.

Key concepts

Understanding Mean Calculation

The mean, often referred to as the "average", is a central value that represents a typical outcome in a data set. In the context of binomial distribution, which is a frequent model in statistics used to describe the number of successes in a sequence of independent experiments, calculating the mean helps in understanding the expected number of outcomes.

For the example of consumers committed to using fewer credit cards, the mean (denoted as \( \mu \)) can be calculated by multiplying the total number of trials (or sample size, \( n \)) with the probability of success (or the proportion, \( p \)). Here, \( n \) is 400 and \( p \) is 0.60, leading to a mean \( \mu = n \times p = 400 \times 0.60 = 240 \).

So, on average, we would expect 240 consumers in this sample to be committed to living with fewer credit cards. This calculation helps set our expectations for what is typical in this data set.

Unpacking Standard Deviation

Standard deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of values. In simpler terms, it tells us how much the values in our data set deviate from the mean. For a binomial distribution, calculating the standard deviation helps to understand how tightly the outcomes are clustered around the mean.

The formula for standard deviation (\( \sigma \)) in a binomial distribution is \( \sigma = \sqrt{n \times p \times q} \), where \( q = 1 - p \). In the credit card usage example, \( q = 1 - 0.60 = 0.40 \). So the calculation becomes \( \sigma = \sqrt{400 \times 0.60 \times 0.40} = \sqrt{96} \approx 9.80 \).

This means that the number of committed consumers will usually vary by about 9.80 from the mean of 240. Understanding the standard deviation gives insights into the reliability and variability of our data.

Exploring Confidence Interval

A confidence interval provides a range of values which likely contains the true population parameter. It offers an idea of the uncertainty around a sample statistic. In the realm of binomial distribution, the confidence interval helps determine the range within which we expect the true value of a parameter to fall.

Using the mean and standard deviation calculated earlier, a common approach is to construct a range that covers approximately 95% of the data by using two standard deviations around the mean. Therefore, for the example given, the lower bound is \( 240 - 2 \times 9.80 = 220.40 \) and the upper bound is \( 240 + 2 \times 9.80 = 259.60 \).

This range from 220 to 260 gives us confidence that the number of committed consumers will typically fall in this interval. Providing a confidence interval is crucial as it incorporates the inherent uncertainty present in any data analysis.

Understanding Statistical Significance

Statistical significance is a term used to determine if the observed outcome of an experiment is unlikely to have occurred due to random chance alone. It helps in assessing whether the results have practical importance or merely occurred without any causal connection.

In our exercise, we determined that 200 committed consumers fall outside the expected range (220 to 260), which suggests that this outcome is unusual. This raises questions about the result's statistical significance.

Such an unusual outcome might lead us to conclude that either the proportion of consumers willing to commit is less than 60%, or that the sample does not accurately represent the population. This consideration is fundamental to statistical analysis, prompting either a reevaluation of the assumptions or attention to possible biases or anomalies in the data. Statistical significance isn't just about the numbers; it's about what those numbers mean in context.