A Credit-card Company Reports That Its Customers' Average Monthly Balance Is $ 10 , 200 \$10,200 $10 , 200 . A Random Sample Of 29 Customers Reveals An Average Monthly Balance Of $ 15 , 100 \$15,100 $15 , 100 , With A Standard Deviation Of $ 6 , 200 \$6,200 $6 , 200 . Is There
Introduction
When analyzing data from a sample, it's essential to determine whether the sample's characteristics are representative of the population. In this scenario, a credit-card company has reported an average monthly balance of for its customers. However, a random sample of 29 customers has an average monthly balance of with a standard deviation of . The question is whether the sample's average monthly balance is significantly different from the company's reported average.
Hypothesis Testing
To determine whether the sample's average monthly balance is significantly different from the company's reported average, we can use a hypothesis test. The null hypothesis (H0) is that the sample's average monthly balance is equal to the company's reported average, while the alternative hypothesis (H1) is that the sample's average monthly balance is not equal to the company's reported average.
Null Hypothesis (H0)
H0: μ = 10,200
Alternative Hypothesis (H1)
H1: μ ≠10,200
Calculating the Test Statistic
To calculate the test statistic, we need to calculate the standard error of the mean (SEM). The SEM is calculated as follows:
SEM = σ / √n
where σ is the standard deviation of the sample and n is the sample size.
Calculating the Standard Error of the Mean (SEM)
σ = 6,200 n = 29
SEM = 6,200 / √29 SEM = 6,200 / 5.385 SEM = 1,150.19
Calculating the Test Statistic (t)
The test statistic (t) is calculated as follows:
t = (x̄ - μ) / SEM
where x̄ is the sample mean, μ is the population mean, and SEM is the standard error of the mean.
Calculating the Test Statistic (t)
x̄ = 15,100 μ = 10,200 SEM = 1,150.19
t = (15,100 - 10,200) / 1,150.19 t = 4,900 / 1,150.19 t = 4.28
Determining the Critical Region
The critical region is the region of the distribution where the null hypothesis is rejected. In this case, we are using a two-tailed test, so the critical region is the region where the test statistic is less than -tα/2 or greater than tα/2.
Determining the Critical Value
The critical value (tα/2) is determined using a t-distribution table or calculator. For a sample size of 29 and a significance level of 0.05, the critical value is approximately 2.045.
Making a Decision
To make a decision, we compare the test statistic (t) to the critical value (tα/2). If the test statistic is less than -tα/2 or greater than tα/2, we reject the null hypothesis.
Making a Decision
t = 4.28 tα/2 = 2.045
Since the test statistic (t) is greater than the critical value (tα/2), we reject the null hypothesis.
Conclusion
Based on the results of the hypothesis test, we can conclude that the sample's average monthly balance is significantly different from the company's reported average. The sample's average monthly balance is , which is higher than the company's reported average. This suggests that the sample may be representative of a population with a higher average monthly balance than the company's reported average.
Implications
The results of this study have several implications for the credit-card company. Firstly, the company may need to re-evaluate its reported average monthly balance to reflect the actual average monthly balance of its customers. Secondly, the company may need to consider offering higher credit limits or other benefits to its customers to reflect their actual average monthly balance.
Limitations
There are several limitations to this study. Firstly, the sample size is relatively small, which may limit the generalizability of the results. Secondly, the study only examines the average monthly balance of a sample of customers, and may not reflect the actual average monthly balance of the entire population of customers.
Future Research
Future research could involve examining the average monthly balance of a larger sample of customers, or examining other characteristics of the sample, such as income or credit score. Additionally, future research could involve comparing the average monthly balance of customers from different demographic groups, such as age or income level.
References
- [1] "Hypothesis Testing." Wikipedia, Wikimedia Foundation, 12 Mar. 2023, en.wikipedia.org/wiki/Hypothesis_testing.
- [2] "t-Test." Wikipedia, Wikimedia Foundation, 12 Mar. 2023, en.wikipedia.org/wiki/T-test.
- [3] "Standard Error of the Mean." Wikipedia, Wikimedia Foundation, 12 Mar. 2023, en.wikipedia.org/wiki/Standard_error_of_the_mean.
Q&A: Understanding the Results of the Hypothesis Test
Q: What is the purpose of a hypothesis test in this scenario?
A: The purpose of a hypothesis test is to determine whether the sample's average monthly balance is significantly different from the company's reported average.
Q: What are the null and alternative hypotheses in this scenario?
A: The null hypothesis (H0) is that the sample's average monthly balance is equal to the company's reported average, while the alternative hypothesis (H1) is that the sample's average monthly balance is not equal to the company's reported average.
Q: How is the test statistic calculated?
A: The test statistic (t) is calculated as follows:
t = (x̄ - μ) / SEM
where x̄ is the sample mean, μ is the population mean, and SEM is the standard error of the mean.
Q: What is the standard error of the mean (SEM)?
A: The standard error of the mean (SEM) is calculated as follows:
SEM = σ / √n
where σ is the standard deviation of the sample and n is the sample size.
Q: What is the critical region in this scenario?
A: The critical region is the region of the distribution where the null hypothesis is rejected. In this case, we are using a two-tailed test, so the critical region is the region where the test statistic is less than -tα/2 or greater than tα/2.
Q: What is the critical value in this scenario?
A: The critical value (tα/2) is determined using a t-distribution table or calculator. For a sample size of 29 and a significance level of 0.05, the critical value is approximately 2.045.
Q: What is the decision based on the test statistic and critical value?
A: Since the test statistic (t) is greater than the critical value (tα/2), we reject the null hypothesis.
Q: What are the implications of the results of the hypothesis test?
A: The results of the hypothesis test suggest that the sample's average monthly balance is significantly different from the company's reported average. The sample's average monthly balance is , which is higher than the company's reported average.
Q: What are the limitations of this study?
A: There are several limitations to this study. Firstly, the sample size is relatively small, which may limit the generalizability of the results. Secondly, the study only examines the average monthly balance of a sample of customers, and may not reflect the actual average monthly balance of the entire population of customers.
Q: What are some potential future research directions?
A: Future research could involve examining the average monthly balance of a larger sample of customers, or examining other characteristics of the sample, such as income or credit score. Additionally, future research could involve comparing the average monthly balance of customers from different demographic groups, such as age or income level.
Frequently Asked Questions
Q: What is the difference between a hypothesis test and a confidence interval?
A: A hypothesis test is used to determine whether a sample's characteristics are significantly different from a population's characteristics, while a confidence interval is used to estimate a population's characteristics based on a sample's characteristics.
Q: What is the significance level (α) in a hypothesis test?
A: The significance level (α) is the probability of rejecting the null hypothesis when it is true. A common significance level is 0.05.
Q: What is the p-value in a hypothesis test?
A: The p-value is the probability of observing a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true.
Q: What is the difference between a one-tailed and a two-tailed test?
A: A one-tailed test is used to determine whether a sample's characteristics are significantly different from a population's characteristics in one direction, while a two-tailed test is used to determine whether a sample's characteristics are significantly different from a population's characteristics in both directions.
Conclusion
In conclusion, the results of the hypothesis test suggest that the sample's average monthly balance is significantly different from the company's reported average. The sample's average monthly balance is , which is higher than the company's reported average. This suggests that the sample may be representative of a population with a higher average monthly balance than the company's reported average.