91Ó°ÊÓ

Large U.S. Companies: Foreign Revenue For large U.S. companies, what percentage of their total income comes from foreign sales? A random sample of technology companies (IBM, Hewlett-Packard, Intel, and others) gave the following information. $$ \begin{aligned} &\text { Technology companies, } \% \text { foreign revenue: } x_{1} ; n_{1}=16\\\ &\begin{array}{llllllll} 62.8 & 55.7 & 47.0 & 59.6 & 55.3 & 41.0 & 65.1 & 51.1 \\ 53.4 & 50.8 & 48.5 & 44.6 & 49.4 & 61.2 & 39.3 & 41.8 \end{array} \end{aligned} $$ Another independent random sample of basic consumer product companies (Goodyear, Sarah Lee, H.J. Heinz, Toys " \(q\) "Us) gave the following information. $$ \begin{aligned} &\text { Basic consumer product companies, } \% \text { foreign revenue: } x_{2} ; n_{2}=17\\\ &\begin{array}{llllllll} 28.0 & 30.5 & 34.2 & 50.3 & 11.1 & 28.8 & 40.0 & 44.9 \\ 40.7 & 60.1 & 23.1 & 21.3 & 42.8 & 18.0 & 36.9 & 28.0 \end{array}\\\ &\begin{aligned} &40.7 \\ &32.5 \end{aligned} \end{aligned} $$ (Reference: Forbes Top Companies.) Assume that the distributions of percentage foreign revenue are mound-shaped and symmetric for these two company types. (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x}_{1} \approx 51.66, s_{1} \approx 7.93, \bar{x}_{2} \approx 33.60\), and \(s_{2} \approx 12.26\). (b) Let \(\mu_{1}\) be the population mean for \(x_{1}\) and let \(\mu_{2}\) be the population mean for \(x_{2}\). Find an \(85 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). (c) Interpretation Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the \(85 \%\) level of confidence, do technology companies have a greater percentage foreign revenue than basic consumer product companies? (d) Check Requirements Which distribution (standard normal or Student's \(t)\) did you use? Why?

Step by step solution

01

Calculate the Means and Standard Deviations

For part (a), verify the given statistics for each group using the calculator:For technology companies:- Calculate the mean (\(\bar{x}_1\)) and the standard deviation (\(s_1\)).For basic consumer product companies:- Calculate the mean (\(\bar{x}_2\)) and the standard deviation (\(s_2\)).This will confirm \(\bar{x}_1 \approx 51.66\), \(s_1 \approx 7.93\), \(\bar{x}_2 \approx 33.60\), and \(s_2 \approx 12.26\).

02

Determine the Confidence Interval Formula

For part (b), to find the confidence interval for \(\mu_1 - \mu_2\), use the formula:\[(\bar{x}_1 - \bar{x}_2) \pm t \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]where \(t\) is the critical value from the t-distribution for an 85% confidence level. The degrees of freedom can be approximated using the smaller sample size minus one.

03

Find the Critical t-Value

For 85% confidence and approximate degrees of freedom (here we'll use \(\min(n_1-1, n_2-1)\), which is 15), find the critical value \(t\) from the t-distribution table. This t-value helps in constructing the confidence interval.

04

Calculate the Confidence Interval

Substitute the values into the formula:\[(51.66 - 33.60) \pm \text{t-value} \cdot \sqrt{\frac{7.93^2}{16} + \frac{12.26^2}{17}}\]Calculate both ends of this interval to find the range for \(\mu_1 - \mu_2\).

05

Interpret the Confidence Interval

For part (c), interpret the confidence interval. If the interval consists only of positive numbers, technology companies generally have greater foreign revenue percentages than basic consumer product companies. If negative, the converse is true. If it includes zero, there is significant overlap.

06

Select the Appropriate Distribution

For part (d), since both sample sizes are relatively small and the population variance is unknown, the Student's \(t\) distribution is used to construct the confidence interval, and it accounts for more uncertainty compared to the standard normal distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean

The concept of the population mean is central to understanding statistics. The population mean, often represented by the symbol \( \mu \), is the average value of a set of data points in an entire population. In the context of large U.S. companies and their foreign revenue, this measure can give insights into the typical percentage of income that comes from international sales. When you take a sample, like from technology companies or consumer product companies, you're trying to estimate their population mean. Your sample mean \( \bar{x} \) is a way to guess what the population mean might be. For instance, in our exercise, we estimated the population means for technology and consumer products. Understanding population means helps companies and analysts get a clearer picture of general trends and behaviors within industries.
It's a pivotal component in forming predictions and making informed decisions. The population mean stands out as a key statistic in understanding and improving business strategies.

T-Distribution

When you have small sample sizes or when the population's standard deviation is unknown, the t-distribution becomes particularly useful. This is why, in exercises like the one involving foreign revenue from large U.S. companies, we use it over the standard normal distribution. The t-distribution is similar to the normal distribution but has heavier tails, which means it gives more room for variability. As a result, it better accommodates the increased uncertainty and sample variation. With larger sample sizes, the t-distribution approximates the normal distribution. Why is this important? In our exercise, the samples from technology and consumer products were relatively small. Because of this, to construct a confidence interval about the difference in foreign revenue means, the t-distribution was appropriate.
It's crucial because it allows for a more accurate estimation of confidence intervals, giving businesses a better understanding of their market positioning.

Foreign Revenue

Foreign revenue refers to the part of a company's total income that is generated from sales in foreign countries. This metric is vital for companies involved in global markets. In our exercise, companies like IBM and Intel represent technology firms, while Goodyear and Sarah Lee represent basic consumer products. Understanding foreign revenue provides insights into a firm's global engagement and its ability to expand beyond home borders. It illustrates not only how diverse a company's income streams are but also how resilient it might be to local market changes. In practical terms, analyzing and comparing foreign revenues across different industries can highlight trends in globalization, economic dependency, and even potential areas for growth. This exercise helps students understand that foreign revenue percentage metrics can guide strategic decisions, including potential expansion or retraction in global markets.

Standard Deviation

Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. In statistical analysis, it is crucial for understanding how much individual data points differ from the mean. For example, in our exercise, the standard deviation of foreign revenue for technology firms and consumer product firms tells us about the consistency or variability of their international sales. A smaller standard deviation indicates that most data points are close to the mean, while a larger standard deviation shows more spread. In business analyses, understanding standard deviation is essential for risk management. Companies with smaller variations in their foreign revenues might have more predictable income streams than those with larger deviations.
It serves as a critical tool for assessing volatility and making informed financial and operational decisions.