Question

Give the correct notation for the quantity described and give its value. Proportion of US adults who own a cell phone. In a survey of 1006 US adults in \(2014,90 \%\) said they had a cell phone. \(^{7}\)

Short answer

Notation: \( p \) , Value: \( 0.90 \) or \( \frac{90}{100} \)

Step-by-step solution

Understand the problem

The problem is asking about the proportion of U.S. adults who own a cell phone. The data given is from a survey of 1006 U.S. adults in 2014 from which 90% confirmed they had a cell phone.

Define the notation

Let's denote the proportion of U.S. adults who own a cell phone as \( p \). This is the standard notation for a proportion in statistical analysis.

Calculate the proportion

To calculate \( p \), we take the given percentage and express it as a decimal. Since 90% of the adults surveyed own a cell phone, the proportion \( p \) is \( 0.90 \) or \( \frac{90}{100} \).

Interpret the results

The notation for the quantity described as 'proportion of U.S. adults who own a cell phone' is \( p \), and its value is \( 0.90 \) or as a fraction \( \frac{90}{100} \). This means that in this survey, 90% of U.S. adults own a cell phone.

Key concepts

Statistical Notation

Statistical notation serves as a universal language in the field of mathematics and statistics, enabling clear communication of quantitative information. In this context, proportions are generally denoted by the letter 'p'.

For example, when denoting the proportion of U.S. adults who own a cell phone, we use the notation '\( p \)'. This allows anyone reading a study or data analysis to quickly understand what '\( p \)' represents without confusion. Proper notation supports clarity and precision, which are essential in any statistical analysis.

Another common aspect of statistical notation is representing percentages as decimals in calculations. This is because mathematical operations can be performed more easily with decimals than with percentages. In our example, the 90% cell phone ownership is represented as '\( p = 0.90 \)', which is clearer for computational purposes than writing '90%'. This standard practice helps avoid errors and makes it easier to perform further statistical analyses, such as hypothesis testing or predictive modelling.

Survey Data Analysis

Survey data analysis involves interpreting responses received from a sample of individuals to draw conclusions about the larger population. In the given exercise, a survey of 1006 U.S. adults in 2014 revealed that 90% owned a cell phone.

To analyze survey data effectively, one must first understand the sample's context and representativeness. In this case, it is implied that the sample of 1006 adults is representative of the U.S. adult population. After understanding the context, it is critical to use the correct statistical notation, as we have assigned '\( p \)' to represent our proportion of interest.

Through proper analysis, we can interpret that the data suggests a majority of U.S. adults owned cell phones in 2014. This kind of analysis is crucial for making informed decisions, such as in marketing strategies, policy-making, or tracking technological adoption trends over time.

Percentage to Decimal Conversion

Converting a percentage to a decimal is an essential skill in both mathematical exercises and real-world applications. The process involves dividing the percentage value by 100. This step transforms the percentage, which can be seen as a portion out of 100, into a proportion represented as a decimal.

For instance, in the given exercise, the percentage of U.S. adults who own a cell phone is 90%. To convert this into a decimal, we simply divide by 100, which gives us '\( p = \frac{90}{100} = 0.90 \)'. This conversion is crucial for calculations in statistics and other disciplines because decimals are much more amenable to computational operations.

In practical terms, understanding this conversion process helps in various scenarios such as calculating discounts, determining interest rates, or even interpreting statistics such as those from surveys. Often, it allows for better comprehension of the data, facilitating more accurate analysis and decision-making based on the results.